Tag: truth maker semantics

Complementation

Each atomic fact has a complement; above I called it its evil twin. The fact that his words were apropos has as its complement the fact that his words were malapropos, and conversely.

Since we are treating the atomic facts as unanalyzed wholes, we cannot define this relationship, but introduce it by means of an operator and a postulate:

Postulate: every atomic fact has a unique complement which is also an atomic fact, and is the complement of its own complement.

The operator ^ maps an atomic fact into its complement:

b^^ = b (idempotent)

Complex facts do not have complements, for there are no disjunctive facts. There is no direct route to defining complementation on propositions. But there is an indirect way.

A direct way would be this: the propositions, which are the closed sets of facts, form (as we found above) a complete distributive lattice — now take on the task of defining a complementation operation on this lattice!

The indirect way that I shall follow is this: I will identify sub-lattices on which there is a well-defined complementation operation, and these will suffice for the semantic analysis of the logic of tautological entailment.

In fact, the clue to this is precisely that the propositions of interest are those expressed by the sentences of a language. It is these that should form (to use the term introduced by Nuel Belnap) an intentionally complemented distributive lattice.

The point about sentences is that they are built up systematically from atomic sentences, in each case from a finite number of atomic sentences. An interpretation of the language which assigns propositions to sentences follows this structure, and so produces a countably generated sub-lattice of the right sort.

Prior to the assignment of propositions to sentences I will associate with each sentence A two bases, its T-base T(A) and its F-base F(A), which are the sets of primary facts that make A true, respectively false.

For each atomic sentence p there is an associated fact e, and T(p) = {e}, F(p) = {e^}. Building on that assignment of atomic facts to the atomic sentences, the definition of the bases follows the recursive definition of the set of sentences of a sentential logic. For all sentences A, B:

• T(~A) = F(A); F(~A) = T(A)
• T(A & B) = T(A) . T(B); F(A & B) = F(A) ∪ F(B)
• T(A v B) = T(A) ∪ T(B); F(A v B) = F(A) . F(B)

where the product X.Y = {e.f: e is in X and f is in Y}.

Next we use this to associate a proposition T*(A) with each sentence A, defining it as the closure of the T-base:

T*(A) = [T(A)].

The theorems about propositions in the previous post apply, so that the meet and join are just intersection and union.

T*(A & B) = T*(A) ∩ T*(B), and T*(A v B) = T*(A) ∪ T*(B)

The following two lemmas show that there exists a complementation operation on these propositions.

Lemma. T*(~~A) = T*(A), F(~~A) = F(A)

Proof. T(~~A) = F(~A) = T(A); and since T(~~A) = T(A) their closures are also the same; similarly for the second part.

Lemma. If T*(A) ⊆ T*(B) then T*(~B) ⊆ T*(~A)

Proof: Suppose that T*(A) ⊆ T*(B). Then all facts subordinate to facts in T(A) are subordinate to facts that are subordinate to facts in T(B). But subordination of facts is transitive, so all facts subordinate to facts in T(A) are subordinate to facts in T(B).

So T*(B) includes both T(A) and T(B), hence also T(A) ∪ T(B), that is, T(Av B). So T*(B) = T*(A v B), since all facts subordinate to facts in T(A) are already included among the ones subordinate to those in T(B).

Hence T*(~B) = T*(~(A v B)) = [F(A vB)] = [F(A).F(B)]. This includes all facts subordinate to facts in F(A), because if e is thus, so is e.f for any fact f.

Hence T*(~B) ⊆ [F(A)] = [T(~A)] = T*(~A).

Definition. The operation ^ is defined on the family of propositions {T*(A): A is a sentence}, and Y = X^ if and only if there is a sentence A such that X = T*(A) and Y = T*(~A).

Theorem. The family {T*(A): A is a sentence} form a countably generated sublattice L+ of the lattice of propositions, which is distributive and intensionally complemented, that is, there is an operation ^ such that for all propositions X in L+, X^^ = A (idempotent) and if X ⊆ Y then Y^ ⊆ X^ (order inverting).

