Tag: semantics

Modality and Negation before 1932

Bas van FraassenLeave a comment

Lewis and Langford’s Symbolic Logic (1932) was the culmination of work over the previous half-century.  This is a well-studied development, and I do not have anything new to say.  But I would like to add some comments on the role of negation.

1.      Exclusion, choice, and strong negation

When Gerrit Mannoury discussed Intuitionism he placed much weight on the distinction between exclusion negation and choice negation.  The latter is a form of negating a proposition that presupposes a definite contrast class, with a meaning that is something like “not this, but one of the others”. For example, “the apple is not ripe” would typically be understood as asserting that the apple is a real apple in one of the stages before ripeness.  The context would supply that definite contrast class.  If we read that statement with “not” as exclusion negation, the contrast would be totally indefinite: the apple is raw, or it is rotten. or it is a fake porcelain apple, or it is a painted apple in a Rembrandt still life, or ….  

That distinction shed some light on the Intuitionists’ approach to infinity.  But Intuitionist negation, as it appears in Heyting’s logic, does not fit either.  Within mathematics, as the Intuitionists present it, a statement can be negated only on the basis that it leads to a self-contradiction. In the logic, the negation of A is A → f, where f is the falsum, the absolute absurdity.

I will call this third form strong negation.  Outside Intuitionism, or at least outside its original presentation in terms of proofs and refutations, we can explain it this way:  

        to assert ¬A, in the sense of a strong negation, is to assert that A is not possibly true. 

Strong negation appears in modal logic generally, and specifically in Lewis’ early work.

However, the principles that govern strong negation are saliently different from the familiar ones characteristic of choice negation.

2.      Choice negation is ‘classical’

In lattice theory the complement x’ of element x is the largest element y such that x ∧ y = 0 (‘x and y are disjoint’), if it exists.[1]  That it is the largest means that if z is any element disjoint from x then z  ≤  x’.   Clearly, if we have several negations in the language they cannot all be like that.  

In view of the many options that have been broached on this subject, it may be best to mention some suggested characteristics in various treatments of negation, and to ask which are satisfied in specific cases.

I will call a negation classical if its corresponding operation ‘ on a lattice of propositions meets three conditions:

x ∧ x’ = 0           (disjointness)

(x’)’ = x                 (involution)

x ≤ y  iff  y’ ≤ x’    (antitone)

So when negation corresponds to an operation of that sort on a lattice of propositions, then negation obeys the principles

            (A & ¬A) implies the falsum                         (Non-Contradiction)

¬¬A entails, and is entailed by A                    (Double Negation)

A entails B if and only if ¬B entails ¬A.        (Contraposition)

Choice negation is classical.  For example, if the contrast class is a range of colors then “the apple is not-not red” is true iff the apple has a color in this range that is not one of the colors other than red, hence is red.  And, taking e.g scarlet to imply red, it is also the case that being other than red (in the intended range) implies being other than scarlet.

We may also note that if negation is defined by the two-valued truth-table, it is a choice negation, hence classical.

Is Exclusion negation classical?  I’ll discuss that in the Appendix.

Strong negation is not classical.  That the law of Double Negation does not hold in Intuitionistic logic is well known.  Contraposition is closely related to Reductio ad Absurdum of which one form (negation elimination) fails in Intuitionistic logic.  Below we’ll see how Contraposition fares in normal modal logic.  

3.      A quick note on Lewis’ relation to MacColl

Lewis’ theory of strong negation and the conditional is the same as MacColl’s, whose work he acknowledges.  (Lewis 1918, p. 292: “The fundamental ideas of the system are similar to those of MacColl’s Symbolic Logic and its Applications”.)

I will go chronologically backward for a moment to discuss Lewis first and then the earlier work by MacColl.

(I will adapt their notation, or use my own, as convenient.)

4.      C. I. Lewis chose strong negation as modal primitive

Lewis published his Survey of Symbolic Logic in 1918, and there presented his logic of strict implication with as modal primitive a strong negation connective ~.  

Notation:  I will now keep ¬ for just the truth-functional (material) negation. 

