Bas van Fraassen's philosophy blog
philosophical ideas, arguments, and diversions
Most of the afternoon my dog Zoe is asleep. When it is getting close to four o’clock she wakes up, and comes to find me — if I don’t seem to get the point she makes it clear that she is calling me for it is time for her afternoon walk. She believes that she will get a walk, if she makes sure that I don’t ignore her.
When I say all this about my dog Zoe, am I just falling victim to anthropomorphism?
[A reflection on some papers by Hallden, Lemmon, Hiz, Makinson, and Segerberg, listed at the end. Throughout I will use my own symbols for connectives, to keep the text uniform.]
Lemmon (1966) proved a theorem that I found quite startling at first sight, and I wondered what to make of it. But then I found that Makinson (1973) used it, to prove an interesting result for modal logics – one that is related to the role of atomic sentences — and it really kindled my interest.
It’s a story that spans several decades, and I will begin with Hallden. But let me note beforehand that each of these authors had the same target, which they specified clearly though informally, namely the class of propositional modal logics.
Some logics, e.g. FDE, have a consequence relation but no theorems. There are also logics that have the same theorems but different consequence relations. So I’ll take a logic to be identified by a consequence relation or operator. I’ll spell this out just for propositional logics.
Definition. A syntax is sentential iff its non-logical vocabulary is a set of zero or more propositional constants and countably infinitely many propositional variables.
The propositional constants and variables are the atomic sentences from which the complex sentences are built.
Definition. If S is a sentential syntax and SS its set of sentences then L is a logical consequence operator (or briefly, a logic) on S exactly if
That is, if X is a set of sentences and X = L(X), and P a propositional variable, and Y the result of substituting a specific sentence A for all occurrences of P in the sentences in X, then Y = L(Y).
So I make closure under the relevant rule of substitution part of the meaning of “logical consequence operator” or “logic”. [1]
The set L(∧) is the set of theorems of L, or the set of L-theorems.
Definition. A logic L on a sentential syntax is classical iff that syntax has at least all the ‘truth-functional’ connectives (primitive or defined), and the set of theorems of L contains (in some form or other) all theorems of the classical propositional calculus, and for all sentences A, B, if A and (A ⊃ B) are in X then B is in L(X). (Compare with Lemmon’s formulation. [2])
For the last clause I will also use “ X is closed under detachment”. I will here abbreviate “classical logic on a sentential syntax” to just “classical logic”, since we will look only at the sentential logic case.[3] A modal logic, as defined by the authors I mentioned, is a classical logic on a syntax that contains a modal propositional operator.
When C. I. Lewis introduced his family of modal logics they were five, but only two of them, S4 and S5, seemed easy to interpret. Sören_Halldén (1951) diagnosed the strangeness of S1,S2, and S3 as due to what we now call
Hallden incompleteness. A logical system L is Hallden incomplete iff there are sentences A and B which share no propositional variables and are not theorems of L while their disjunction (A v B) is a theorem of L.
The incompleteness, Hallden suggests, is this: “for S1, S2, and S3 the class of true formulas cannot coincide with the class of theorems” (Hallden 1951: 127). His reason is straightforward: if the class of true formulas includes a disjunction then it also includes the disjuncts.
But this looks quite beside the point: a logic is not meant to capture all truths, but all logical truths. Hallden is not attending to the difference between a logic and a theory.
Definition. If L is a classical logic on syntax S then a set of sentences of S is an L-theory iff X contains all L-theorems and is closed under detachment.
The rule of substitution does not enter into it: we cannot deduce q from atomic sentence p. Unlike detachment, the rule of substitution is not a rule designed to preserve truth but to preserve validity. The set of L-theorems is an L-theory, but unlike most L-theories it is closed under substitution.
We can do better than Hallden, I think, with the reflection that if A and B have propositional variables, but have none in common, then they could be interpreted to be about just any two different things altogether, with no connection at all – so if they are not logical truths, how could their disjunction be? If they are not logical truths then they could be false, and if there is no connection between them they could both be false. We’ll see arguments like this below.
It was soon shown, after Hallden’s paper, that our familiar normal modal logics T, B, S4, S5 are all Hallden complete.
What Lemmon (1966) proved was this:
A classical logic L is Hallden incomplete if and only if there are classical logics L1 and L2, distinct from each other and from L, such that L = L1 ∩ L2.
Attention to closure under substitution is important for this theorem. For example, If L i s a classical logic, X is any L-theory, and P a propositional variable, then X may well be the intersection of two least L-theories that contain (X ∪ {P}) and (X ∪ ~P) respectively. But the least logic L’ that contains the theorems of a logic L, and also P, has as theorems all sentences of the syntax – because of closure under substitution. And similarly for the one with ~P, if the logic is classical, due to double negation, so those two extensions of L are then identical.
In the Appendix I will sketch Lemmon’s proof. But we can see from the definitions that if L is Halden-incomplete, with A v B the relevant disjunction, then we can take L1 and L2 to be the least logics containing L that have A, B respectively as theorems.
The proof that in that case L1 and L2 are distinct from each other, as well as from L, is straightforward. For suppose that B is a theorem also of L1 as well. In that case there are certain substitution instances A1, …, Am of A, which are of course not in L, such that (A1 & …& Am) ⊃ B is a theorem of L. But L also contains all of A1 v B, …, Am v B, since they are substitution instances of its theorem A v B. Their conjunction is tautologically equivalent to (A1 & …& Am) vB. By Modus Ponens and disjunction elimination, it follows that B is a theorem of L, contrary to supposition.
Hiz’s paper that I discussed in the previous post was called “A Warning …”.
David Makinson also called his 1973 paper “A Warning …”. Segerberg titled a section in his book “Makinson’s warning”. I won’t say just now what the warning was, I’ll come to that later.
Instead I’ll begin by present Makinson’s argument in a general form, somewhat generalized from his presentation, and only afterward turn to his specific conclusion.
Makinson defines the class of modal logics, his target, as follows. (I will use my terminology as set out at the beginning here.) The sentential syntax S has as all ‘truthfunctional’ connectives (whether primitive or as defined, we will leave this open for now), as well as the unary connective □, and it has no propositional constants.
Definition. A modal logic is a classical logic on S.
So modal logics will differ by including some formulas in which □ occurs (axioms for specific modal logics like K or S5). Any intersection of modal logics is a modal logic. The smallest is L0, the intersection of all, and its theorems are all and only instances of classical tautologies.
But here is a variant: we introduce the notion of a propositional constant to form a slightly different family of logics. The sentential syntax S+ is exactly like S except that it has one propositional constant, q.
Definition. A modal+ logic is a classical logic on S+.
The smallest modal+ logic on S+, let us say, is L1. Its theorems too are just the instances of classical tautologies. Propositional constants and propositional variables are both atomic sentences. As a syntactic distinction it may look arbitrary, but the difference comes in the logic: the rule of substitution applies only to the propositional variables.
Theorem. L1, the smallest modal+ logic is Hallden incomplete.
Relying on Lemmon’s theorem this follows from the proof that L1 is the intersection of two logics that are distinct from each other and from L1.
The first part is interesting but easy. The proof is by display of an example. Let La be the least modal logic that contains L1 and □~q, and let Lb be least modal logic that contains L1 and ~ □~q. Obviously L1 is part of both La and Lb. Also, since the added formulas are not classical tautologies, and are contraries, La and Lb are distinct from each other and from L1.
For the converse, suppose that A is a theorem of both La and Lb. Note that there are no propositional variables in either □~q or ~ □~q. Therefore, if formula A belong to both La and Lb then (□~q ⊃ A) and (~□~q ⊃ A) are theorems of L1. But then, A is a theorem of L1, by classical sentential logic.
Hence by Lemmon’s theorem, this means that L1 is Hallden-incomplete. And the point is very general, no axioms for □ were assumed. The result is just due to the presence of a propositional constant. Illustrative examples of an ‘offending’ disjunction in L1 are easy to find. (□~q v ~ □~q) is a tautology, hence a theorem of L1, but its disjuncts are not. For an example that has some propositional variables involved, let r and s be distinct propositional variables. Then L1 has theorem
(r ⊃ □~q) v (s ⊃ ~□~q)
for that too is a truth-functional tautology, but neither disjunct (which have propositional variables but share none) is a theorem of L1.
The full title of Makinson’s paper is “A Warning about the Choice of Primitive Operators in Modal Logic”. Makinson’s logic L0 is like my L0 (the least modal logic on S) except that Makinson considers two specific options for syntax. The first is that the primitive ‘truth-functional’ connectives are ⊃ and ⊥ (the falsum) and the second option is that they are & and ~. Then he proves that if we take the first option, then L0 is the intersection of two logics L and L’, distinct from each other and from L. (And so, we may note, Hallden incomplete.)
His proof is the one of which I gave the general version above, except that “q” is replaced by “⊥”. We can classify the falsum as a logical sign, a 0-adic operator, but syntactically it (also) plays the role of a sentence: ~⊥ is a sentence, ⊥&⊥ is a sentence, ⊥ is a sentence …. The proof does not rely on any specific features of the falsum, but only on the fact that it plays the same role as any propositional constant.
On the second option L0 is not Hallden-incomplete (see Appendix for a sketch of his proof). But the two options yield languages that are entirely inter-translatable. As Segerberg (1982: 104) comments, the two options give us languages that “even though [they] have the same ‘internal’ properties, they do not share all the ‘external’ ones”.
Makinson recognizes that his result about the choice of primitives does not affect any of the more familiar modal logics. In those, □~⊥ is a theorem.[4] He took the result as being important for an insight into the structure of the lattice of modal logics. But as we also saw, his main argument generalizes to the presence of any propositional constant. So it also gives an insight into the curious roles that atomic sentences can play.
Outline of Lemmon’s proof that a classical logic L is Hallden incomplete if and only if there are classical logics L1 and L2, distinct from each other and from L, such that L = L1 ∩ L2.
Lemmon’s proof of the ‘only if’ part is straightforward, but made lengthy by the need to take the rule of substitution into account. Suppose that L is Hallden incomplete. Let A v B be an L-theorem, while A, B are not L-theorems and share no propositional variables. Let L1 and L2 be the extensions of L made by adding A, B respectively. Clearly the L-theorems are all L1-theorems as well as L2-theorems.
Suppose now that C is both an L1-theorem and an L2 theorem. The proofs for C in L1 and in L2 must be from premises that are substitution instances A1, …, Am of A, and substitution instances B1, …, Bn of B respectively. So the following are both L-theorems:
(A1 & …& Am) ⊃ C
(B1 & … & Bn) ⊃ C
therefore [(A1 & …& Am) v (B1 & … & Bn)] ⊃ C is an L-theorem. But that is tautologically equivalent to [(A1 v B1 ) & …& (Am v Bn)] ⊃ C. Since the L-theorems include all substitution instances (Ai v Bi) of L-theorem (A v B), it follows that C is an L-theorem.
The proof that if L is the intersection of two distinct logics L1 and L2 then it is Hallden incomplete is shorter but more interesting. Select a theorem A of L1 that is not a theorem of L2, and a theorem B of L2 that is not a theorem of L1. Clearly, neither is a theorem of L. Since L1 is closed under substitution it will contain an ‘isomorphic’ substitution instance A’ of A formed by substituting propositional variables foreign to B, for the propositional variables in A. Both L1 and L2 contain the disjunction (A’ v B). Therefore so does their intersection L.
Suppose per absurdum that sentences A and B have no propositional variables in common, and that (A v B) is a theorem of L0, while A, B are not theorems. So (A v B) is a classical tautology. Let f assign truth-values 0, 1 to the atomic sentences in A such that f(A) = 0. This is possible since A is not a tautology and does not contain the falsum or any propositional constant. Similarly let g assign truthvalues to the atomic sentences in B such that g(B) = 0. Since the domains of f and g do not overlap, we can combine them to yield a function h such that h(A) = h(B) =0. Thus h(A v B) = 0 which contradicts the supposition that (A v B) is a tautology.
Sören Halldén (1951) “On The Semantic Non-Completeness Of Certain Lewis Calculi”. The Journal Of Symbolic Logic 16: 127-129.
E. J. Lemmon (1966) “A Note On Hallden-Incompleteness”. Notre Dame Journal Of Formal Logic VII 1966: 296-300
David Makinson (1973) “A Warning about the Choice of Primitive Operators in Modal Logic”. Journal of Philosophical Logic 2: 193- 196.
[1] I do not mean that it is a substantive constraint. Rather, we classify logics by what is substitutable. A propositonal logic is one for which the class of substitutables is a set of sentences, a predicate logic is one where the substitutables are or include a set of primitive predicates. If a closure operator on a syntax has no substitutables at all, however, I do not think it can count as a logic, whatever else it may be.
[2] Compare Lemmon (1966: 300) “Throughout this paper, a logical system is understood to be a propositional logic whose class of theorems is closed with respect to substitution as well as detachment, and which contains (in some form or other) the classical propositional calculus.”
[3] Since I refer to Segerberg below, I should note that this is not the same as his definition of “classical logic”, though it is not far. [4] Makinson makes the stronger point that choice of primitive truth-functional operators will not make a difference in any congruential modal logic.[A reflection on papers by Hiz and Thomason, listed at the end. Throughout I will use my own symbols for connectives, to keep the text uniform.]
Atomic sentences, we say, are not a special species. They could be anything; they are just the ones we leave unanalyzed. What we study is the structures built from them, such as truth-fuctional compounds.
But that innocuous looking “They could be anything” opens up some leeway. It allows that the atomic sentences could have values or express propositions that the complex sentences cannot. I will discuss two examples of how this leeway can be exploited for proofs of incompleteness.
The story I want to tell starts with a small error by Paul Halmos in 1956.
In his 1956 paper Paul Halmos wanted to display the classical propositional calculus with just & and ~ as primitive connectives. (Looks familiar, what could be the problem?) As guide he took the presentation in Hilbert and Ackermann, with v and ~ as primitives. For brevity and ease of reading they had introduced “x ⊃ y” as abbreviation for “~x v y”.
