What exactly is classical sentential logic? (2)

Prior to 1934 a logic would be presented as a set of axioms and rules, together with the definition of a derivation or deduction as a finite sequence of statements constructed in accordance with the rules.

The Original Classical Sentential Logic (OCSL) was a special case, which could do with just one rule, Modus Ponens, and whose axioms were easily interpreted as representing bi-valent truth-functions.

Alternatives to the axioms are possible of course; for example Intuitionistic Logic would reject some of the classical ones. But if we stay in a classical mood, we should think about such logics which at least contain OCSL, not rejecting either the axioms or the rule.

Extending the logic to a specific field of study

If we study a particular subject, some principles that do not belong to pure logic will be natural additions. To begin we can add special axioms, like “All men are mortal”, or “The whale is a mammal”. These logically contingent statements will then be allowed entry into the derivations, without requiring justification. Reasoning with extra axioms is not so different from reasoning from extra premises. The only difference would be that the additions would be fixed, as background knowledge, so to speak.

But what if, to characterize the subject matter, new rules are added? The effect is drastic. For if that is done the admissibility of such meta-rules as Conditional Proof, Disjunctive Syllogism, and Reduction Ad Absurdum is newly put into question. Either new proofs are to be supplied for their admissibility, or the natural deduction formulation of classical logic (NDFSL) is not sound.

To make this concrete, I want to recall two topics from the 1970s where this becomes real.

Example 1: Existence and reference failure

The first is Peter Strawson’s response to Russell’s theory of definite descriptions. The question “Is the king of France bald?” has two direct answers, yes and no. But the king does not exist (at that time France did not have a king), so Strawson insisted that “the question does not arise”. Neither answer is true. Questions have presuppositions, and so Strawson proposed that each of the sentences Bk (“The king of France is bald”) and ~Bk (“The king of France is not bald”) presupposes that E!k (“The king of France exists”).

Intuitively: When a statement’s presuppositions are not all true, that statement itself is neither true nor false.

At first blush it may seem that Strawson was rejecting classical logic. But he did not mean to do so, he just wanted to add this logical notion of presupposition. It can’t be done by adding axioms, using the conditional, for in OCSL the conjunction of (Bk ⊃ E!k) and (~Bk ⊃ E!k) implies E!k. So if those two conditionals were added as axioms then E!k would be a tautology. That would certainly take the wind out of Strawson’s sails.

What can be added instead are the two rules, to say that E!k can be inferred from Bk as well as from ~Bk:

Bk ├ E!k ~Bk ├ E!k

Remember that for a rule to be admissible means that adding it will not result in additions to the body of theorems or body of deductive inferences. We can see at once that the meta-rule of Conditionalization cannot be admissible — for that would produce those two conditionals again, and lead to the new theorem├E!k.

We can quickly note that there is another meta-rule, that we have not mentioned yet, that must also become inadmissible. OCSL has the theorem, generally referred to as the principle of contraposition

├ (A ⊃ B) ⊃ (~B ⊃ ~A)

But the meta-rule of Contraposition, which says that if A ├ B then ~B ├ ~A will be inadmissible for the same reason as Conditionalization. For it could be used it to argue:

1] Bk ├ E!k

2] ~Bk ├ E!k

3] ~Ek ├ ~Bk from 1 by Contraposition

4] ~Ek ├ E!k from 2 and 3 by transitivity of ├

5] ├ E!k from 4

There is no way to block the move from 2 and 3 to 4 because ├ is a defined relation, based on the definition of what counts as a derivation (see Appendix). It may be possible to block the move from 4 to 5; that would involve rejecting another meta-rule, Reductio ad Absurdum. So at least one of these two meta-rules will no longer be admissible.

Example 2. The ins and outs of Truth

The second illustration takes off from the first one, reflecting on the fact that there is no longer a simple reading of the logic in terms of truth-functions: Bk was said to be neither true nor false if E!k is not true. This suggests that we should look carefully at the theory of truth, familiarly announced with the slogan, from Tarski,

“Snow is white” is true if and only if snow is white.

That is surely correct, if properly understood, but how is it to be understood? The two sentences

“Snow is white” is true

snow is white.

surely cannot have different truth-values. The inference of either from the other must be valid.

But if we are to have a theory of truth that allows for some statements being neither true nor false, then “It is true that A or it is true that ~A” cannot be a tautology.

However, if we read the “if and only if” as the usual ⊃ conditional going both ways, classical logic will lead to the conclusion that every sentence must be true or false, it cannot cannot be neither. Suppose we write TA for the statement that statement A is true. Then the deduction would go like this:

1] A ⊃ TA

2] ~TA ⊃ ~A

3] ~A ⊃ T(~A)

4] ~TA ⊃ T(~A)

5] TA v T(~A)

Here 1 and 3 come from the reading of Tarski’s principle in terms of the conditional connective, and the rest follows in OCSL. But you will notice that the move from 1 to 2 was by contraposition (in its form that just requires the theorem and Modus Ponens).

If we only accept the rules

A ├ TA TA ├ A

then it is possible for line 5] not to follow because the meta-rule of Contraposition is not admissible when those rules are added to OCSL.

So what are we to think?

The main conclusion is clear: if OCSL is augmented with additional rules, then the proofs of admissibility for our familiar meta-rules, our natural deduction rules, do not go through. And if the logic is not to be trivialized, those meta-rules must in fact not be admissible.

The main open question is clear too: how are we to understand the language if the usual reading in terms of truth-values and truth-functions does not work any more? How can we understand this without giving up on classical logic altogether?

That is the question for the next post.

APPENDIX: about logics in general

The original (early 20th century) concept of a logic was that it consists of a set of axioms and rules, together with the definition of a derivation or deduction as a finite sequence of statements constructed in accordance with the rules. Specifically, the definition of deductive inference is this:

Definition. If X is a set of sentences and B a sentence then X ├ B exactly if there is a finite sequence of sentences such that each of its members is either an axiom, or a member or X, or follows from preceding members of the sequence by one of the rules.

Note well that “rules” refers here to basic inferences, and does not include meta-rules which allow for “sub-derivations”.

Given this, what is the relation ├ like? It follows from the Definition that, for any such logic whatever, it has the following characteristics (often referred to as Structural Rules, they are meta-rules but not ones that can possibly become inadmissible):

• if B is a member of X then X├ B
• if X├ E for every member E of set Y, and Y├ B then X├B
• (Corollary) if Y is part of X and Y├ B then X├ B

Since the axioms can be defined as the sentences which can be deduced from any set of sentences whatsoever, even the empty set, it is the deductive inference relation ├ which encodes all there is to the logic.

Tarski introduced another way to present that relation, using a notion familiar in topology. We say that a set X is closed under a relation R when everything related by R to members of X is already in X. And then the closure of X, under that relation, is the set {B: there are members of X which bear relation R to B}.

The deductive inference relation ├ gives rise then to the notion of deductive closure, as a consequence operation:

Cn(X) = {B: X├ B}

and the basic characteristics of ├ correspond to the basic characteristics of Cn:

• X is part of Cn(X)
• Cn(Cn(X)) is part of Cn(X)
• if X is part of Y then Cn(X) is part of Cn(Y)

This has a corollary as well: The first entails that Cn(X) is part of Cn(Cn(X)). So we have an identity here:

Cn(Cn(X)) = Cn(X),

the operation is idempotent.

The deductive inference relation and the consequence operator determine each other completely, so either one can be taken as all there is to the logic.

All of the above applies to classical logic of course, but also continues to apply when that logic is augmented with new axioms and with new rules of inference.