 The sentence ‘~p’ is to be read as ‘it is impossible that p’.  Then the strict conditional → is defined by:  

p → q =def ~(p & ¬q)

Lewis’ strong negation is intuitively at least the same as the Intuitionist’s.    For by the above definition,

            p → (q & ~q) = ~(p & ¬ (q & ¬q)) = ~p

given that logically equivalent formulas are mutually substitutable everywhere (and in normal modal logic the falsum is equivalent to any self-contradiction).

But Lewis was not guided by any precise semantics for modality,  and his vague intuitions led him astray.  Specifically he assumed that strong negation obeys Contraposition.  He had the, at first sight so innocent looking, principle

            2.21 (~q → ~p) → (p → q)

Two years later Lewis  published an emendation (Lewis 2020).  Emil Post had pointed out to him that this principle led to the theorem

            ~p = ¬ p

thus collapsing strong negation into material negation.  Lewis commented “Mr. Post’s example which demonstrates the falsity of 2.21 is not here reproduced, since it involves the use of a diagram and would require considerable explanation.” 

With our current preference for □ as primitive, we understand his 2.21 as

            2.21*.  □(¬◊q ⊃ ¬◊p) → □(p ⊃ q)

In the Appendix I will sketch a possible world model that gives us a counterexample to 2.21* [Hint: focus on the case of  ¬ ◊q being false.]

Dropping the ill-fated 2.21 did not do great damage to the logic of strict implication, which then eventually appeared in Lewis and Langford.

5.      Oskar Becker to C. I. Lewis:  nesting and iteration

But the theory of strict implication was not unchanged in other ways.  In 1930 there had appeared Oskar Becker’s monograph (Becker 1930) which discussed iterations and nestings of modal operators, with questions about how they are to be understood.  Lewis and Langford acknowledged this in their preface and wrote their famous Appendix II in response.  

Becker had proposed the following principles as possible additions to Lewis’ logic:

C10. □□p = □p

C11.  ◊p → □◊p

C12.  p → □◊p  (which Becker called the Brouwersche Axiom)

Lewis and Langford then defined five modal logics, S1-S5, in which (our familiar) S4 is S3 plus C10, and S5 is (our familiar) S4 plus C11 and C12.

But Becker’s questions about how C10.-C12. are to be understood were not very well answered.  We’ll see below how MacColl made a determined effort to establish the truth conditions for one case of an iterated modality, one that involves negation.

6.      Tracing it all back to Hugh MacColl

From a distance MacColl’s symbolism looks rather like arithmetic.  He introduces five modalities: true, false, certain, impossible, and variable.  Each has a Greek letter as its symbol, with ε (epsilon) for certainty  and η(eta) for impossibility, for example.  

But there seems at first to be a curious ambiguity.  The proposition that A is certain or necessary is symbolized Aε.  But then we also see ε entering as a propositional constant, for the absolutely certain, the ‘top’ or ‘unit’ of the algebra.

I think we can understand that this is not just a notational quirk.  In arithmetic, consider the sentences “3 + 2 = 5” and “32 = 9”.  The numeral “2” appears first as the name of a number, and then, in superscript, as the name of the squaring function.  We can give a good account of this: the symbol “32” the denotes the value of a certain function (exponentiation) applied to the ordered couple <3, 2>.  Superscripting is a convention to symbolize exponentiation.

modal statements, strong negation

In the same way we should understand MacColl’s formulation of modal statements.  In his commentary and examples they are clearly taken as de dicto: the assertion that it is certain that A is the assertion that A has a certain modal property, namely certainty.  The symbol “Aε” denotes the proposition which is the value of a certain function (we may call it propositional exponentiation) applied to the ordered couple <A, ε>.  The function is symbolized by the superscripting convention.

In the same way the assertion that A is impossible is symbolized Aη.  This is strong negation, it is what Lewis later symbolized as ~A.

propositional constants

But impossibility too is itself a proposition, the ‘bottom’ of the algebra, and we have the laws (MacColl uses ‘ for ordinary negation and . for conjunction):

A . ε = A                (ε is the unit, the top, the tautology)

A . η = η                (η is the bottom, the absurdity)

(A . A’)η                 (Law of Non-Contradiction)

Aε . (A’)ε = η

6.3 the strict conditional

MacColl defines a conditional:

            A : B =def  (A . B’)η   (It is impossible that A and not B)

That is exactly the definition that Lewis then chose for his strict conditional.  MacColl, who has the connective + for disjunction, points out that, equally, A : B = (A’ + B)ε, it is certain that either not-A or B.  