Knowing how truth functions work, Halmos (1956: 368) treated “x v y” as abbreviation of “~(~x & ~y)” and “x ⊃ y” as abbreviation of “~(x & ~y), to read Hilbert and Ackermann’s axioms. That means that his formulation, with ~ and & primitive, was this:
But, unlike what it translates (Hilbert and Ackermann’s), this set of axioms is not complete!
Henryk Hiz (1958) showed why not. (He mentioned that Halmos had raised the possibility himself in a conversation, and Rosser had done so as well, in a letter to Halmos.)
Let’s look for a difference in the roles of atomic sentences and of complex sentences in Halmos’ axiom set. What springs to the eye in Axiom b. is that there is an occurence of x that is preceded by ~, and one that is not so preceded but ‘stands by itself’. So we can make trouble by allowing an atomic sentence x to take values that a negated sentence ~x cannot have.
That is what Hiz does, with this three-valued truth-table where an atomic sentence x could have value 1, 2, or 3, but ~x can only have values 1 or 3.
(He writes A and N for my & and ~.)
So if x has value 2 then ~(~x & x) has value ~ (~2 & 2) = ~(1 & 2) = ~1 = 3, which is not designated. So there is a classical tautology, the traditional Non-Contradiction Principle, that does not receive a designated value.
In this three-valued logic neither conjunction nor negation behaves classically, but all of Halmos’ axioms have the designated value 1. So his formulation of classical sentential logic is sound but not complete.
Thomason’s (2018) argument and technique, which I discussed in a previous post, were very close to Hiz’, but applied to modal logic.
In modal logic the basic K axiom can be formulated in at least these three ways:
The third is a translation of the first with “□” translated as “~◊~”. In the previous post (“Is Possiblity-Necessity Duality Just a Definition”, 07/17/2025) I explained Thomason’s model in which that third formulation of K is satisfied, but the Duality principle is shown to be independent. Here I will show that satisfaction of Axiom (iii) is compatible with a violation of Axiom (ii).
Thomason presented a model with 8 values for the propositions. I’ll use here the smaller 5-valued model which I described in the post. My presentation here, in a slightly adapted form, is sufficient for our purpose.
This structure (matrix)is made up of the familiar 2-atom Boolean lattice B = {T, 1, ~1, ⊥} with the addition of an ‘alien’ element k. The meet and join on B are operators ∧ and +. The operator ~ is the usual complement on on B. The only designated element is T.
To extend the operators to the alient element, we set ~k = ~1. So x can take any of the five values but ~x can only have a value in B.
What about the joins and meets of elements when one of them is alien? They are all in B too, with these definitions:
Define. x* = ~~x, called the Twin of x. (Clearly x = x* except that k* = 1.)
Define. For any elements x and y: x & y = x* ∧ y*, and x v y = x* + y*.
Finally the possibity operator is defined by: ◊x = T iff x = 1 or T; ◊x = ⊥ otherwise.
Instances of Axiom (iii.) always get the desigated value (by inspection; note that every non-modal sentential part starts with ~).
But in Axiom (ii) we see the leeway, due to the fact that x can be any element. The negation, join, or meet of anything with anything can only take values in B. So Axiom (ii) does not always get a designated value, for if we set x = y = k, we get the result:
◊(k v k) = ◊(k* + k*) = ◊1 = T
(◊k v ◊k) = ⊥* + ⊥* = ⊥
In Thomason’s article this technique is used to show that with formulation (iii) of K, the duality ¬◊¬x = □x is independent, and needs to be added as an axiom rather than a definition.
Axiom (ii.), with the attendant rules changed mutatis mutandis, and the Duality introduced as a definition, is a complete formulation of system K (cf. Chellas 1980: 117, 122). A formulation that has Axiom(iii) instead of Axiom (ii) is not.
Hiz’ warning was well taken.
Chellas, Brian F. (1980) Modal Logic: An Introduction. Cambridge.
Hiz, Henryk (1958) “A Warning about Translating Axioms”. Am. Math. Monthly 65: 613-614.
Thomason, Richmond H. (2018) “Independence of the Dual Axiom in Modal K with Primitive ◊”. Notre Dame Journal of Formal Logic 59: 381-385.
Appendix and Bibliographical Note 7
After the initial astonishment that self-contradictions need not be logical black holes, there is a big question: how can classical reasoning find a place in a subclassical logic, such as the minimal subclassical logic FDE?
Classical propositional logic is, in a fairly straightforward sense, the theory of Boolean algebras. In the same sense we can say that the logic of tautological entailment, aka FDE, is the theory of De Morgan algebras.
In previous posts I have also explored the use of De Morgan algebras for truthmakers of imperatives and for the logic of intension and comprehension. So FDE’s algebraic counterpart has some application beyond FDE.
How, and to what extent, can the sub-classical logic FDE accommodate classical logic, or classical theories, as a special case? A corresponding question for algebraic logic is how, or to what extent, Boolean algebras are to be found inside De Morgan algebras.
Terminology. Unless otherwise indicated I will restrict the discussion to bounded De Morgan algebras, that is, ones with top (T) and bottom (⊥), these are distinct elements and ¬T = ⊥. If L is a De Morgan lattice, an element e of L is normal iff e ≠¬e, and L is normal iff all its elements are normal. Both sorts of algebras are examples of distributive lattices. From here on I will use “lattice” rather than “algebra”.
There are some simple, trivial answers first of all, and then two answers that look more important.
First, a Boolean lattice is a De Morgan lattice in which, for each element e, (e v ¬e ) = T (the top), or equivalently, (e ∧¬e) = ⊥ (the bottom).
Secondly, in a De Morgan lattice, the set {T, ⊥} is a Boolean lattice.
Thirdly, if L is De Morgan lattice and its element e is normal, then the quadruple {(e v ¬ e), e, ¬e, (e ∧ ¬e)} is a Boolean sub lattice of L.
More important is this: If L is a De Morgan lattice then B(L) = {x in L: (x v ¬ x) = T} is closed under ∧, v, and ¬ and is therefore a sub-lattice of L. It is a Boolean lattice: the Boolean Center of L.
Mighty Mo:
Figure 1 The eight-element De Morgan lattice Mo
The Boolean Center B(Mo) = {+3, +0, -0, -3}.
My aim here is to display Boolean lattices that ‘live’ inside De Morgan lattices. My general term for them will be Boolean families. They will not all be of the type, and I hope that their variety will itself offer us some insight.
The fact that a De Morgan lattice has a Boolean center can be generalized:
Suppose element e is such that (e v ¬e) is normal, and define B(e) = {x in L: (x v ¬ x) = (e v ¬e)}. Then (see Appendix for proof) B(e) is a Boolean family, with top = (e v ¬e) and bottom = (e ∧ ¬e).
The big question: are there examples of non-trivial Boolean families distinct from the Boolean center?
We can construct some examples by adding ‘alien’ points to a Boolean lattice. For example this, which I will just call L1.
Figure 2 L1, augmented B3
This lattice L1 is made up from the three-atom Boolean lattice B3 by adding an extra top and bottom. This sort of addition to a lattice I will call augmentation, and I will call L1 augmented B3. For the involution we keep the Boolean complement in B3, and extend this operation by adding that T = ¬ ⊥ , and ⊥ = ¬T.
L1 is distributive, hence a De Morgan lattice (proof in Appendix). The clue to the proof is that for all elements e of L1, T ∧ e = e and T v e = T.
The Boolean center B(L1) = {T, ⊥} is trivial, and the sublattice B3 is a non-trivial Boolean family.
In the above reasoning nothing hinged on the character of B3, taken as example. Augmenting any Boolean lattice B in this way will result in a De Morgan lattice with trivial Boolean center and B as a Boolean sublattice. But this still does not go very far. For the concept of Boolean families in De Morgan lattices to be possibly significant requires at least that there is a large variety of non- or not-nearly trivial examples.
To have a large class of examples with more than such a single central Boolean sublattice, we have to look for a construction to produce them. And this we can do by ‘multiplying’ lattices. I will illustrate this with B3, and then generalize.
B3 as a product lattice
The Boolean lattice B3 is the product B1 x B2 of the one-atom and two-atom Boolean lattices. The product of lattices L and L’ is defined to be the lattice whose elements are the pairs <x,y> with x in L and y in L’, and with operations defined pointwise. That is:
<x,y> v <z,w> = <x v z, y v w>
<x,y> ∧ <z,w>= <x ∧ z, y ∧ w>
¬<x, y> = < ¬x, ¬y>
<x,y> ≤ <z,w> iff <x,y> ∧ <z,w> = <x, y>, iff x ≤ z, and y ≤ w
Any such product of Boolean lattices is a Boolean lattice.
Figure 3. B3 as a product algebra.
It looks a bit like ordinary multiplication: B1 has 2 elements, B2 has 4 elements, 2 x 4 = 8, the number of elements of their product B3.
Inspecting the diagram, and momentarily ignoring the involution, we can see that B3 has two sublattices, that are each isomorphic to B2. (The definition of ‘sublattice’ refers only to the lattice operations ∧ and v.) That is to say, the components of the product construction show up as copies in the product. And that is also the case once we take the involution into account, given a careful understanding of this ‘copy’ relation.
The way we find these sublattices: choose one element in B1 to keep fixed and let the second element vary over B2:
sublattice B3(1) has elements T2, T1, T ⎯ 1, T0
sublattice B3(2) has elements ⊥2, ⊥1, ⊥⎯ 1, ⊥0
Sublattices so selected will be disjoint, for in one the elements have T as first element and in the other the elements have ⊥ as first element.
These sublattices are intervals in B3, e.g. B3(1) = {x in B3: T0 ≤ x ≤ T2}.
What about the involution? The restriction of the operator ¬ on B3 to interval B3(1) is not well-defined, for in B3, ¬ T2 is not in B3(1), it is ¬ T2 = ⊥0 which is in B3(2).
However, there is a unique extension to a Boolean complement on B3(1): start with what we have from B3, namely that ¬T1 = T ⎯1 and ¬ (T ⎯1) = T1, then add that ¬T2 = T0 and ¬T0 = T2 (“relative complement”; cf. remark about Theorem 10 on Birkhoff page 16, about relative complements in distributive lattices). It is this relatively complemented interval that is the exact copy of B2, which is a different example of a Boolean family.
(Looking back to section 5, we can now see that the example there was a simple one, where the restriction of the lattice’s involution, to the relevant sublattice, was well-defined on that sub lattice.)
Thus if we have a product of many Boolean algebras, that product will contain many Boolean families:
If L1, L2, …, LN, are Boolean lattices and L = L1 x L2 x … x LN, then L has disjoint Boolean families isomorphic to L1, L2, …, LN
For example, if e1, e2, …, eN are elements of L1, L2, …, LN respectively, then the set of elements S(k) = {<e1, e2, …, ek-1, x, ek+1, …eN>: x in Lk} form a sublattice of L that is (with the relative complement on S(k) defined as above) isomorphic to Lk. (See Halmos, page 116, about the projection of L onto Lk, for precision.)
And if we then augment that product L, in the way we formed L1, we arrive at a non-Boolean De Morgan lattice, augmented L. The result contains many Boolean families, but (e ∧ ¬e) is in general not the bottom, so it lends itself to adventures in sub classical logic.
But we need to turn now to a less trivial form of augmentation.
A product of distributive lattices is distributive (by a part of the argument that a product of Boolean lattices is Boolean).
The product of De Morgan lattices is a De Morgan lattice. To establish that, we need now only to check that the point-wise defined operation ¬ on the product is an involution (see Appendix for the proof).
So suppose we have, as above, a Boolean lattice product B, that has many Boolean families, and we form its product with some other, non-trivial non-Boolean De Morgan lattice, of any complexity.
The result is then a non-trivial non-Boolean De Morgan lattice with many Boolean families.
The basic sub-classical logic FDE has as non-logical signs only ¬ , v, and ∧. That is not enough to have Boolean aspects of De Morgan lattices reflected in the syntax.
For example, the equation (a v ¬ a) = (b v ¬ b) defines an equivalence relation between the propositions (a, b). But the definition involves identity of propositions, which for sentences corresponds to a necessary equivalence. To express this, a modal connective, say <=>, could be introduced, in order to identify a fragment of the language suitable for formulating classical theories.
There is much to speculate.
[1] Define, for any De Morgan lattice L, B(e) = {x in L: (x v ¬ x) = (e v ¬e)}.
Theorem. If L is a De Morgan lattice and its element (e v ¬e) is normal, then B(e) is a Boolean sublattice of L.
First, all elements of B(e) are normal. For (e v ¬e) is normal, and if x is not normal then (x v ¬x) = x.
For B(e) to be a sublattice with involution of L it suffices that B(e) is closed under under the operations ∧, v, and ¬ on L.
Define t = (e v ¬e). If d and f are in B(e) then
[(d v f) v ¬(d v f) ] = [(¬ d ∧ ¬f) v (d v f)]
= [(¬ d v d v f)] ∧ (¬f v d v f)]
= t ∧ t
(d ∧ f) is also in B(e) because
[(d ∧ f) v ¬(d ∧ f) ] = [(d ∧ f) v (¬ d v ¬f)]
= [(d v ¬ d v ¬ f)] ∧ (f v ¬ d v ¬ f)]
= t ∧ t
So B(e) is a sublattice of L, and hence distributive. It has involution ¬, its top is t and bottom ¬t. So B(e) is a bounded De Morgan lattice. B(e) is Boolean, because all its elements x are such that (x v ¬ x) = t.
B(T) is the Boolean center of the lattice.
[2] Theorem. The lattice L1, the augmented lattice B3, is a De Morgan lattice.
