Can the law of non-contradiction exist without the law of identity?

Question

Lately I've been reading about Quentin Meillassoux, and it seems that the only law of logic he doesn't see as contingent is the law of non-contradiction, because if the world is what it is not, then there cannot be absolute contingency, and contingency is absolute. Charles Williams Johns explains it like this:

no contradiction can exist or else it would be both what it is and what it is not and hence has no room for real contingency and subsequently becomes necessary (i.e. it cannot be destroyed if an entity is both what it is and what it is not simultaneously)

I do not, however, understand how he could not therefore accept the law of identity as absolute. If A cannot not equal A, then does that not mean that A=A necessarily? Could this somehow be fixed by the law of the excluded middle being contingent?

Answer 1

Can the law of non-contradiction exist without the law of identity?

in what follows, we'll tackle the 0th-order law of identity, the scheme 'p → p' for some binary connective for which modus ponens holds (here we're taking modus ponens to be the defining property/characteristic of any 'implication-like' connective): the the 2nd-order cases involving '=' can then be analysed via leibniz, and the 1st-order cases may be dealt with in appropriate theories, so the crux of the matter seems to be the 0th-order case

Could this somehow be fixed by the law of the excluded middle being contingent?

consider intuitionistic propositional logic, and define a binary connective @ as

p @ q := ¬p ∨ q

then MP holds, that is

p, p @ q ⊢ q

so @ is, at least in this (weak) sense, an implication, but of course

p @ p := ¬p v p

does not necessarily hold

this may be considered 'cheating' somewhat as ¬ is in principle defined in terms of the usual intuitionistic implication...

a certain variation on the previous example provides an example in which excluded middle still holds:

consider some (normal) modal logic in which at least axiom T/reflexivity holds, and that 'p → □p' does not in general hold, define

p @ q := ¬p ∨ □q

and check that MP holds (using T), plus also LNC and LEM, but that 'p @ p' does not necessarily so

you also may find interesting stuff regarding "implication problems" in dalla chiara and giuntini's chapter on "quantum logics" in gabbay's handbook of philosophical logic

Answer 2

In the way the terms LNC and identity are standardly used within logic, they are independent. Classical logic includes both, but you could have a non-classical logic that lacks either one or lacks both.

If someone wishes to talk about contingency, then it is important to say what kind. Contingency, like possibility, comes in many varieties. There is logical possibility, physical possibility, epistemic possibility, deontic possibility, metaphysical possibility, actual possibility, and no doubt many others. Ditto for contingency and for absoluteness.

Much confusion follows from not being specific. In fact, the two pieces of text that you quoted sound like claptrap to me. Maybe if I knew the context they would be comprehensible. What is "contingency is absolute" supposed to mean? What does a "contradiction... cannot be destroyed" supposed to mean? Contradictions are propositions; they are not things that might or might not exist in the world but features of how we describe the world and reason about it. In classical logic all contradictions are false; that doesn't mean they "don't exist".

The text you quote also seems confused about whether it is talking about propositions, things, or beliefs. Usually LNC is understood to be about propositions: it states that ¬(P ^ ¬P) is a logical truth. This should not be confused with talk about things. It is different to say that a thing never both exists and does not exist. It is also different from saying that it is irrational both to believe and disbelieve the same proposition at the same time.

Answer 3

Can the law of noncontradiction exist without the law of identity?

I am going to say no. Here is a summary of the three laws of thought, from the online Encyclopedia Britannica (formatting altered):

—————

[T]raditionally, the three fundamental laws of logic [are]: (1) the law of contradiction, (2) the law of excluded middle (or third), and (3) the principle of identity.

The three laws can be stated… as follows.

(1) For all propositions p, it is impossible for both p and not p to be true[.]

(2) Either p or ∼p must be true, there being no third or middle true proposition between them[.]

(3) [A] thing is identical with itself, or (∀x) (x = x), in which ∀ means “for every”; or simply that x is x.

—————

There are different ways to play with these ideas. In the absence of the assumption that something is what it is, then something (p) can be anything at all; p can still be itself, but it does not have to be. In short, p can be not-p. This conclusion does indeed destroy the law of noncontradiction, for it becomes possible for p and not-p to be true simultaneously.