6.4 iterated and nested modal operators

Especially interesting is iterated propositional exponentiation, to symbolize nested modalities, a subject which, as we saw above, did not return in the literature till 1930.  MacColl writes (section 9, page 7):

The symbol ABC means (AB)C; it asserts that the statement AB belongs to the class C, in which C may denote true, or false, or possible, &c. Similarly ABCD means (ABC)D, and so on.  (MacColl 1906: 7)

So for example, (Aε)η is the statement that it is impossible that A is certain: what we would write as ~◊□A.

MacColl returns to this in section 22, where he writes “But, it may be asked, what is meant by statements of the second, third, &c., degrees, when the primary subject is itself a statement?”   What follows in that section, and in many other pasages, points to an understanding of logic as pertaining to information processing.  There is the suggestion that, for example, Aη may be a revision if A, made when new data arrive that are incompatible with A.  But I will leave that aside for now.

There is an interesting discussion of a nested modality in a later paper (MacColl 1910).  Here he finds what he takes at first blush to be an antinomy.  In his system the symbol θ stands for the modality variable, it is his term for what is possible but not certain, or equivalently, what is neither impossible nor certain.  So it could be defined:

Aθ = (Aε)’.(Aη)’ 

MacColl then writes:

The symbol Aθθ, in my system, is short for  (Aθ)θ and asserts that the statement Aθ is a variable.  The antinomy consists in the conflict of two arguments, of which the one professes to prove that the second-degree proposition Aθθ is an impossibility or self-contradiction; while the other professes to prove that it is not. (MacColl 1910: 196, with the initial “Aθ” corrected to “Aθθ”)

A first blush its natural reading looks entirely intelligible, or at least perfectly grammatical:

“It is neither certain nor impossible that it is neither certain nor impossible that A”.

But it becomes puzzling when we try to see under what conditions, or for what sort of statement A, this would be true or definitely false.

To prove the first option in the antinomy MacColm argues, first of all:

 A will itself be either certain, or impossible, or neither certain nor impossible.  If A is either certain or impossible then Aθ is clearly false.  

That seems correct, but then MacColl argues:

if A is neither certain nor impossible then it is certain (and hence not neither certain nor impossible) that Aθ.  And so again, Aθθ is false.  

Therefore Aθθ is false under all conditions, hence impossible. 

The reasoning about the case in which A is itself neither certain nor impossible, is not obviously correct.  The inference appears to rely on an unstated modal principle, perhaps one as strong as the S5 principle, that statements asserting or denying certainty or possibility are certain if true.  With such an assumption added, the first horn of the dilemma is established.

Accordingly, MacColl’s contrary argument, that Aθθ is possible, is not needed if we can push back to a prior question:  

whether, or under what conditions, Aθ is certain if true. 

The most reasonable attitude to take would seem to be that there are many modes of modality, pertaining to different subject matters – in some cases S5 is the correct logic and in other cases not.

7.      APPENDIX. 

[1] Is Exclusion negation classical?

Suppose we assert that the apple is not red, while intending no definite contrast whatsoever.  One option may be to assert that nevertheless, all the ways the apple might or could be, ways that do not involve its being red, form a class.  Then, just as for choice negation it will follow that not being in any of those ways implies being red.  

This ostensibly simple view may have difficulties, with vagueness and the vagueness of vagueness, or with Russel-type paradoxes, or with more general questions about whether the class could or could not be a set.  

If we are uneasy with the idea of reifying ways things might or could be as a definite class, then we will certainly begin to doubt inference by Reductio ad Absurdum (as Intuitionists do), and almost certainly Contraposition and Double Negation as well, whenever the negation in play is exclusion negation.  