(a) The operation ¬ extended from B3 to L1 by adding that T = ¬ ⊥ , and ⊥ = ¬T, is an involution. The addition cannot yield exceptions: each element e of L1 is such that ⊥ ≤ e ≤ T, which is equivalent to, for all elements e of L1, ¬ T ≤ ¬ e ≤ ¬⊥.
(b) To prove that L1 is distributive, we note that, for all elements e of L1,
T ∧ e = e and T v e = T.
⊥ ∧ e = ⊥ and ⊥ v e = e.
To prove: If x, y, z are elements of L1 then x ∧ (y v z) = (x ∧ y) v (x ∧ z).
The remaining cases are similar.
[3] Theorem. A product of De Morgan lattices is a De Morgan lattice.
The product of any distributive lattices is a distributive lattice (Birkhoff 1967: 12)
To establish that the product of De Morgan lattices is a De Morgan lattice, we need then only to check that the point-wise defined operation ¬ on the product is an involution.
Let L1 and L2 be De Morgan lattices and L3 = L1 x L2. Define ¬<x. y> = <¬x, ¬y>
For the relation between FDE and De Morgan lattices see section 18 (by Michael Dunn) of Anderson, A. R. and N. D. Belnap (1975) Entailment: The Logic of Relevance and Necessity. Princeton.
For distributive lattices in general and relative complementation see Birkhoff, Garrett (1967) Lattice Theory. (3rd ed.). American Mathematical Society and Grätzer, George (2009/1971) Lattice Theory: First Concepts and Distributive Lattices. Dover.
For products of Boolean lattices see section 26 of Halmos, P. R. (2018/1963) Lectures on Boolean Algebras. Dover.
There are traditional examples that move us quickly away from the idea that our language is just extensional. And there are some that put into doubt that our language is only intensional, with no distinctions between any concepts that are necessarily co-extensional. These examples suggest that predicates have, or may have, three distinct semantic values: extension, intension, and comprehension.[1]
What the examples leave largely open is the character of relations between these three levels or modes of meaning. Seemingly best understood is the relation between extension and intension. I shall explore that first, and then explore how the same sort of relationship may obtain between intension and comprehension. There will be a connection with paraconsistent logic.
From Medieval philosophy we have a notable theory of distinctions. There is a distinction de re between featherless biped and rational animal: Diogenes the Cynic displayed a plucked chicken as real instance of the one and not the other. Today we say: these two concepts do not have the same extension. But there is only a distinction of reason between woman and daughter: there are no real entities instantiating the one but not the other, but there could be, as are represented in pictures, stories, and myths. Today we say: these concepts have the same extension but not the same intension.
Are there examples that push us still further? We can symbolize has a property, as x̂ (∃F)Fx, and is identical with something, as x̂(∃y)(y = x). Let’s refer to these concepts as being and existence. Then we can puzzle over them: there surely could not be, or even be fantasized to be, an instance of one that would not be an instance of the other.[2] As the Medievals would say, not even God could create something that has one but not the other. If there is a distinction nevertheless, it is what Duns Scotus called a formal distinction. Today we would say, if anything, that the two concepts do not have the same comprehension, and that if so, our language is hyper-intentional.
Terminology. I will use “property” only for what can be an intension of a predicate in a given language. Only different terms are to be used for extensions and comprehensions.[3]
There are well-known ways in which properties (intensions of predicates) are represented in semantic analyses of modal logics.
Let’s abstract from the details. Properties, in this sense, form an algebra AI. As a working hypothesis I will take this to be a Boolean algebra. It has a top, T, and a bottom, f. In a given world, our world say, any property x has an extension |x|, which is a set of entities in the world. Necessarily, T‘s extension includes everything and f‘s extension is empty. A distinction of reason is then a distinction between properties that have the same extension.
The function | | is a homomorphism: if x, y are properties and x ≤ y then |x| ⊆ |y|, ⎯ |x| = | ⎯ x|, and |x . y| = |x| ∩ |y|. As a result the sets, which are the extensions of properties, form a Boolean algebra, AE.
To formulate the distinction of reason we can define an equivalence relation, extensional equivalence, on the properties: (x ≡ y) iff |x| = |y|. Then AE is isomorphic to the quotient algebra AI modulo ≡ : the elements of this are the equivalence classes of elements of AI, [x] = {y: x ≡ y}, with [x] ≤ [y] iff x ≤ y, ⎯[x] = [⎯x], and |x| ∧ |y| = |x ∧ y|.
But there is another nice way of thinking about this relation between properties and their extensions. Think about the properties that | | maps into the empty set. Let us call these the ignorable properties from a extensionalist point of view.
properties x and y are co-extensional iff they differ only by an extensionally ignorable part, that is, there is some extensionally ignorable property z such that x v z = y v z.
The distinction of reason pertains then to exceptions to co-extensionality. For example, woman = (woman who has parents) v (woman who has no parents), and that is the join of daughter with an extensionally ignorable property.[4]
Now the ignorable properties form an ideal in the Boolean algebra BI of properties: if x ≤ y and y is ignorable then so is x, and moreover, (x v y) is ignorable iff both x and y are ignorable. This is not coincidental: for any equivalence relation E on a Boolean algebra there is an ideal J such that x E y iff for some z in J, x v z = y v z.
Are co-extensional and extensionally equivalent the same relation? In other words, if x and y have the same extension must there then be an ignorable property z such that x v z = y v z?
The answer is yes: z is the symmetric difference between x and y, that is, the join of (x ⎯ y) and (y ⎯ x). If either of those had a non-empty extension then |x| and |y| would not be the same.
I would like to explore this idea of ‘ignorables’ to get at the relationship between comprehension and intension.
The idea is that the above abstract form of the relationship between extension and intension obtains also for the relationship between intension and comprehension. That is, the comprehensions of concepts form an algebra AC, and AI is (isomorphic to) the quotient algebra formed by reducing AC by an appropriate equivalence relation.
In view of the above, the way to identify that appropriate equivalence relation is to specify an ideal in AC: the ideal of intensionally ignorable comprehensions.
If x is in AC then it has an intension ||x|| in AI, and x is intentionally ignorable exactly if ||x|| = the absurd property f. Define x and y to be intensionally equivalent (x ⇔ y) exactly when x v z = y v z for some ignorable comprehension z. And I submit that AI is (is isomorphic to) AC modulo ⇔.
Let’s see how this plays out with the example of being and existence. A quick check shows that
(∃y)(y = x) v [(∃y)(y = x) & ~(∃F)Fx] v [(∃F)Fx & ~(∃y)(y = x)]
is logically equivalent to
[(∃F)Fx] v [(∃y)(y = x) & ~(∃F)Fx] v [(∃F)Fx & ~(∃y)(y = x)]:
that is
existence or [existence and non–being] or [being and non–existence]
= being or [existence and non-being] or [being and non-existence]
Therefore to complete the example we must declare that [existence and non-being] as well as [being and non-existence] are intensionally ignorable comprehensions. The mapping || || sends these into the absurd property .
In this case the three comprehensions being, existence, and being cum existence have the same intension. Necessarily, any real things that have being exist, and any that exist have being. So the intension is the summum genus among properties, T.
The example of being and existence already shows that the logic pertaining to comprehension cannot be classical, where the definitions of those properties are tautologically equivalent.
Medieval discussions of such concepts as being, (‘transcendentals’), included also examples such as finite or infinite. While distinct from being, that property it is clearly not distinguishable from being by any real or possible instances. From a classical logic point of view, finite or infinite is just an ‘excluded middle’, hence tautological, and there is no logical leeway.
To do justice to the formal distinction between being and ‘excluded middles’, therefore, the logic pertaining to comprehension must allow for ‘excluded middles’ that do not imply each other.
The most modest logic of this sort is FDE, which corresponds in algebra to De Morgan lattices: distributive lattices equipped with an involution. The involution ⎯ is like a Boolean complement:
x = ⎯ ⎯ x
if x ≤ y then ⎯y ≤ ⎯x,
and from these two, given distributivity, the De Morgan laws follow:
⎯(x v y) = (⎯x ∧ ⎯y)
⎯(x ∧ y) = (⎯x v ⎯y)
But (x v ⎯x) may not be the top, (x ∧ ⎯x) may not be the bottom, and it can happen that x = ⎯x.
As models for the logic of comprehension I propose the De Morgan lattices with top (Θ), bottom ⊥ , and such that ⎯⊥= Θ. The function || || assigns an intension to each comprehension; || ⊥ || = f, and ||Θ|| = T.
Boolean algebras are De Morgan lattices, with the characteristic that their involution ⎯ is such that for each element x, (x v ⎯ x) = the top element. Happily the connection between ideals, homomorphisms, and equivalence relations holds as well for De Morgan lattices.[5]
To explore how comprehensions fare in this landscape, let us take a simple example of a De Morgan lattice for a model.
The 8-element De Morgan lattice DM8 (aka Mighty Mo) looks quite like the three atom Boolean algebra B3, but the involution is different:
Suppose we take as our ideal of intensionally ignorable elements the set of elements marked with ⎯, which consists of ⎯ 0 and everything below that. To represent the two classical tautologous ‘excluded middles’ of finite or infinite and round or not round let us choose +2 for finite and +1 for round. Then we see that:
finite or infinite = (+2 v ⎯2) = +2
round or not round = (+1 v ⎯1) = +1
+2 v ⎯0 = +1 v ⎯0 ( = +3),
so the two ‘excluded middles’ differ by the ignorable element ⎯ 0, hence are intensionally equivalent though distinct. And yes, they are also equivalent to the top +0 and +3, their intension is the property T, the summum genus.
There is a first-blush troubling question about the insistence that AC must be a De Morgan lattice, and not Boolean. Above I had advanced as working hypothesis that AI is Boolean. But now we have AI as a quotient algebra, namely AC modulo ⇔, which implies that AI is a De Morgan lattice as well. That does not rule out that AI is Boolean. But is it?
I will take this question as the clue to how to identify the intensionally ignorable comprehensions.
First, to have a clear example, let’s suppose again that AC is Mighty Mo, DM8. We chose an ideal of ignorables, namely the ideal generated by ⎯0. Then we already saw that with that choice +0, +1, +2, +3 are intensionally equivalent. A quick check shows that ⎯1 v ⎯0 = ⎯2 v ⎯0 = ⎯0 v ⎯0 = ⎯3 v ⎯ 0 = ⎯3. So ⎯0, ⎯1, ⎯2, and ⎯3 are all equivalent as well. Therefore in this case AI has just two elements, a top and a bottom (‘the True’ and ‘the False’). That is the two element Boolean algebra.
Can this be the case in general? What candidates do we have for intensionally ignorable comprehensions? Clearly the elements (x ∧ ⎯x). They are mapped to the absurd property: ||x ∧ ⎯x|| = f. So let us choose for the ideal of ignorables an ideal that includes {z: for some x in AC, z = x ∧ ⎯x}.[6]
But then in AI, whose members are exactly the properties ||x|| with x in AC,
||x ∧ ⎯x|| = f = ||x|| ∧ ⎯||x||
hence by De Morgan’s laws, ⎯||x|| v || x|| = ⎯ f = T, the summum genus.
Therefore AI is a Boolean algebra.
I have not so far mentioned comprehensions that are not ignorable although they strike us at once as self-contradictory. At first blush
brother of someone with no siblings,
sister of someone with no siblings
are different concepts, despite their apparent self-contradictory-ness. For one is a concept of a male and the other is a concept of a female.
What we can say about it is only this, I think: they do not have the same intention, and are not intensionally equivalent, but each is the meet of something with the intentionally ignorable comprehension being a sibling of someone who has no siblings.
I doubt that this is the end of the matter. The logic of comprehension must be, with respect to self-contradictions at least as liberal as FDE … yes, but who knows what else lurks in these deep-black logical waters, yet to be appreciated?
INDEX TO SYMBOLS
≤ , v, ∧ , ⎯ : partial order, join, meet, involution in an algebra
the absurd property: f, the summum genus (top property): T
equivalence class of property x: [x]
assignment of extensions to properties: | |
assignment of intensions (properties) to comprehensions: || ||
extensional equivalence of properties: ≡
intensional equivalence of comprehensions: ⇔
top of a De Morgan lattice: Θ
bottom of a De Morgan lattice: ⊥
NOTES
[1] Terminology varies. Alonzo Church’s review of C. I. Lewis’ The Modes of Meaning begins with “As different mtodes, or kinds, of meaning of terms the author distinguishes the denotation of a term, the comprehension, the connotation or intension, the signification, the analytic meaning.” (JSL 9 (1944): 28-29). Lewis’ terminology was not standard, and as Church shows, not clear; though influential, his work did not standartize usage in this respect.
[2] More familiar is the example of a distinction between triangle and trilateral, discussed by Leibniz among others. His oft quoted passage on the matter: “[T]hings that are conceptually distinct, that is, things that are formally but not really distinct, are distinguished solely by the mind. Thus, in the plane, Triangle and Trilateral do not differ in fact but only in concept, and therefore in reality they are the same, but not formally. Trilateral as such mentions sides; Triangle, angles.”
[3] I am tempted to adopt “concept” for the comprehension of a predicate. But its associations to the mental might become a constant worry.
[4] To help my intuition and imagination I keep in mind reduction modulo sets of measure 0 as paradigm example.
[5] Thm. 5 on p. 27 of Birkhoff 1967 Lattice Theory 3rd edition
[6] Minimally, the ideal generated by that set. But we may want additions to the ignorables, like the meet of being and non-existence.
[A reflection on Richmond Thomason’s “Independence of the Dual Axiom in Modal K with Primitive ◊”]
Within modal logic we customarily take necessary to be equivalent to not possibly not. Thomason shows that the answer to my title question is NO: if we formulate the logic with ◊ as primitive, we need to add that equivalence as an axiom. There is an interpretation of K in which the duality fails. The interpretation, he says, is exotic …. It is, but also startling and provocative.
I’ll explain his model and reasoning, and then explore his method with a smaller model (which gives a weaker result). His method: construction of a small language that is classical (truth-functional connectives) and hyper-intensional.