Since the family L+ is closed under both intersection and union it is a lattice, and the proof of distributivity is the same as in the preceding post. But it can only be a sublattice, since each element is a proposition generated by a finite base.

The above lemmas show that the operation ^ is idempotent and order inverting.

The logic of tautological entailment is sound and complete for countably generated intensionally complemented distributive lattices (Nuel Belnap 1967). This guarantees then that this logic is sound under the present interpretation.

The claim of completeness for this logic under our interpretation does not follow in the same way. What is peculiar about the sublattice L+ is that its set of atoms (the propositions which are closest to the zero element) is closed under complementation: [{e}]^ = [{e^}]. That is not typical: for example, if a Boolean lattice has three atoms, the complement of one of them is the join of the other two.

Nevertheless our lattices of propositions suffice for the logic of tautological entailment. We need to take the bull by the horns to prove completeness! I will put that in the Appendix, it is the more technical part.

NOTE. That completeness proof is in my 1969 article “Facts and Tautological Entailment”. What is new in these posts, and continued in the Appendix, is to present this semantic analysis in the form that aligns it with the lattice-theoretic approach which Nuel Belnap introduced. At the same time, by doing this, I can show how it fits into the general framework that I presented in the posts called “An Oblique Look at Propositions”.

APPENDIX

Recall from the first post on Tautological Entailment its basic principle (where an atom is either an atomic sentence or the negation of an atomic sentence):

If A is a conjunction of atoms, and B a disjunction of atoms, then A tautologically entails B if and only if at least one conjunct in A is a disjunct in B.

Definition. A primitive entailment (A, hence B) is an argument whose single premise A is a conjunction of atoms and its conclusion B is a disjunction of atoms.

Theorem. A primitive entailment (A, hence B) is valid if and only if A tautologically entails B.

This means that A =(p(1) & … & p(k)) while B = (q(1) v … v q(m)), with each p(i) and q(j) an atom, and hence each has as its T-base a set containing a single atomic fact. The T-base of A is then a single molecular fact e and the T-base of B is a set containing m atomic facts f(1), … ., f(m). For T*(A) to be included in T*(B) requires that at least one component of e is identical with one of the facts f(j). And this requires that at least one conjunct in A is a disjunct in B.

So what about all the other arguments, that don’t have this simple form? In the full exposition of the logic of tautological entailment by Anderson and Belnap, a number of principles are added, which together guarantee that to check any argument will consist in checking a family of primitive entailments. (“conversion into normal form”)

These principles include the basic lattice laws as well as distribution, which we have already dealt with along the way. In addition of course they include the principles of double negation and De Morgan’s Law, which correspond to the idempotency an order inversion of intensional negation. The only thing that we still have to address then, is De Morgan’s Law, which allows the replacement of ~ (A & B) by (~A v ~ B)) and ~(A v B) by (~A & ~B).

Lemma. T(~(A & B)) = T(~A v ~B); T(~(A vB) = T(~A & ~B)

Proof. T(~(A & B)) = F(A & B) = F(A) ∪ F(B) = F(~~A) ∪F(~~B) = T(~A) ∪ T(~B) = T(A v B). Argument for the second part is similar.

In the preceding post I outlined, intuitively, what Anderson and Belnap’s logic of tautological entailment is, and sketched an accompanying semantics framed in a rudimentary theory of facts as truth-makers. Here I will make this explicit to the extent of governing conjunction and disjunction. We’ll see that the lattice of propositions is rather a simple one, but also that it has the additional property of distributivity, corresponding to the classical logical law of Distribution.

I said that there are atomic facts and complex facts, the latter being something like bundles of atomic facts. There is no order in these bundles, so you can think of them as sets or as mereological sums. But nothing hinges on that, so for generality I will just indicate how facts combine to make more facts.

A model has as first ingredient a set of facts, and I specify that the facts are countable, there are no more than there are natural numbers. There is, as second ingredient, an operation (combination) on facts to produce other facts, symbolized by a dot. So if b, c are facts, then so is b.c. (See Appendix for some clarifying remarks.)

Postulate: every fact is a combination of atomic facts; and facts are identical if and only if they are combinations of the same atomic facts.