In a many-valued logic that defines the connectives in terms of a ‘many-truthvalue-table’, it is possible to have a negation that is not classical.  For example, if ¬1 = 0, ¬2 = 1, ¬0 = 1 then ¬¬2 ≠ 2.  We could reckon this as an exclusion negation (‘value other than the designated value’), though a simple case.  In a language with truth-value gaps, where conjunction and disjunction are typically not functional (not compositional), it may also be possible to have a non-functional negation.

It is perhaps a failing in formal semantics to take negation (in our language in use) for granted as understood and unequivocal.  

[2] Counterexample to Lewis 2.21:  (~q → ~p) → (p → q) in normal modal logic

In our preferred notation, and assuming the Duality □¬ = ¬◊, we can write 2.21 as

*. □{□(¬ ◊q ⊃ ¬ ◊p)  ⊃ □(p ⊃ q) }

Recall that in a normal modal logic possible world model M = <W, R>, the sentence □A is true at world w in W iff A is true in all worlds in R(w).  

I will sketch a model, describing only just enough to show that it provides a counterexample to *.

The worlds w1, w2, w3, w4 in W are related as follows:

  1. w2 is in R(w1)
  2. w3 and w4 are in R(w2)
  3. For any world w in W, if w is in R(w2) then R(w) = R(w2)
  4. p and ¬q are true in w3
  5. p and q are true in w4

Argument

  1. q is true in w4
  2. ◊q is true in all members of R(w2)                             
  3. (¬ ◊q ⊃ ¬ ◊p) is true in all members of R(w2)
  4. □(¬ ◊q ⊃ ¬ ◊p)  is true in w2
  5. (p & ¬q) is true in w3
  6. □(p ⊃ q) is false in w2                                   since w3 is in R(w2)
  7. □(¬ ◊q ⊃ ¬ ◊p)  ⊃ □(p ⊃ q) is false in w2,                by 4. and 6.
  8. □{□(¬ ◊q ⊃ ¬ ◊p)  ⊃ □(p ⊃ q) } is false in w1          since w2 is in R(w1)

Note that, for simplicity, I made clause c. stronger than it needs to be to justify line 2.  Apart from that I have left <W, R> with as little constraint as possible.

 

8.      REFERENCES 

Becker, Oskar (1930)  “Zur Logik der Modalitäten”.  Jahrbuch  für Philosophie und Phänomenologische ForschungXI: 497-548.

Gabbay, Dov M. and John Woods (2006)  Handbook of the History of Logic. Vol. 7: Logic and the Modalities in the Twentieth Century.  Amsterdam: Elsevier.

Lewis, Clarence I. and C. H. Langford (1932) Symbolic Logic. New York: The Century Company. 

Lewis, Clarence Irving (1920) “Strict implication – an emendation”. The Journal of Philosophy, Psychology, and Scientific Methods 17: 300-302. 

MacColl, Hugh  (1906) Symbolic Logic and ‘its Applications. London: Longmans, Green and Co.

MacColl, Hugh (1910) “Linguistic misunderstandings (I)”. Mind, n.s., 19: 186–199. 

Read, Stephen (1998) “Hugh MacColl and the algebra of strict implication”. Nordic Journal of Philosophical Logic 3: 59-84.

Wolenski, Jan (1998) “MacColl on Modalities”. Nordic Journal of Philosophical Logic 3:133-140.

[1] In a Boolean algebra it exists, is unique, and x ∧ x’ = 0, x v x’ = 1.  

Wilfrid Sellars and the overabundance of things

Bas van FraassenLeave a comment

It was in Sellars’ seminars that I came to sense the enormous, mysterious, gap between names and statements. For Sellars, as I remember him, this is the first thing we need to understand about language. For me, there is a mystery: I was puzzled then, and it is something that I still find very puzzling.

This post is not mainly about Sellars, one of my teachers and a fascinating person, but about what I find so puzzling.

Sellars’ guiding text here was a passage in Wittgenstein’s Tractatus (4.1432), “We must not say: ‘The complex sign ‘aRb’ says ‘a stands in relation R to b‘”. We must say instead “That ‘a’ stands in a certain relation to ‘b’ says that aRb.”