In Thomason’s paper the family of propositions is represented by the union B ∪ B’ of two Boolean algebras with two atoms each:
The partial order within this union is just within each of the parts: if x and y belong to B and B’ they are not ordered relative to each other, they have no meet or join. Think of this as a matrix, with only V and V’ designated values (‘true’).
There is a sort of complement operator, which I will symbolize as ¬, that is ordinary in B but takes elements in B’ to B.
As a result, of course, ¬¬ x is always in B, and is called Twin(x). Specifically, ¬1 = 2, ¬2 = 1. But ¬ 2’ = 1 as well, so Twin(2’) = ¬ ¬ 2’ = 2. Similarly, ¬1’ = 2, so Twin(1’) = 1.
Then there is sort of conditional operator (x → y) = (¬Twin(x) v Twin(y)); its values are always in B. Thus when sentences are given these propositions as their semantic values, and “not” and “if then” are assigned ¬ and →, classical propositional logic is sound.
Clearly ¬ is not an involution on B ∪ B’, but like in Intuitionistic logic, doubly complimented propositions act classically.
So, as far as that is concerned, the strange algebraic structure of the family of propositions is hidden from sight. But that strangeness can be utilized in the interpretation of ◊.
That interpretation is simple: each of B and B’ is divided into ‘possible’ and ‘impossible’ regions: the ‘possible’ propositions are those inside the dotted ellipses.
The possibility operator, too, when applied to an element of B’, shifts attention to B:
◊1 = V, ◊V = V, but if x is in B’, we still refer to V in B: ◊2’ = V, ◊V’ = V. In all other cases ◊x= ∧. Notice that for all x, ◊x is in B, so this addition cannot affect the soundness of classical sentential logic.
With this interpretation we can verify the K axiom formulated with ◊ as primitive:
¬ ◊¬ (x → y) → [¬◊¬x → ¬◊¬y]
To check that no assignment of values to x, y yields a counterexample is straightforward.
What has happened to duality? If we define □ x as ~◊~x, what is the status of
Duality. ◊x → ~□~x, and ~□~x → ◊x ?
The first part is the same as ◊x → ~~◊~~x, which is the same as ◊x → ◊~~x. If x = 2’ then this is (◊2’ → ◊~~2’), which is (V → 2) = 2, hence not true.
The second part is similarly seen not to be true, using 1’ rather than 2’.
Adding Duality as an axiom will eliminate the ‘exotic’ interpretation.
Thomason’s method has a general form:
choose a structure and interpretation in such a way that all the semantic values of complex sentences belong to a Boolean algebra, and use extra structure in the interpretation of non-Boolean operators.
That makes it possible to construct non-standard interpretations of even normal modal logics.
I thought I’d try my hand it with a small familiar lattice that has a Boolean sublattice. As it turns out (not surprisingly) it (only) gets half of Thomason’s result.
This is N5, the smallest non-modular (hence non-distributive) lattice, the ‘pentagon lattice’. Let us define operations on it in the way Thomason did:
a sort of complement: ┐1 = ⎯ 1, ┐ ⎯ 1 = 1, ┐k = ⎯ 1, ┐T = ⊥, ┐⊥ = T.
We define the Twin x* of x to be ┐┐x. Note that k* = 1.
N5 has a Boolean sublattice B = {T, 1, ⎯ 1, ⊥} = {x*: x in N5}
a sort of conditional: (x → y) = (┐x* v y*)
Only T is designated (‘true’). Interpretation of the syntax: as above; once again all semantic values of complex sentences are located in the Boolean sublattice, so classical sentential logic theorems are all true.
a sort of possibility operator: ◊x = T iff x = 1 or T; ◊x = ⊥ otherwise.
Verification of the K axiom
¬ ◊¬ (x → y) → [¬ ◊¬x → ¬◊¬y]
Note:
¬◊¬T = T, ¬◊¬1 = T
¬ ◊¬k = ¬◊(⎯ 1) = T ¬◊¬(⎯ 1) = ¬◊1 = ⊥
¬◊¬ ⊥ = ⊥
For the consequent of the K axiom to be ⊥ means that [◊¬x v ¬◊¬y] = ⊥, so:
x is k or T or 1, and y is (⎯1) or ⊥
In these cases the antecedent is ¬◊¬ followed by:
(k → ⎯ 1) = (¬k* v ⎯ 1) = ⎯ 1
(k → ⊥) = (¬k* v ⊥) = ⎯ 1
(T→ ⎯ 1) = ⎯ 1
(T → ⊥ ) = ⊥
(1 → ⎯ 1) = ⎯ 1
(1 → ⊥ ) = ⎯ 1
and the result of prefixing ¬◊¬ is in each case ⊥. So any attempt at a counterexample fails.
Now for the duality axiom:
Duality. ◊x → ¬¬□~x, and ¬□¬x → ◊x
The second part is the same as ¬¬◊¬¬x→ ◊x, which is the same as ◊¬¬x → ◊x. But (◊¬¬k → ◊k) = (◊1 →◊k) = (T → ⊥) = ⊥.
However, the converse holds, so only half of Duality is refuted.
PART THREE. How can we generalize this method?
Note that in the above, for all x, ¬x = ¬¬¬x = ¬Twin(x). So the mapping of B’ into B does not need to be defined in terms of ¬.
So we can simply say: we have a map Twin, and we define ¬ and → to be as usual on B, and for x, y in B’ we define (x → y) = (Twin(x) → Twin(y)), and define ¬x = ¬Twin(x).
For x in B, set Twin(x) = x to make it a map of the entire structure into itself.
So that is one map, then choose another map of the structure into itself, call it α , whose values are all in B. (That is necessary to make sure that classical propositional theorems remains valid.) Any other properties you like.
Now you have a model of a classical sentential calculus extended with addition of a single unitary propositional operator, which will satisfy any axioms of your desire.
For example, thinking about the K□ axiom for α, you could specify that if x ≤ y then Twin(x) ≤ Twin(y). But you could do the opposite, so that if p implied q then αq would imply αp, acting like negation (but perhaps not just like negation). Or you could want α to be read “It is so in the story that …” and the story could be by Graham Priest.
Remark: hyper-intentionality
It is remarkable that by such simple means Thomason created a language that is at once classical and hyperintensional:
Hyper-intensionality. For all x in B and in B’, x → ¬ ¬x and ¬¬x → x. But it is not the case that ◊x → ◊¬ ¬x. For example, ◊2’ = V but ◊¬ ¬2’ = ◊2 = ⊥.
Note that if x is in B then so is ¬ ¬x. For how Thomason interprets the language, we can add that if A is any complex sentence then ◊A will imply ◊¬ ¬A. Only by using an atomic sentence (with value in B’) there is a counterexample to Duality.
In the case of axiom ¬ ◊¬ (A → B) → [¬◊¬A → ¬◊¬B], the ◊ operator is applied only to sentences that begin with ¬, hence are complex. So no such way of providing counter-examples applies there.
The created language, at once classical and hyper-intensional, is intriguingly unusual!
REFERENCE
Thomason, Richmond (2018) “Independence of the Dual Axiom in Modal K with Primitive ◊”. Notre Dame Journal of Formal Logic 59: 381-385.
This is a reflection on John Bacon’s article “First Order Logic based on Inclusion and Abstraction”.
Aristotle’s syllogistic was a logic of general terms, and the same could still be said of traditional logic in the 19thcentury. Writing in 1900, Bertrand Russell explained the failures he saw in Leibniz’s philosophy by the simple point that Leibniz was held back by the form of Aristotle’s syllogistic, the only logic then in play.
Ironically, we still see the struggles and birth pangs of a new logic coming into being in Russell’s 1903 Principles of Mathematics. What sense could there be in the new ideas about sets or classes? A set is meant to be a collection. But by the very meaning of the word, a collection is a collection of things, “things” plural. Does it make sense to speak of a collection of just one thing? What is the difference between “I put an orange on the table” and “I put a collection of one orange on the table”? It does make sense to say “I put nothing on the table”, but what about “I put a collection of nothing on the table”?
One might say, not unfairly, that all these puzzles were put behind us when Frege (in effect) introduced set-abstraction. (Even if he did so a little injudiciously, even inconsistently, the idea was soon tamed and made practical.) Within limitations, x̂Fx denotes the set of all things F, the singleton or unit set appears as extension of x̂(x = y), and the empty set as x̂(x 
But then, can’t we imagine that traditional logic could have developed equally successfully, through that same innovation?
I read John Bacon’s paper as answering this question with a resounding Yes.
As Bacon shows, the new logic that we enjoy today can be seen as a straightforward development of the very ‘logic of terms’ that Boole and De Morgan (and Schroeder and Peirce and …) turned into algebra. And so I would like to explain in a general way just how he does that.
To begin, it is a recurrent complaint about syllogistic that it is really only a logic of general terms, and has no good place for singular propositions. “Socrates is mortal” is accommodated awkwardly and artificially .
But as soon as singletons are respectable denizens of the abstract world, there is no problem for a logic of terms:
“Socrates is mortal” is true if and only if {Socrates} ⊆ {mortals}.
The three primitive notions that Bacon uses to simultaneously construct a contemporary form of the logic of terms and reconstruct first-order logic in just that form are these: the relation of inclusion, an abstraction operator, and the empty set.
Vocabulary: a set of variables, a set of constants, x̂, ⊆, and 0. Grammar: variables and constants are terms, and andx̂A is a term if A is any sentence and x a variable; if X, Y are terms then X ⊆ Y is a sentence.
The constants are names of sets, they cannot include “Socrates” but only a name for the set {Socrates}. The variables stand for singletons. So “Socrates is mortal” can be symbolized as “x ⊆ Y”.
Reconstruction of propositional calculus
If A is a sentence then x̂A is the set of all things x such that A. In this part we’ll only look at the case where x does not occur in A or B, for example x̂[2 + 2 = 4]. All things are such that 2+2 = 4! So that set is the universe of discourse, it has everything in it. We have a neat way to introduce the truth values:
A is true if and only if x̂A = the universe
A is false if and only if x̂A = 0.
This already tells us how to denote the universe:
U = x̂(O ⊆ 0)
A quick reflection shows that, accordingly, (x̂A ⊆ 0) always has the opposite truth-value of A. So we define:
~A = (x̂A ⊆ 0)
What about material implication, the sentence A ⊃ B? That is false only if A is true and B is false. But that is so exactly and only if x̂A = U and x̂B = 0, so we define:
(A ⊃ B) = (x̂A ⊆ x̂B)
Since nothing more is needed to define the other truth-functional connectives, we are finished. It is at once useful to employ the definable conjunction & at once for a definition:
X = Y iff (X ⊆ Y) & (Y ⊆ X)
Reconstruction of quantificational logic
Let’s to begin think only about unary predicates. The variables here can occur in the sentences we are looking at.
“Something is F” we symbolize today as “(∃x)Fx”. But this is clearly true if and only if the set x̂[Fx] is not empty. We define:
(X ≠Y) = ~(X = Y)
(∃x)A = (x̂A ≠ 0)
Similarly, the universal quantification (∀x)Fx is true precisely if x̂A = U.
Symbolizing “All men are mortal”:
x̂[(x ⊆ X) ⊃ (x ⊆ Y)] = U
Relations
The famous puzzle for 19th century logic was to symbolize
If horses are animals then heads of horses are heads of animals.
Bacon had no difficulty explaining how the system can be extended to relational terms P2, Q3, and so on, with the superscript indicating the addicity. The semantics can be straightforward in our modern view: P2 is a set of couples, Q3 a set of triples, and so on, with the simple idea that (ignoring use/mention):
(*) Rab is true if and only if <a, b> is in R.
Then Bacon allows strings of variables as n-adic variables. So if x stands for {Tom} and y for {jenny} then xy stands for {<Tom, Jenny>}. The sentence (xy ⊆ P2) then means that {<Tom, Jenny>} ⊆ P2.
But Bacon does it in a particular way that has some limitations. For example, in his formulation, inclusion is significant only between relations of the same degree. That raises a difficulty for examples like:
If Tom is older than Jenny then Tom is in age between Jenny and Lucy.
But there is a simple way, Tarski-style, to do away with this limitation. When Tarski introduced his style of semantics he gave each sentence as its semantic value the set of infinite sequences that satisfy that sentence.
Above we took terms to stand for sets of things in the universe of discourse. Now we can take them to stand for sets of countably infinite sequences of things in that universe. For example, the predicate “is older than” we take to stand, not for a set of couples, but for the set OLDER of infinite sequences s whose first element is older than its second element.
We represent any finite sequence t by the set of infinite sequences of which t is the initial segment. The notation <a, b> now stands not for the ordered pair whose first and second members are a and b respectively, but for the set of all infinite sequences whose first and second member are a, b respectively.
The predicate “is in age between … and —” we take to stand for the set BETWEEN of infinite sequences whose first member is in age between its second and third member. So the intersection of OLDER and BETWEEN is the set of sequences u such that the first element is older than the second, and the third element is someone older than both of them. This intersection can be the extension of a predicate P2 and if Q3 has BETWEEN as its extension then P2 ⊆ Q3is true.
But this is a minor point. There are always different, equally good, ways of doing things. The important moral of Bacon’s article is that it is not right to say that our contemporary logic had to replace the deficient logic of the tradition. Rather, the innovations that made our logic possible would equally improve the traditional logic, to precisely the same degree of adequacy.
REFERENCES
Bacon, John (1982) “First-Order Logic Based on Inclusion and Abstraction”. The Journal of Symbolic Logic 47: 793-808.
When someone in that crowded theatre shouted “Phlogiston escaping!” we knew that it was false, but of course we ran out at once.
This is a good example to illustrate Paul Feyerabend’s pragmatic theory of observation, as I will explain below. Feyerabend did leave some questions unanswered. Thinking about those questions led me to something that I had found perplexing, in Wilfrid Sellars’ correspondence with Roderick Chisholm about intentionality.
