# Can you achieve the same level of 'consistency' or similar if you remove or modify the law of identity in a logical system?

Can you achieve the same level of 'consistency' or similar if you remove or modify the law of identity in a logical system? Well, first, I consider it to be impossible to maintain the same level of consistency if you remove or modify the law of identity, but I was wondering if it was possible to do so if you modify it a little bit.

Instead of saying A = A, modifying it to A = A or B, or maybe a similar statement, I was wondering if it was possible to achieve the same or a similar level of consistency or not.

I use the word consistency, because I am assuming if A = A is not necessarily true, then there might be some inconsistency resulting from this lack of one-to-one correspondence.

• It's puzzling that you seem interested in logic and yesterday you just accepted quantum logic as formalized logic without LOI which is consistent since it directly models observable quantum behaviors for over 56 years since Birkhoff's proposal without any known inconsistency proof. If you want to talk about level of consistency aka relative consistency strength like PA expressed in ZFC vs ZFC then you'd need to specify another system to compare with... Sep 25, 2023 at 23:23

Off the top of my head, the closest thing I can imagine along the indicated lines is to suspend AA on the propositional level, but even so, this would be on pragmatic, not semantic, grounds (i.e. "it's dialogically silly to infer a proposition from itself," although in terms of abstract validity, hardly any inference seems more obvious!O).

Alternatively, consider hyperintensionality, e.g. the difference between, "Sam believes that Dean is his brother," and, "Sam believes that a fictional character named 'Dean' in a TV show, is his brother," or, "Sam believes that Sam Clemens is Sam Clemens," vs., "Sam believes that Mark Twain is Sam Clemens." If we drop the axiom scheme of term substitution, then, for example, or modify it at least (to get at what you're aiming to hit upon), we might generate hyperintensionally paraconsistent sets of sentences (I suppose so, anyway).

We might also modify identity in terms of relativizing such frameworks (c.f., as the article discusses, the myriads of thin and thick equivalence relations in mathematics) or by traversing the dark anfract of transworld identity. Questions such as of predicativity also come to mind, and then the matter of object-vs.-definition circularity. You might also be interested in connexive logic:

The name ‘connexive logic’ was introduced by Storrs McCall (1963, 1964) and suggests that systems of connexive logic are motivated by some ideas about coherence or connection between the premises and the conclusions of valid inferences or between the antecedent and the succedent (consequent) of valid implications. The kind of coherence in question concerns the meaning of implication and negation (see the entries on indicative conditionals, the logic of conditionals, counterfactuals, and negation). One basic idea is that no formula provably implies or is implied by its own negation. This conception may be expressed by requiring that for every formula A,

⊬ ~AA and ⊬ A → ~A

Note that if we defined some sentence S such that S = S + T (letting "+" be the conjunction mark), then either S = T already, or somehow T by itself doesn't equal S. I don't know how such a sentence would work, and it seems as if it might not really work very well, ultimately, so these kinds of modifications of identity formulae, in whichever intended system, must be undertaken carefully, if at all.

OIndeed, if (deductive) validity occurs when it's the case that if the premises are true, then the conclusion is true, then in case A is true (any A), then A is true. But dialogically, we object to if the premise is true as a premise, then the premise is true as a conclusion too, since this is tantamount to circular reasoning after a mere half-step through one's thoughts.