There are complex names, and to understand them we have to take into account what their parts are and how they are related to each other. We can say the very same thing about statements. But there must be a glaring difference beyond that similarity, for a statement is most certainly not a complex name. That glaring difference is there even if in English, or any natural language with its own peculiarities, we find ourselves looking at the very same overt expression.

Take “The deer run”. This can be used as a complex name, to denote something, similar to the expression “the dog run”, as in “The deer run, where we can admire their graceful prancing, is just at the edge of the forest”. Or it can be used as a sentence, as in “The deer run swiftly along the edge of the forest”. In both cases we have to understand their grammatical parts and to appreciate the relation between them, that is, their auditory or visual configuration. But what we do in this appreciation, or what it is that is appreciated, must be very different. For in the first use, the expression denotes, it designates something, it is a complex name. In the second use it asserts, it says that something is the case, it is a statement.

How the difference disappears

That illustration is striking precisely because it forces us to notice a difference between two identical appearances. In a fully disambiguated language, a logician’s dream, I could not give it. But perhaps that is exactly why in formal semantics, and more generally in some examples of mainstream philosophy of language, where the disambiguation is presupposed, the difference appears to disappears.

In formal semantics, for the usual sort of language studied in logic courses, a name denotes a thing, an element in a domain of discourse. Or in intensional logic, it denotes a function from worlds into elements of a domain or domains. In both cases, the ‘semantic value’ is a thing, whether concrete or abstract. Just right!

But a statement is something assertable, which a name is not. A predicate is something applicable to a thing, which a name is not. So why is it that in the formal semantics we teach and use, their semantic values are things as well?

For Carnap in Introduction to Semantics the denial at is explicit: he simply asks what a statement designates, and says that it is a proposition. A proposition is a thing. An abstract thing, but a thing nevertheless.

In more detailed accounts the semantic value of a statement — which Carnap said is a proposition — is a truth-value, or a function (from worlds into truth-values), or a set (of worlds). These are all things, abstract things. The semantic value of a predicate is a set (subset of a domain of discourse) or a function (from worlds into subsets of a domain or domains). These too are things, abstract things.

Thus in semantic analysis of language everything is reduced to things and relations among things. Things are not assertible, nor applicable. What has happened to real statements and predicates in use, after we start mentioning them and talking about them?

Retrenchment, possibly

The clue may be in my preceding sentence: once we mention, rather than use, a piece of language, we have literally reified it. Or replaced it by a simulacrum: rather than talk about the English statement that snow is white, we talk about the expression “Snow is white”, a sequence of concatenated letters, thus, a thing.

Peter Strawson insisted on distinguishing statements from sentences. Above I was careful to us “statement” throughout (although sometimes it felt more natural to use “sentence”). It seems to me that we are forced to go along with Strawson. Let us use “sentence” for an expression belonging to a certain syntactic category, and identified as an ordered concatenation of letters.

And then we can say that the sentence is a stand-in for the English statement. Sentences are things we use to represent statements. Thus, a sentence is not what we naturally, as speakers and hearers, take a statement to be. Instead, this stand-in, the sentence, is a complex name of a thing, more generally of some relational structure (thus, an abstract thing), namely its semantic value. What is the interest of this representational procedure? In formal semantics it is that they allow us to catalogue certain inference patterns in natural language.

Fine. We have worked enough in that vineyard that we’d have to feel acutely embarrassed to deny its value. But …

Is there any way for us, in philosophy of language, to get behind this representational procedure, and talk about the English statements and predicates themselves, directly, and not by proxy, not only by talking about the things which stand in proxy for them?

(Threat of paradox? To explain that proxy relation it seems we would have to be able to talk about the statements as well as about the sentences. So then we will find ourselves in quite a predicament if we can only talk about statements by talking about the sentences that are their proxies!)

This I find all very puzzling.

Appendix

For Sellars’ discussion see his “Naming and Saying”, in Philosophy of Science 1962, reprinted as Chapter 7 of his Science, Perception, and Reality. For a quick discussion and critique of this point in Carnap’s Introduction to Semantics see the review by Alonzo Church, The Philosophical Review 52 (1943): 298-304.