When Paul Feyerabend presented his “Attempt at a realistic interpretation of experience” to the Aristotelian Society in February 1958, much of the new scientific realism was already in place.[1] While Feyerabend presents his ideas in an explicit, detailed critique of positivist views of science and experience, we can (and his peers then could) proceed at once to his positive contribution.
This begins with a presentation of the pragmatic theory of observation, which is in the first place about what counts as an observation language.[2] There are four pragmatic conditions for observation reporting, that we can summarize (using Feyerabend’s own terms) as:
Definition. L is an observation language for a community C of observers, set S of situations, and set A of sentences of L exactly if there is a function F (‘association’) which maps S into the powerset of A, such that given a situation s in S, the members of C are able to come to a quick unanimous decision about whether to accept to reject the sentences in the set F(s), and their acceptance of any of these sentences in F(s) is a reliable indicator of their being in situation s.[3]
These conditions include nothing at all about the meaning of those sentences. The role of observation report is entirely separated from any reference to meaning or reference.
Suppose that in community C the utterance of p is a reliable indicator of the presence of fire to the utterer. The syntax of p is irrelevant: p can be learned to have this use by conditioning. It could be “boojum!” or “fire!” or “phlogiston escaping!” or “rapid oxidation!”.
“Observability is a pragmatic concept” (Feyerabend 1958:146). That is, it is a concept that belongs to the analysis of the conditions and contexts of the use of language. The distinction is Charles Morris’: in semantics we abstract from use, to concentrate on the word-world relation, while in pragmatics the relation studied is three-fold: word, world, user.
In observation reportage, humans function as measuring instruments. Perhaps in your car, a red light goes on if and only if the engine is overheating. There is no logical connection between the color of the light and the temperature of the engine. But if the light goes on and I ask you “what does that mean?” or “what does that signify?” you will answer “that the engine is overheating”.
When this little dialogue is transposed, from measurement output to observation reportage, it becomes an example of what Feyerabend calls interpretation.
For L to be not just an observation language but a fully-fledged observation language, Feyerabend submits, it must have an interpretation which determines what its sentences “are supposed to assert” (Feyerabend 1958: 145-46).
As mentioned above, in making an observation, an organism is acting as a measuring instrument:
“What the observational situation determines (causally) is the acceptance or the rejection of a sentence, i.e. a physical event. In so far as this causal chain involves our own organism we are on a par with physical instruments. But we also interpret the indications of these instruments … and this interpretation is an additional act.” (Feyerabend 1958: 146)
How is that done? Suppose again that that in community C the utterance of p is a reliable indicator of the presence of situation s to the utterer. When you, who may or may not be a member of C, interpret that utterance of p, you will describe situation s in your own language, and in accordance with your assumptions, presuppositions, theories, and linguistic practices of the community to which you belong.
Accordingly, examples of interpretation must take the following sort of form:
[1] Observers in community C reliably reliably agree to “Phlogiston is escaping” in the presence of fire and reject it in the absence of fire.
[2] Observers in community C reliably agree to “There is fire” in the presence of phlogiston escaping and reject it in the absence of phlogiston escaping.
[3] Observers in community C reliably agree to “Phlogiston is escaping” in the presence of rapid oxidation, and reject it in the absence of rapid oxidation.
Could any of this be said by a member of community C?
Yes, in the case of [1] or [2], and definitely not in the case of [3]. So in each case we have to take into account who could offer such an interpretation, and what language that person would be speaking.
I think this important, and to be emphasized: to understand [1]-[3] properly we must in each case imagine ourselves inside the community – possibly, but not generally community C – where we make the statement, or are addressed by someone making that statement. For in each case the imagined speaker takes for granted or presupposes that the addressees understand the words used to describe the relevant situation. More: the speaker takes for granted that the addressees would describe those situations in the same way.
Speakers in a given community report on their experience, by means of words which may apply either correctly or incorrectly (or not be descriptive at all) to what they are actually experiencing. This is very far from the idea that the semantic content of the observation report describes anything like an immediately, unmediated content of the ‘given’.
Standing outside a certain community we say that they reported reliably on an experience, which they most certainly had, by asserting that there was a phlogiston escape. They took it to be that. In that case, how can we think of observation reports as providing the data to which our theories are accountable?
In philosophy of language elsewhere the corresponding question was: how could there be successful reference by means of a false description? The response was a turn from semantics to pragmatics. On Russell’s theory of descriptions, the phrase “the so-and-so” denotes entity x if and only if x is so-and-so and nothing else is so-and-so. Keith Donnellan (1966) argued that we may keep the term “denote” for this, but then must recognize another use for which he offered the term “refer”. That is, someone may use “the so-and-so” to denote what that phrase denotes (whenever it denotes anything), but will be using it to refer to something which it does not denote, when the conditions are felicitous for that use.
Plausible ordinary examples abound. For example, there is in the room exactly one man who appears to be drinking a martini, and in discussing him we refer to him as “the man with the martini”. Our communication about him may be entirely successful, all (or most of) we say about him may be true, although in fact what is in his glass is just water. David Lewis offered an especially nice example: “Help me, Stephanie, the cat is fighting with the other cat again!” In this example the phrase “the cat” does not actually denote anything, but Lewis used it successfully to refer to a specific cat – a cat he falsely described as being the only cat there! – nevertheless.
In Donnellan’s terminology, denotation is a (semantic) relation between words and things while reference is a (pragmatic) relation between users and use, words, and things.
We can make a similar distinction about truth and a related notion of correctness under the circumstances:
an observer may be making a correct observation report of fire by asserting “there is phlogiston escaping”, although that statement is literally false.
How we interpret the output of a measuring device depends on the theories we currently accept. Galileo designed an instrument to measure the force of the vacuum; today we interprets its results as measuring atmospheric pressure. Feyerabend insists that the same goes for our observation reports, and codifies this as:
“thesis I: the interpretation of an observation-language is determined by the theories which we use to explain what we observe, and it changes as soon as those theories change.” (Feyerabend 1958: 163, his italics)[4]
When it comes to such an observation report as “Fire!”, that was once interpreted, in a certain community, as reporting a rapid phlogiston escape. Members of that community could equally well shout “Phlogiston escaping!”, and if they did, everyone, in theoretical agreement or not, would be well advised to prevent themselves from getting burned.
If thesis I. is correct then the way we understand observation reports will change as our theories change. Is that consequence of the thesis in accord with our history?
Observational language in use may appear not to change as theories change, because the syntax does not change. From this we should not infer that meaning is invariant.[5] One example Feyerabend offers is the changing interpretation of color-reports (Feyerabend 1958: 160-162). Once the Doppler effect for light is discovered, the report “x is red” is interpreted as a relational statement, with the relative velocity of observer and observed entered as the additional parameter. On the “human” level, the actual practice of color-reporting does not change, for there the velocities are too small for the effect to be noticed. But the interpretation does: the scientifically literate will interpret color observation reports by describing the situation in question with reference to that relative velocity.
The realist account of experience which Feyerabend submits is therefore along the following lines:
The interpreter, speaking his own current natural language and from within his own cluster of accepted scientific theories, has no difficulty referring to the relevant situations, generally by means of descriptions that he takes to be correct, and classifying the putative observation statement as a report about that situation if the above pragmatic conditions are satisfied.
It seems to me that we should understand Feyerabend’s lecture with reference to the background in which he developed these ideas. Salient in this respect is the, by then already accepted, new scientific realism in the Minnesota Center for the Philosophy of Science, which Feyerabend had joined in 1957.
Wilfrid Sellars, Thomas Kuhn, Paul Feyerabend, and Norwood Russell Hanson all insisted on the theory-laden-ness of our language in use. There was a great difference between their approach to the language of science and the way logical positivists had thought about language. The new realists’ lack of interest in the logical syntax of language is understandable. For scientific theories are presented in our current natural language, though augmented with mathematics. The language of science is not, in itself, an uninterpreted calculus that needs to have meaning bestowed on it!
And so, when Feyerabend says that to be full-fledged language the observation language needs to have an interpretation, it can be taken for granted that this interpretation can be given in natural language, and be based on the currently accepted scientific theories which were formulated in our natural language in use.
But there is still an important difference between our four protagonists. Wilfrid Sellars, unlike the other three, was intent on engaging with a wide-ranging diversity of traditional issues in philosophy. His rejection of the earlier realism of Roy Wood Sellars’ generation, as well as of Carnap’s logical positivism, came along with systematically developed responses to those issues. What an interpretation is, what meaning is, taken as a general question, Sellars could not leave unaddressed.
What is clear enough, and in 1957 had already been clear for some time, is that there is no simple relation between observation terms and theoretical terms. Not even for the speaker who formulates the interpretation. The idea of an operational definition of such terms as “oxidation” does not get anywhere. What about the converse? Do the sorts of interpretation exhibited above as [1]-[3] offer anything like a definition of the terms used for observation reports?
If someone in our community says “Fire is rapid oxidation”, could we parse that as “Our observation term “fire” means rapid oxidation”?
That is plausible at first blush. To make it plausible to us today may not be easy, used as we are to the sparsity of semantic accounts that take only truth and reference into account. We could certainly say that any actual proper reporting use of “fire” will refer to an instance of rapid oxidation. But actual reference is not enough to determine meaning.
Nevertheless it would seem that to interpret “Fire!” in Feyerabend’s sense, is to say that it means that there is rapid oxidation. For that is just a parallel to the example of the driver who, seeing a red light in his car, says “this means that the engine is overheating”.
But what is meant by “means” in that sort of assertion?
Feyerabend criticizes two accounts of the meaning of observation terms, ones he attributes to positivist philosophers of science, but does not give one of his own.[6]
Willard Van Orman Quine had thrown a wrench into this topic of meaning, with his “Two dogmas of empiricism” (1951) and its trenchant, though rather behavioristic, critique of meaning, analyticity, and synonymy.
In response, Rudolf Carnap pointed insistently to the distinction between pragmatics and semantics, in his “Meaning and Synonymy in Natural Languages” (Carnap 1955). Denotation, extension, and truth, studied in abstraction from use, are the subject of semantics, and Quine is right to find a solid basis for philosophical analysis there. But meaning, intension, and intensional relations like synonymy are the subject of study in pragmatics, which brings in patterns in usage. Carnap insists that these patterns, based in dispositions to use words in certain ways, fix considerably more than extension. As an example he suggests that different linguists might translate “Pferd”, as used by a German speaker, Karl, as “horse” or instead as “horse or unicorn”. While the extensions of those English phrases are the same, that there is a difference can be put to the test by asking for Karl’s response to pictures and stories.
While I (and I think Sellars) would speak of linguistic commitments rather than linguistic dispositions, I take Carnap’s to be an adequate response to Quine’s main arguments (and have nothing good to say about the others). Nevertheless, Carnap’s response does not throw much light on the basic concepts of pragmatics, and does not go far toward providing pragmatics with a sound theoretical basis.[7] In the Minnesota Circle, where Feyerabend resided at the time, there was a much farther reaching attempt to do so, in progress, at the hands of Wilfrid Sellars.
Sellars’ correspondence with Roderick Chisholm about intentionality had been published just then, as an Appendix to the Minnesota Studies in Philosophy of Science (Sellars and Chisholm 1957 ). There an enigmatic, and perhaps somewhat unfortunate, assertion by Sellars introduces what I see as the central theme in Sellars’ analysis.
Chisholm held that we have thoughts, and the meaning our statements have is the thoughts they express. As Sellars understands this, it implies that such a sentence as
[a] “Hund” (in German) means dog
has the form
“Hund” expresses t, and t is about dogs.
which states that there are certain relations between three things. Each of these relations is a word-thing relation. So this way of understanding meaning statements remains solidly within semantics rather than pragmatics.[8]
The dialogue between these two eminently subtle thinkers is eminently subtle, but I think that the crucial clue to Sellars’ view arrives in this passage in Sellars’ letter of August 31, 1956:
“Thus, while I agree with you that the rubric
” .. . ” means – – –
is not constructible in Rylean terms ( ‘Behaviorese,’ I have called it), I also insist
that it is not to be analysed in terms of
“. . .” expresses t, and t is about – – -.
My solution is that “‘ .. .’ means – – -” is the core of a unique mode of discourse which is as distinct from the description and explanation of empirical fact, as is the language of prescription and justification.” (Sellars and Chisholm 1957 : 527)
Chisholm is puzzled. Prescriptions, he writes, are neither true nor false. But isn’t such a semantic statement as [a] “’Hund’ (in German) means dog” true?
Sellars agrees that it is true. That admission introduces a negative analogy to prescriptions. But Sellars insists there is more to [a] than that it is true, so some positive analogy remains.
To teach someone a bit of German by saying “’Hund’ (in German) means dog” requires that this person is, like the teacher, a user of the English word “dog”. Sellars writes that in such a case,
“there is an important sense in which this statement does not describe the role of “Hund” in the German language, though it implies such a description. (Remote parallel : When I express the intention of doing A, I am not predicting that I will do A, yet there is a sense in which the expression of the intention implies the corresponding prediction.)” ” (Sellars and Chisholm 1957 : 532)[9]
There is indeed an important distinction between the expression of an intention, and the statement that one has that intention. Imagine asking someone “Will you marry me?” and receiving as answer the statement “I do in fact have the intention to marry you, and such intentions are typically followed by marriage”. Imagine, in contrast, that the answer had been the expression of intention in the words “Yes, I will marry you”. The latter is surely what the suitor hoped, not the former. Expressing the intention is different from stating that she has that intention. Nevertheless, if she expressed the intention to do so, the suitor would have warrant to infer the statement that she does in fact have that intention.
We may still, like Chisholm, be at a loss as to how this clarifies the discourse about meaning. Sellars then goes on to explain his point in a different way, by recourse to Church’s translation test. I think we can see that employed here as follows, in an attempt to teach someone a bit of German. Suppose that
[A] “Hund” is a word for dogs
were just a statement of fact. Then its German translation would also be a statement of fact, with the same information content. That translation is
[B] „Hund“ ist ein Wort für Hunde.