Note that this Postulate does not imply that a combination must be finite; there is even a fact that is the combination of all atomic facts. Because there is no significant ordering, if A is a set of facts, we can legitimately refer to the combination of all A’s members, and denote it ΠA.

As third ingredient there is a relation ≤ among facts, subordination, that we can define:

Fact b is subordinate to fact c ( b ≤ c) if and only if there is a fact d such that b = d.c

By this definition, it follows that b ≤ b, as well as that ≤ is transitive, and in view of the Postulate, if b ≤ c and c ≤ b then b=c..

Fourth ingredient: The propositions are the sets of facts that are closed under subordination. Finally, fifth, the relation of making true, e » Q, holds just iff fact e is a member of proposition Q. So, if e makes proposition Q true, then so does every fact that is subordinate to e.

Being the closed sets of facts, as we know from earlier discussion, the propositions form a complete lattice L, partially ordered by set inclusion. The meet of Q and R is their intersection, and their join is the closure of their union. But this lattice construction is simpler than some that we saw before, and so has some special properties.

Distributivity

Theorem 1. If Q, R are propositions then fact g is in Q ∩ R if and only if there are facts e and f such that g = e.f, where e is in Q and f is in R.

In other words, Q ∩ R = {e.f : e is in Q and f is in R}.

Proof. It is clear that if e is in Q then e.f is as well, because e.f ≤ e, and similarly that e.f is in R if f is in R. So if e is in Q and f in R then e.f is in Q ∩ R.

Conversely, suppose that g = (e.f) is in Q ∩ R. Then g is in Q and also in R. But then we can recast that as: there are facts h, k such that g = h.k with h in Q and k in R, namely, g = h = k = (e.f), recalling that combination is idempotent.

Theorem 2. If Q and R are propositions then [Q ∪ R]= Q ∪ R

For suppose that e ≤ f for some member of Q ∪ R. If f is in Q, then e is also in Q, for it is subordinate to f. And similarly if f is in R. So all the facts subordinate to any member of Q or R is already in Q ∪ R. Since ≤ is transitive, this applies also to any facts that are subordinate to facts that are subordinate to … subordinate to facts in Q ∪ R. Hence all of the members of [Q ∪ R] are already in Q ∪ R.

So we do not need a special symbol for the join operation on propositions, it is just union. Note well though that this does not go for set of facts in general; a set is not typically its own closure! Note also that this holds because there is here no subordination relation of facts to sets of facts that is more complicated than their subordination to specific members of those sets.

Theorem 3. L is distributive: if Q, R, S are propositions then Q ∩ (R ∪ S) = (Q ∩ R) ∪(Q ∩ S).

That is obvious now, given that we are dealing with just set union and set intersection, for propositions. (But given how fact-combination works, it is as well to spell this out properly: see the Appendix.)

The next, final post in this series, will take up negation, and complete the semantics for tautological entailment.

APPENDIX

So that order is automatically absent, combination has the properties:

b.b = b (idempotent)

b.c = c.b (commutative)

b.(c.d) = (b.c).d (associative)

Call b an atomic fact exactly if for all facts c, b = b.c implies that b = c. (Note that, in view of the postulate, b = c is the case if and only if there is a set A of facts such that both b and c are the fact ΠA.

Long, perhaps more insightful, proof of Theoremm 3: Q ∩ (R ∪ S) = (Q ∩ R) ∪(Q ∩ S).

By theorem 1, fact g is in Q ∩ (R ∪ S) exactly if there are facts e, f such that g = e.f while e is in Q and f is in (R ∪ S), that is to say, in R or in S. So g = e.f with e in Q and f in R or e in Q and f in S, thus g is either in Q ∩ R or in Q ∩ S), that is, g is in (Q ∩ R) ∪ (Q ∩S).

Conversely, if g is in (Q ∩ R) ∪(Q ∩ S) then g is in either (Q ∩ R) or (Q ∩ S). The two cases are similar, so suppose the former. In that case g = e.f with e in Q and f in R, for some facts e, f. This g = e.f for some fact e in Q and f in (R ∪ S), and therefore g is in Q ∩ ( R ∪ S).