But, although that is certainly true, [B] does not have the same status as [A]. Indeed, the student might already know enough German to realize that sentences of form [B] are always true, even while not knowing the reference of “Hund”. For [B] has the status for a German speaker which
[C] “Dog” is a word for dogs
has for a speaker of English, while [A] does not have that status. [10]
It may seem that I have gone astray, into something not related to Feyerabend’s provocative Thesis I. But not so.
In his answer to Chisholm, Sellars refers unavoidably to distinct communities with different languages. The informative [A] must be presumed to be addressed to someone sufficiently far in the English speaking community to understand “means dog in German” or “is a word for dogs”. The speaker of [A] takes that for granted, presupposes that, and we are here at the crucial point also made above about assertions [1]-[3].
It is a crucial point for meaning or interpretation in general. In his lectures collected as The Metaphysics of Epistemology, Sellars clarifies this with the distinction between
[D] “und” (in German) means and
and
[E] “und” (in German) means the same as “va” in Sanskrit”. (Cf. Sellars 1989: 240)[11]
The important difference between [D] and [E] lies in their presuppositions, when taken as items in a dialogue or communication. The assertion of [D] presupposes understanding of the English “and”, it is addressed to someone taken to have in his own vocabulary all that follows the word “means” . In contrast, [E] does not presuppose understanding of any Sanskrit. The assertion of [E] conveys factual information only, a relation between elements of two languages outside the addressee’s community.
If we tried to deal with these examples of ‘meaning’ discourse solely within semantics – that is, with attention only to the relation words bear to the world, independent or abstracted from contexts of use — we would be at a loss. Ignoring what is presupposed when a speaker addresses someone with [D], we would have to construe [D] as
[F] “und” (in German) means the same as “and” in English.
But [F], although it is an English sentence and must be assumed to be addressed to an English speaker, does not presuppose that the addressee has “and” in their vocabulary. If that is implausible (for how can someone have a significant amount of English, and not have “and”?), an example in which the target words has some unusual synonyms will serve:
[F*] “Hund” (in German) means the same as “canine” in English.
That information would not suffice for addressees who knew English but did not have the word “canine” in their vocabulary.
Within pragmatics, then, [D] does not have the status of a simple assertion of the form “a is related R-ly to b”. Instead [D] is part of an intra-communal discourse, meaningful in certain contexts and meaningless in others.
Let us go back now to Feyerabend’s realist interpretation of experience and its reportage, as I codified it in
[1] Observers in community C reliably agree to “Phlogiston is escaping” in the presence of fire and reject it in the absence of fire.
[2] Observers in community C reliably agree to “There is fire” in the presence of phlogiston escaping and reject it in the absence of phlogiston escaping.
[3] Observers in community C reliably agree to “Phlogiston is escaping” in the presence of rapid oxidation, and reject it in the absence of rapid oxidation.
Imagine ourselves in a distinct community C*, where we speak an English that is by now so thoroughly, relevantly theory-laden, that we would be entirely at a loss if we heard any apparent difference in usage between “fire” and “rapid oxidation” .
To begin, we would have no qualms about rejecting [2] altogether, while parsing [1] as
[1*] “phlogiston is escaping” (in C language) means that fire is present.
This would be on a par, for us, with
[1**] That the red light is on means that the engine is overheating,
though we would definitely reject as false:
[1***] “The red light is on” means the same as “the engine is overheating”.
We would also say that
[4] “fire is present” is a phrase for episodes of rapid oxidation,
and, if pressed, we would have to agree to the rather awkwardly worded
[3*] “phlogiston is escaping” (in C language) means there is rapid oxidation occurring.
Note well that [4] is in our community a pragmatic tautology, and that [1*] and [3*] make sense only as intra-communal discourse by us, as accurate statements about another community’s observation language.
At the same time we would surely reject:
[3**] “phlogiston is escaping” (in C language) means the same as “there is rapid oxidation occurring” in our language.
For the meaning of “phlogiston is escaping” can only be explained in terms of phlogiston theory, which we do not accept, and which we take to be false. It is [3*], and not the falsehood [3**] that motivates us to leave the theatre if someone shouts “Phlogiston escaping!”, even while we judge the shouter to be shouting a falsehood.
Formal semantics did not develop along the route charted by Wilfrid Sellars.[12] In the above account of meaning there is a crucial distinction between
and
It is not assumed that the language of another community is unintelligible to us, or incommensurable with our own.
Quite the contrary: someone whose own language is English may be a teacher, teaching German to a French student, who is still learning English while enrolled in that teacher’s German class.
Equally, someone in our own community, whose language is current chemistry-theory-laden, may teach a history of science class, and depict how persons in a certain historic community reported the presence of fire, using language that was phlogiston-theory-laden.
Fine so far, but ….
As I reflect on the above account of meaning and interpretation, it seems to me that a great deal is left to rest on the distinction between what is my language, and what is a language that I understand.
And that raises a further question, that remains to trouble us: what is my language?
Carnap, Rudolf (1947) Meaning and Necessity: A Study in Semantics and Modal Logic. Chicago: University of Chicago Press.
Carnap, Rudolf (1955) “Meaning and synonymy in natural languages”. Philosophical Studies 6: 33-47.
Keith S. Donnellan (1966) “Reference and definite descriptions”. The Philosophical Review 75: 281-304).
Feyerabend, Paul (1958) “Attempt at a realistic interpretation of experience”. Proceedings of the Aristotelian Society 58: 143-170.
Feyerabend, Paul (1962) “Explanation, Reduction, and Empiricism”. PP. 103-106 in H. Feigl and G. Maxwell (ed.), Minnesota Studies in the Philosophy of Science, 3: 28-97.
Feyerabend, Paul (1981) Realism, Rationalism & Scientific Method. Philosophcial Papers Volume I. Cambridge: Cambridge University Press.
Kuhn, Thomas (1962) “The Structure of Scientific Revolutions”. pages 1-173 in the International Encyclopedia of Unified Science II-2. Chicago: University of Chicago Press.
Sellars, Wilfrid and Roderick Chisholm (1957) “Intentionality and the Mental: a Correspondence”. Minnesota Studies in the Philosophy of Science 2: 507- 539.
Sellars, Wilfrid (1989) The Metaphysics of Epistemology. Ed.: Pedro Amaral. Atascadero: Ridgeview Pub. Co.
[1] In a footnote Feyerabend acknowledges his debt to discussions at the Minnesota Center for Philosophy of Science, where he was a member in 1957. (Note: in the published paper that is footnote 22, in the 1981 book reprint it is 31.) Thomas Kuhn’s The Structure of Scientific Revolutions would appear in the International Encyclopedia of Unified Science in 1962, with an acknowledgement to Feyerabend in its preface. The Journal of Philosophy (54: 709-712 notes and news, 1957) reported: “A conference, sponsored by the National Science Foundation, was conducted at the Minnesota Center for Philosophy of Science from August 12 to September 14, 1957. The participants were: H. Gavin Alexander, Eva Cassirer, H. Feigl (Director of the Center), P. Feyerabend, C. G. Hempel, G. Maxwell, H. Mehlberg, E. Nagel, H. Putnam, W. Rozeboom, M. Scriven, and W. Sellars. Daily group discussions and essays, circulated as memoranda, treated, extensively and in detail, the logical and philosophical issues of quantum mechanics in particular and of scientific theories in general.”
[2] The term “pragmatic theory of meaning” does not occur in this lecture, but Feyerabend used it afterward; see e.g. Feyerabend 1981: 51, 125.
[3] The inclusion of “unanimous” is meant to indicate that these reactions by the community are reliable or consistent in certain respects, which Feyerabend indicates but does not clarify very far.
[4] This is offered in opposition to the Stability Thesis, that the meaning of observation terms is the same before and after scientific theory change.
[5] Feyerabend (1962: 30) introduces the term “principle of meaning invariance” for what he disputes, whereas in the 1958 lecture he used “Stability Thesis”.
[6] The two accounts he criticizes are the principle of pragmatic meaning (the interpretation of an observational term is determined by its use) and the principle of phenomenological meaning (the interpretation of an observational term is determined by what is ‘given’ by way of feelings and sensations in the appropriate circumstances).
[7] Carnap’s main achievement, in the development of what he calls the method of extension and intension (Carnap 1947) was the development of a formal semantics for modal logic, still in abstraction from context- or use-dependence of modal locutions.
[8] While Chisholm has a quasi-psychological account, with thoughts as central characters, the form of his view is that of the Platonist construal of language and meaning: the word’s meaning is an entity, to which the word bears a certain relation.
[9] The remote analogy is not very remote, for Sellars asserts without qualification that “semantical statements about linguistic episodes do not describe, but imply a description, of these episodes” (Sellars and Chisholm 1957: 536).
[10] Sellars makes the point in a slightly different way: someone might be told that “Hund” plays in German the same role as “dog” plays in English, and still not know the reference of “Hund”. namely if he has not learned the referring use of “dog”.
[11] The lectures collected in this book were delivered in 1975, and edited in collaboration with Sellars.
[12] Much work was done to develop formal pragmatics, adapting models of modal logic by adding parameters for contexts, speakers, and agents. My hope is that this can be complemented by reference to the early discussions of the language of science in practice.
We tend to have wrong beliefs about many things. The criteria for having a belief do not stop at introspection and so we may be wrong also about what beliefs we have. We are not fully self-transparent, and so it may not be right to blame us for such mistakes.
But it is still appropriate to point out debilitating forms of error, just as we would for a distracted or forgetful accountant. After all, the success of our practical projects may depend on the beliefs we had to begin.
As a most minimal norm, beyond mere logical consistency, I would propose this:
our belief content should not include any awowal of something we have definitely disavowed.
We can avow just by asserting, but to disavow we need to use a word that signifies belief in some way. For example, to the question: “Is it raining?”, you can just say yes. But if you do want to demur, without giving information that you may not have, the least you must do is to say “I don’t believe that it is raining”.
Definition. The content of someone’s beliefs is B-inconsistent if there it includes some proposition p and also the proposition that one does not believe that p.
B-consistency is just its opposite.
I am modeling belief content as a set of propositions, and minimally consistent belief contents are B-consistent sets of propositions. I will also take it that belief can be represented as a modal operator on propositions: Bp is the proposition that encodes, for the agent, the proposition that s/he believes that p.
Now the study of belief systems has often focused on problems of consistency for updating policies. Whatever you currently believe, it may happen that you learn, or have warrant to add, or just an impulse to add, a new belief. That would be a proposition that you have not believed theretofore. The updating problem is to do without landing in some inconsistency. That is not necessarily easy, since the reason that you did not believe it theretofore is because you had contrary beliefs. So there is much thought and literature about when such a new belief can just be added, and when not, and if not, what to do.
However, responses to the updating problem generally begin by mapping out a safe ground, where the new belief can just be added. Under what conditions is that unproblematic?
A typical first move is just to require consistency: that is, if a new proposition p is consistent with (consistent) belief content B then adding p to B yields (perhaps part of) a (consistent) belief content. I think we had better be more conservative, and so we should require that the prior beliefs include an explicit disavowal of any belief both of p and of it its contraries.
So here is a modest proposal for when a new belief can just be added without courting inconsistency of any sort:
Thesis. If a belief system meets all required criteria of consistency, and it includes disavowal of both p and not-p, then the result of adding p while removing its disavowal, does not violate those criteria of consistency.
We might think of the Thesis as articulating the condition for a system of belief to be updatable in the normal way under the best of circumstances.
A pertinent example then goes like this:
I have no idea whether or not it is now raining in Peking. I do not have the belief that it is so, nor the belief that it is not so. For all I know or believe, it is raining there, or it is not raining there, I have no idea.
The Thesis then implies that if I were to add that it is raining in Peking to my beliefs (whether with or without warrant) the result would not be to make me inconsistent in any pertinent sense.
But now we have a problem. In that example, I have expressed my belief that I do not believe that it is raining in Peking – that part is definite. But whether it is raining in Peking, about that I have no beliefs. Let’s let p be the proposition that it is raining in Peking. In that case it is clear that I neither believe nor disbelieve the following conjunction:
p & ~Bp
So according to the thesis I can add this to my beliefs, while removing its disavowal, and remain consistent.
But it will take me only one step to see that I have landed myself in B-inconsistency. For surely I believe this conjunction only if I believe both conjuncts. I will be avowing something that I explicitly disavow.
Dilemma: should we accept B-consistency as a minimal consistency criterion for belief, or should we insist that a good system of beliefs must be one that is updatable in the normal way, when it includes nothing contrary, and even disavows anything contrary, to the new information to be added?
(It may not need mentioning, but this dilemma appears when we take into account instances of Moore’s Paradox.)
If we represent our opinion by means of a subjective probability function, then (full) belief corresponds to probability 1. Lack of both full belief and full disbelief corresponds to positive probability strictly between 0 and 1.
Normal updating of a prior probability function P, when new information E is obtained, consists in conditionalizing P on E. That is to say, the posterior probability function will be
P’: P’(A) = P(A|E) = P(A & E)/P(E), provided P(E) > 0.
So this is always the normal update, whenever one has no full belief either way about E.
In a passage famous in certain quarters David Lewis wrote about the “class of all those probability functions that represent possible systems of beliefs” that:
This class, we may reasonably assume, is closed under conditionalizing. (1976, 302)
In previous posts I have argued that probabilistic versions of Moore’s Paradox raise the same problem for this thesis, that a class of subjective probability functions represent possible systems of belief only if it is closed under conditionalization.
( “Stalnaker’s Thesis –> Moore’s Paradox” 04/20/2023; “Objective Chance –> Moore’s Paradox” 02/17/2024).
Lewis, David K. (1976) “Probabilities of Conditionals and Conditional Probabilities”. The Philosophical Review 85 (3): 297-315.
Seminal text: Nicholas Rescher, The Logic of Commands. London: 1966
In deontic logic there was a sea change when imperatives were construed as default rules (Horty, Reasons as Defaults: 2012). The agent is conceived as situated in a factual situation but subject to a number of ‘imperatives’ or ‘commands’.
Imperatives can be expressed in many ways. Exclamation marks, as in “Don’t eat with your fingers!”, may do, but are not required. Adapting one of Horty’s examples, we find in a book of etiquette:
These are declarative sentences. But in this context they encode defeasible commands, default rules. Reading the book of etiquette, the context in question, we understand the conditions in which the indicated actions are mandated, and the relevant alternatives that would constitute non-compliance.
In this form of deontic logic, what ought to be the case in a situation is then based on the facts there plus the satisfiable combinations of commands in force.[1]
Imperatives have a three-fold pattern for achievement or lack thereof:
In the first example above, the case will be ‘moot’ if there is no food, or if you are not eating. Success occurs if there is food and it is not eaten with the fingers, Failure if there is food and it is eaten with the fingers.
Whenever this pattern applies, we can think of that task as having to be carried out in response to the corresponding imperative. There are many examples that can be placed in this form. For example, suppose you buy a conditional bet on Spectacular Bid to win in the Kentucky Derby. Doing so imposes an imperative on the bookie. He is obligated to pay off if Spectacular Bid wins, allowed to keep the money if she loses, and must give the money back if she does not run.
An imperative may be identified in the form ‘When A is the case, see to it properly that B’. This way of identifying the imperative specifies just two elements of the three-fold pattern, Success and (the opposite of) Moot.
But the opposite of Moot is just the disjunction of the two contraries in which the condition is present. Therefore it is equally apt to represent the imperative by a couple of two contraries, marking Success and Failure. Doing so gives us a better perspective on the structure of imperatives and their relation to ‘ought’ statements.
So I propose to identify an imperative with an ordered pair of propositions <X, Y>, in which X and Y are contraries. Intuitively they correspond respectively to Success (and not Moot), and Failure (and not Moot).
Our examples point quite clearly to a view of imperatives that goes beyond truth conditions of the identifying propositions. What makes for success or failure, what makes for the truth of the statement that the imperative has been met or not met, are specific events.
That Spectacular Bid wins, or that you close the door when I asked you to, are specific facts or events which spell success. That I eat the asparagus with a fork is a distinct event which spells a failure of table etiquette.
Consider the command
(*) If A see to it that B!
as identified by its two contraries, Success and Failure. For each there is a class of (possible) events which ‘terminate’ the command, one way or the other.
The statement “Spectacular Bid wins” states that a certain event occurs, and encodes a success for the bookie’s client. The statement that encodes Failure is not “Spectacular Bid does not win”. Rather it is “Spectacular Bid runs and does not win”, which is, for this particular imperative the relevant contrary.
To symbolize this identification of imperatives let us denote as <X| the sets of events that make X true, and as |X> the set of events that make the relevant contrary (Failure) true.[2] The imperative in question is then identified by an ordered couple of two sets of events, namely (<X|, |X>). I will abbreviate that to <X>.
In (*), <X> is the imperative to do B if A is the case, so X = the statement that (A and it is seen to that B), which is made true by all and only the events in set <X|. Its relevant contrary in this particular imperative is the statement that (A but it is not seen to that B), and that relevant contrary is whatever it is that is made true by all and only the events in set |X>.
There is an obvious sense in which E, “Close the door and open the window!” entails F, “Close the door!” Success for E entails success for F. But that is not all. Failure for F entails failure for E. The latter does not follow automatically from the former, if there is a substantial Moot condition: not winning the Derby does not, as such, imply losing.
So entailment between imperatives involves two ‘logical’ implications, going in opposite directions, and we can define:
Definition. Imperative A entails imperative B exactly if <A| ⊆ <B| and |B> ⊆ |A>.
“Open the door!” is a ‘strong’ contrary to “Close the door!”. There is a weaker contrary imperative: if someone looks like he is about to close the door, you may command “Do not close the door!”.
Negation. In the logic of statements, the contradictory is precisely the logically weakest contrary. For example, yellow is contrary to red and so is blue, but to be simply not red is to be either yellow or blue or … and so forth.
So I propose as the analogue to negation that we introduce
<┐A>: <┐A| = |A> and |┐A> = <A|
Whatever makes ┐A true is what makes A false, and vice versa. Here the symbol “┐” does not stand for the usual negation of a statements, because imperatives generally have significant, substantial conditions. The relevant contrary to Success is not its logical contradictory (that would be: either Failure or Moot) but Failure (which implies not-Moot), and that is whatever counts as Failure for the particular imperative in question.
Conjunction. “Close the door and open the window” we can surely symbolize as <A & B>. Success means success for both. In addition, failure means failure for one or the other or both. So there is no great distance between conjunction of Success statements and the ‘meet’ operation on imperatives:
<A & B>: <A & B| = <A| ∩ <B|, |A & B> = |A> ∪ |B>.
Disjunction. Similarly, dually, for disjunction and the ‘join’ operation:
<A v B>: <A v B| = <A| ∪ <B|, |A v B> = |A> ∩ |B>
We can already see that some familiar logical relations are making an appearance.
[1] <A & B> entails <A>, while <A> entails <A v B>.
For example, <A & B| ⊆ <A| and |A> ⊆ |A & B>.
(All proofs will be provided in the Appendix.)
We could go a bit further with this. But answers to the really interesting questions will depend on the underlying structure of events or facts, that is, of the truthmakers.
Events combine into larger events, with an analogy to conjunction of statements. So the events form a ‘meet’ semilattice. Important are the simple events:
Postulate: Each event is a unique finite combination of simple events.
Is it reasonable to postulate this unique decomposability into simple events?
At least, it is not egregious. Think of how we specify a sample space for probability functions: each measurable event is a subset of the space. The points of the space may have weights that sum up to the measure of the event of which they are the members. Two events are identical exactly if they have the same members.
In any case, the idea of truthmakers is precisely to have extra structure not available in possible worlds semantics.
Combination we can conceive of as a ‘meet’ operation. Besides combining, we need an operation to identify contraries among events, in order to specify Success and Failure of imperatives.
Definition. A event structure is a quadruple E = <E, E0, ., ° >, where E is a non-empty set, . is a binary operation on E, E0 is a non-empty subset of E, and ° is a unary operation on E0, such that:
This last clause implies along the way that if e is an element of E then there is a set a1, …, an of elements of E0 such that e = a1 … an. That is part, but only part, of what the Postulate demands, and would not by itself imply unique decomposability.
The involution operates solely on simple events. A particular imperative could have a simple event b to identify Success; in that case simple event b° will be identify its Failure.
The following definitions and remarks refer to such an event structure E.
Definition. e ≤ e’ if and only if there is an event f such that e’.f = e.
Analogy: a conjunction implies its conjuncts, and if A implies B then A is logically equivalent to (A & C) for some sentence C.
The definition is not the standard one, so we need to verify that it does give us a partial order, fitting with the meet operator.
[2] The relation ≤ is a partial ordering, and e.f is the glb of e and f.
That is, we have the familiar semilattice laws: e.g. if e ≤ e’ and f is any other event then f.e ≤ e’.
So <E, ., ≤ > is a meet semilattice. Note also that if a and b are simple events then a ≤ b only if a = b. For if b.f = a, the Postulate implies that b = f = a.
So far we have a relation of contrariety for simple events only. For events in general we need to define a general contrariness relationship.
Definition. Event e is contrary to event e’ if and only if there is an event a in E0 such that e ≤ a and e’ ≤ a° .
Contrariety is symmetric because a°° = a.
At this point we can see that the logic we are after will not be classical. For contrariety is not irreflexive.
That is because (a.a°) ≤ a and (a.a°) ≤ a°, so (a.a°) is contrary to itself. But (a.a°) is not the bottom of the semilattice. If a, a°, and b are distinct simple events then it is not the case that (a.a°) ≤ b. For if b.f = a.a° and f = a1 … an then the Postulate requires {b, a1, …, an} = {a, a°} so either b = a or b = a° .
It is tempting to get rid of this non-classical feature. Just reducing modulo some equivalence may erase the distinction between those impossible events, a.a° and b.b° . Such events can never occur anyway.
But there are two reasons not to do so. The first is that the history of deontic logic has run on puzzles and paradoxes that involve apparent self-contradictions. The second is more general. We don’t know what new puzzles may appear, whether about imperatives or related topics, but we hope to have resources to represent whatever puzzling situation we encounter. Erasing distinctions reduces our resources, and why should we do that?
More formally now, let us introduce a language, and call it LIMP. Its syntax is just the usual sentential logic syntax (atomic sentences, &, v, ┐). The atomic sentences will in a specific application include sentences in natural language, such as ‘”One does not eat with one’s fingers”. The interpretations will treat those sentences not as statements of fact but as encoding imperatives. In each case, the interpretation will supply what a context (such as a book of etiquette) supplies to set up the coding.
An interpretation of language LIMP in event structure E = <E, E0, ., ° > begins with a function f that assigns a specific event to each atomic sentence in each situation. Then there are two functions, < | and | >, which assign sets of truth-makers to each sentence:
Definition. A set X of events is downward closed iff for all e, e’ in E, if e ≤ e’ and e’ is in X then e is in X.
[3] For all sentences A, <A| and |A> are downward closed sets.
Now we can also show that our connector ┐, introduced to identify the weakest contrary to a given imperative, corresponds (as it should) to a definable operation on sets of events.
Definition. If X ⊆ E then X⊥ = {e: e is contrary to all elements of X}.
I will call X⊥ the contrast (or contrast class) of X.
Lemma. X⊥ is downward closed.
That is so even if X itself is not downward closed. For suppose that f is in X⊥. Then for all members e of X there is a simple event a such that f ≤ a and e ≤ a°. Thus for any event g, also g.f.e ≤ a while e ≤ a°. Therefore g.f is also in X⊥.
[4] For all sentences A, <┐A| = |A> = <A|⊥ and |┐A> = <A| = |A> ⊥ .
The proof depends De Morgan’s laws for downward closed sets of events:
Lemma. If X and Y are downward closed sets of events then
(X ∩ Y) ⊥ = X ⊥ ∪ Y ⊥ and (X ∪ Y) ⊥ = X ⊥ ∩ Y ⊥.
In view of [4], there is therefore an operator on closed sets of events that corresponds to negation of imperatives:
Definition. If A is any sentence then <A> ⊥ = (<A| ⊥ , |A> ⊥ ).
[5] <A> ⊥ = <┐A>
This follows at once from [4] by this definition of the ⊥ operator on imperatives.
We will concentrate here, not on the connections between sentences A, but on connections between their semantic values <A>. These are the imperatives, imperative propositions if you like, and they form an algebra.
Recall the definition of entailment for imperatives. It will be convenient to have a symbol for this relationship:
Definition. <A> ⇒ <B> exactly if <A| ⊆ <B| and |B> ⊆ |A>.
The following theorems introduce the logical principles that govern reasoning with imperatives.
[6] Entailment is transitive.
To have the remaining results in reader-friendly fashion, let’s just summarize them.
[7] – [11]
COMMENTS. In order for these results to make proper sense, each of the connectors ┐, &, v needs to correspond to an operator on imperatives, modeled as couples of downward closed sets of events. This was shown in the previous section.
The logic of imperatives is not quite classical. We can sum up the above as follows:
The logic of imperatives mirrors FDE (logic of first degree entailment); the imperatives form a De Morgan algebra, that is, a distributive lattice with De Morgan negation.
Definition. Imperative A entails imperative B exactly if <A| ⊆ <B| and |B> ⊆ |A>.
[1] <A & B> entails <A>, and <A> entails <A v B>.
For <A & B| = <A| ∩ <B| ⊆ <A| while |A > ⊆ |A|> ∪ |B> = |A & B>. Similarly for the dual.
Postulate: each event is a unique finite combination of simple events.
Definition. A event structure is a quadruple E = <E, E0, ., ° >, where E is a non-empty set, . is a binary operation on E, E0 is a non-empty subset of E, and ° is a unary operation on E0, such that:
[2] The relation ≤ is a partial ordering, and the meet e.f of e and f is the glb of e and f.
For e ≤ e because e.e = e (reflexive), and if e = e’.f and e’ = e”.g then e = e”.f.g (transitive).
(Perhaps clearer: For if e = e’.f then e.g = e’.f.g, so if e ≤ e’ then e.g ≤ e’, for all events g.)
Concerning the glb:
First, e.f ≤ e because there is an element g such that e.f .g = e. g, namely g = f.
Secondly suppose e’ ≤ e, and e’ ≤ f. Then there are g and h such that e.g = e’ and f.h = e’. In that case e’ = g.h.f.e, and therefore e’ ≤ e.f.
Definition. Event e is contrary to event e’ if and only if there is an event a in E0 such that e ≤ a and e’ ≤ a° .
Contrariness is symmetric because a°° = a. But it is not irreflexive for (a.a°) ≤ a and (a.a°) ≤ a°.
Lemma 1. If a and b are simple events then a ≤ b only if a = b.
That is because decomposition into simple events is unique. For suppose that a.f = b. Then there are simple events c1, …, ck such that f = c1….ck and a.f = a. c1, …, ck = b, which implies that a = c1 = … = ck = b.
Interpretation of the imperatives expressed in language LIMP, in event structure E = = <E, E0, ., ° >, relative to function f from atomic sentences to simple events. Then there are two functions, < | and | >, which assign sets of truth-makers to each sentence:
Definition. A set X of events is downward closed iff for all e, e’ in E, if e ≤ e’ and e’ is in X then e is in X.
[3] For all sentences A, <A| and |A> are downward closed sets.
Hypothesis of induction: this is so for all sentences of length less than A.
Cases.
A has the form (B & C) or (B & C). Here it follows from the fact that intersections and unions of downward closed sets are downward closed.
Definition. If X ⊆ E then X⊥ = {e: e is contrary to all elements of X}
Lemma 2. X⊥ is downward closed.
Suppose that e is in X⊥. Then for all e’ in X, there is a simple event a such that e ≤ a and e’ ≤ a . This implies for any event f, that f.e ≤ a and e’ ≤ a . Hence f.e is also in X⊥.
[4] For all sentences A, <┐A| = |A> = <A|⊥ and |┐A> = <A| = |A> ⊥ .
Hypothesis of induction: If B is a sentence of length less than A then <┐B| = |B> = <B|⊥ and |┐B> = <B| = |B> ⊥ .
Cases.
We prove first that <┐A| = |A> = <A| ⊥
<A| = <B| ∩ <C|, while <┐A| = |B & C> = |B> ∪ |C>. If e is in <┐A| then it is in |B> ∪ |C> so by hypothesis e is contrary either to all of <B| or to all of <C|, and hence to their intersection.
Suppose next that e is in <A|⊥ = (<B| ∩ <C|)⊥ . To prove that this is <┐A| = <┐(B & C)| = |B & C> = |B> ∪ |C> = <B|⊥ ∪ <C|⊥ it is required, and suffices, to prove the analogues to De Morgan’s Laws for downward closed sets. See Lemma below.
We prove secondly that |┐A> = <A| = |A> ⊥ . The argument is similar, with appeal to the same Lemma below.
(4) A has form (B v C). The argument is similar to case (3), with appeal to the same Lemma below.
Lemma 3. De Morgan’s Laws for event structures: If X and Y are downward closed sets of events then (X ∩ Y) ⊥ = X ⊥ ∪ Y ⊥ and (X ∪ Y) ⊥ = X ⊥ ∩ Y ⊥.
Suppose e is in X ⊥. Then e is contrary to all of X, hence to all of X ∩ Y, hence is in (X ∩ Y) ⊥. Similarly for e in Y ⊥. Therefore (X ⊥ ∪ Y ⊥ ) ⊆ (X ∩ Y) ⊥.
Suppose on the other hand that e is in (X ∩ Y) ⊥. Suppose additionally that e is not in X⊥. We need to prove that e is in Y ⊥.
Let e’ be in X and not contrary to e. Then if e’’ is any member of Y, it follows that e’.e’’ is in X ∩ Y, since X and Y are both downward closed. Therefore e is contrary to e’.e’’. We need to prove that e is contrary to e”.
Let b be a simple event such that e ≤ b and e’.e” ≤ b°. By our postulate, e’ and e’’ have a unique decomposition into finite meets of simple events. So let e’ = a1…ak and e’’= c1…cm, so that e’.e” = a1…ak.c1…cm. Since e’.e” ≤ b°, there is an event g such that a1…ak.c1…cm = e’.e’’= g.b°. The decomposition is unique, so b° is one of the simple events a1, …, ak, c1, …, cm. Since e is not contrary to e’, it follows that none of a1, …, ak is b°. Therefore, for some j in {1, ..,m}, cj = b°, and therefore there is an event h such that e” = h. b°, in other words, e” ≤ b°. Therefore e is contrary to e”.
So if e is not in X ⊥ then it is in Y ⊥, and hence in X ⊥ ∪ Y ⊥.
The argument for the dual equation is similar.
In view of the above, there is an operator on closed sets of events that corresponds to negation of imperatives:
Definition. If A is any sentence then <A> ⊥ = (<A| ⊥ , |A> ⊥ ).
[5] <A> ⊥ = <┐A>
(<A| ⊥ , |A> ⊥ ) = (<┐A|, |┐A>), in view of [4].
Definition. <A> ⇒ <B> exactly if <A| ⊆ <B| and |B> ⊆ |A>.
The following theorems introduce the logical principles that govern reasoning with imperatives.
[6] Entailment of imperatives is transitive.
Suppose <A> ⇒ <B> and <B> ⇒ <C>. Then <A| ⊆ <B| and <B| ⊆ <C|, hence <A| ⊆ <C|. Similarly, |C> ⊆|A>.
[7] <A & B> ⇒ <A>, and if <X> ⇒ <A> and <X> ⇒ <B> then <X> ⇒ <A & B>, Also <A> ⇒ <A v B>, and if <A> ⇒ <X> and <B> ⇒ <X> then <A v B> ⇒ <X>
First, <A| ∩ <B| ⊆ <A| and |A> ⊆ |A> ∪ |B>, hence <A & B> ⇒ <A>.
Secondly, suppose that X is such that <X| ⊆ <A| and <X| ⊆ <B| while |A> ⊆ |X> and |B> ⊆ |X>. Then <X| ⊆<A| ∩ <B| = <A& B| while |A & B> = |A> ∪ |B> ⊆ |X>. Hence <X> ⇒ <A & B>.
The dual result for disjunction by similar argument.
[8] Distribution: <A &(B v C)> ⇒ <(A & B) v (A & C)>.
<A &(B v C)| = <A| ∩ <B v C| = <A| ∩ (<B| ∪ <C|) = [<A| ∩ <B| ] ∪ [<A| ∩ <C|)] = <(A & B) v (A & C|. Similarly for the other part.
[9] Double Negation: <A> ⇒ < ┐ ┐ A> and < ┐ ┐ A> ⇒ <A>.
< ┐ ┐ A| = |┐ A> = <A| and |┐ ┐ A> = <┐ A| = |A>
[10] Involution. If <A> ⇒ <B> then <┐B> ⇒ <┐A>.
<┐B> ⇒ <┐A> exactly if <┐B| ⊆ <┐A|, i.e. |B> ⊆ |A>, and |┐A> ⊆ |┐B>, i.e. <A| ⊆ <B|. But that is exactly the case iff <A> ⇒ <B>
[11] De Morgan. < ┐ (A & B)> ⇒ < ┐A v ┐B> and vice versa, while < ┐ (A v B)> ⇒ < ┐A & ┐B> and vice versa.
< ┐ (A & B)| = |A & B> = |A> ∪ |B> = < ┐A| ∪ < ┐B| = < ┐A v ┐B|. Similarly for |┐(A & B> . Therefore < ┐ (A & B)> = < ┐A v ┐B>.
Similarly for the dual.
Curry, Haskell B. (1963) Foundations of Mathematical Logic. New York: McGraw-Hill.
Lokhorst, Gert-Jan C. (1999) “Ernst Mally’s Deontik”. Notre Dame Journal of Formal Logic 40 : 273-282.
Mally, Ernst (1926) Grundgesetze des Sollens: Elemente der Logik des Willens. Graz: Leuschner und Lubensky
Rescher, Nicholas (1966) The Logic of Commands. London: Routledge and Kegan Paul
[1] Rescher traces this analysis of ‘ought’ statements to Ernst Malley (1926) who coined the name Deontik for his ‘logic of willing’. Since the logic of imperatives we arrive at here is non-classical, note that Lokhorst (1999) argues that Mally’s system is best formalized in relevant logic.
[2] We can use Dirac’s names for them, “bra” and “ket”, with no reference to their original use.
This is a reflection on Kit Fine’ (2017) survey of truthmaker semantics. I will describe the basic set-up presented by Fine, aiming to clarify the logical relations between exact and inexact truthmakers.
1 Exact truthmakers p. 1
2 Sentences and truthmaking p. 2
3 Propositions and logic: an isomorphism p. 3
Exact truthmakers are entities of some sort (‘states’, ‘facts’, ‘events’) which combine in just one way (‘fusion’, I’ll use symbol +). Kit Fine gives as examples wind, rain, and their fusion wind and rain. That fusion is associative is assumed:
So we can just write e.g. wind and rain and snow without ambiguity. In mathematical terms, this property makes the exact truthmakers a semigroup. But it has two additional properties:
So the exact truthmakers form an idempotent, commutative semigroup. That structure is at the same time a semilattice by the following:
Definition. x ≤ y iff x + y = y and x ≥ y iff x + y = x
So wind is part of wind and rain, for the fusion of wind and rain is just wind and rain. That the exact truthmakers thus form a a semilattice means first of all that ≤ , and its converse ≥, are partial orders.
I find it convenient to focus attention on the converse order. For that will make it easier to see the relation between exact and inexact truthmakers, when we discuss that below.
In addition, a fusion is the least upper bound of its parts in the ≤ ordering (as Fine has it). That is the same as the greatest lower bound in the ≥ ordering:
By anti-symmetry it follows quickly that x + y is the unique element for which (7) and (8) hold in general, hence
Lemma 1. {z: z ≥ x + y} = {z: z ≥ x} ∩ {z : z ≥ y}
For by (7), if z ≥ s + t then z ≥ s and z ≥ t. And by (8), if z ≥ s and z ≥ t then z ≥ s + t.
Idempotent commutative semigroup and semilattice are really one and the same. For starting with the semilattice and defining x + y to be the least upper bound of x and y in the ≤ ordering, we arrive at properties (1)-(3) above for the defined notion.
I have followed Fine here in defining the ≤ order so that wind and rain looks in the mathematical representation like a disjunction rather than like a conjunction. So this semilattice <S, ≤ >is a join semilattice, while equivalently <S, ≥ > is a meet semilattice, in which the operator selects the greatest lower bound. (Ignore the visual connotation of symbol “≥” in the latter case.)
Kit Fine added to his definition of a state space that the family of exact truthmakers with the fusion operation forms a complete semilattice. That is, each set of exact truthmakers (and not just pairs or finite sets) has a least upper bound. That is audacious, and may introduce difficulties if probabilities are eventually introduced into this framework. So I will just stay with the finitary operation here.
I will make the definition redundant, so as to display the two equivalent ways of looking at this structure.
Definition. A state space is a triple S =(S, +, ≥), with S is a non-empy set (the exact truthmakers), + is a binary operator on S, and ≥ is a binary relation on S such that (S, +) is an idempotent commutative semigroup, (S, ≥) is a semilattice, and for each x, y in S, x + y is the greates lower bound of x and y in that semilattice.
Although this is more elaborate, to make some details explicit, this definition is equivalent to Fine’s definition of a state space (S, ≤) as a semilattice in which the fusion of x and y is the least upper bound of x and y in the ≤ ordering.
Let language L have sentence connectors &, v, to be read as ‘and’ and ‘or’. An interpretation of L in state space S is a function |.|+ assigns to each sentence A the set |A|+ (the exact truthmakers that verify A, that make A true), subject to the conditions
Kit Fine defines (page 565) the notion of an inexact truthmaker for sentence A so that e.g. wind and rain inexactly verifies “It is raining” just because rain is an exact truthmaker of that sentence. That is, if s exactly verifies A then s + t inexactly verifies A.
Definition. s inexactly verifies A if and only if there is some element t of |A|+ such that t ≤ s.
I will use the notation || . ||+ for the set of inexact truthmakers of a sentence:
|| A ||+ = {s: s ≥ t for some t in |A|+}
that is, the set of inexact truthmakers of A is the upward closure of its set of exact truthmakers in the ≤ ordering.
This notion is so far defined only in terms of the relation to language. But there is obviously a corresponding language-independent notion, and the two go well together. For each exact truthmaker engenders a specific set of inexact truthmakers:
Clearly u ≥ t exactly if 𝜙(u) ⊆ 𝜙(t), and so t is in |A|+ exactly if 𝜙(t) is part of || A ||+. In fact,
|| A ||+ = ∪{ 𝜙(t): t is in |A|+}
This is an upward closed set in the ≤ ordering.
What should we take to be the proposition expressed by sentence A, should we take it to be |A|+ or || A ||+ ? We can just say there are two, the ‘exact proposition’ and the ‘inexact proposition’.
Just how different are the families { |A|+: A a sentence of L} and {|| A ||+: A a sentence of L}? They are certainly different; for example be |A & B|+ is not in general part of |A|+, while || A & B ||+ ⊆ || A ||+. So as far as exact truthmakers are concerned, A & B does not entail A, while for the inexact truthmakers A & B does entail A.
But the relationship that we saw just above shows that from a structural point of view there is an underlying identity.
Lemma 2. For s, t in S, 𝜙(s) ∩ 𝜙(t) = 𝜙(s +t)
This is just Lemma 1 transcribed for function 𝜙.
Lemma 3. The system Φ(S) = < {𝜙(t): t in S} , ∩, ⊆ > is a (meet) semilattice.
The Lemma follows Lemma 2 and the informal discussion at the end of section 2. In fact, by our earlier definition, this system is also a state space.
Theorem. The function 𝜙 is an isomorphism between state space S = <S, +, ≥ > and state space Φ(S) = < {𝜙(t): t in S} , ∩, ⊆ >
So the inexact truthmakers provide a set-theoretic representation of the semilattice of exact truthmakers. The proof is standard textbook fare for semilattices.
Sketch of the proof.
To begin, the ordering in S is mirrored in the range of 𝜙 because ≥ is also a partial ordering:
This establishes an order isomorphism between S and the range of 𝜙. In addition:
𝜙(x + y) = {z: z ≥ x + y} = {z: z ≥ x and z ≥ y} = 𝜙(x) ∩ 𝜙(y)
by Lemma 2.
How much should we conclude from this?
Perhaps not much. As Fine shows, there are many distinctions to be made in terms of exact truthmakers that can be exploited in different ways, and might be obscured by turning to the inexact truthmakers alone. And many examples can be illuminated by focusing just on the exact truthmakers of atomic sentences, with propositions, in either sense, built up from there. In addition, we have only been discussing the most basic set-up, and there are many interesting complications when negation and modal operators are introduced, as Kit Fine shows.
But at the same time, it may help to reflect that the state space of exact truthmakers does have a ‘classical’ structure, with a set-theoretical representation.
Kit Fine (2017) “Truthmaker Semantics”. Pp. 556-577 in A Companion to the Philosophy of Language (eds B. Hale, C. Wright and A. Miller)
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