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Can there be logic without the law of identity?

If a = a doesn't always hold, can there be logic? I heard there are logical laws where the law of non-contradiction doesn't always hold, so I am wondering if there were also other logics where the law of identity isn't held as an universal truth.

armand
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Sayaman
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    Sure. Standard propositional and predicate logics do not include identity = as a symbol, so a = a is not among their laws. When "=" is added "a = a" is typically postulated, but as part of convention for using the symbol, it is not exactly a "law" either. In fact, it is hard to say what "identity law" means substantively. Classically, it is associated with permanence through change, or lack thereof (that Heraclitus and Hegel advocated), which can be modeled by temporal logic, for example. A more radical approach is to exclude object variables, to which "=" applies, altogether, as in PFL. – Conifold Jun 02 '21 at 05:10
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    Requiring consistent interpretation of bound variables and requiring 'identity' are different things. Mathematics has no ultimate sense of identity, only equivalence relations, and it gets along just fine. The requirements upon an equivalence relation test that the logic will work correctly. That does not mean any two things are ever identical, it requires that your axioms be tested every time you create a new use for the equals sign, the same way you need to test the axioms when you define anything else to which you will apply a generalization. – hide_in_plain_sight Jun 02 '21 at 16:52
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    I don't know the details but apparently category theory gives a formal way to replace equality with "equivalence" under a "category", see here & here with the comment "Instead of calling two things exactly equal, Eilenberg and Mac Lane urged mathematicians to embrace sophisticated new mathematical structures that captured the many ways in which two things might be the same, or equivalent." – Hypnosifl Jun 02 '21 at 20:07
  • Are you referring to the reflexive axiom? – forest Jun 04 '21 at 23:35

8 Answers8

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Let's say you've demonstrated something about A. Now let's say you want to use this result in another proposition about A. Ah, but is the A in the first proposition the same A than in the second one? If A=A is not always true, you can't say, and therefore you can't link two propositions sharing the same symbol.

Without the law of identity you can pretty much go nowhere.

Example for clarity:

  1. Socrates is a human.
  2. All humans are mortal.

Therefore what? Well, nothing, because to conclude Socrates is mortal you would need the symbol "human" to refer to the same concept in 1 and 2, but we don't know that anymore. This goes further than mere semantic trickeries where the same symbol is used to refer to 2 different ideas and pass some fallacy under the audience radar. We simply can't assume a symbol always refers to the same concept anymore.

Jens
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armand
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    Well, in Aristotelian logic there is actually no symbol "=" and no rules for it. We meta-think about Aristotelian logic using words like "the same concept in 1 and 2," but this is not formally part of the logic. – causative Jun 02 '21 at 01:18
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    Point taken. But as much as i love hair splitting about the notation of ideas, I think we should concentrate on the ideas here. – armand Jun 02 '21 at 01:33
  • Obviously there's no logic that we can't "meta-think" about using ideas of equality, because we can meta-think about anything that way. The more pertinent question IMO is what logics formally, internally lack symbols of equality. – causative Jun 02 '21 at 01:58
  • Except the OP didn't ask about the symbol =, he asked about the law of identity. – armand Jun 02 '21 at 02:14
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    Having the law of identity is effectively the same as having the symbol; if a logic has "=" without the law of identity, then "=" does not mean equality in that logic, and if a logic has the law of identity with a different symbol than "=", then that other symbol is really a symbol of equality. – causative Jun 02 '21 at 02:17
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    No. Historically the law of identity predates the symbol = and formal logic. – armand Jun 02 '21 at 02:20
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    I said it is effectively the same, in the sense that whenever another symbol was used in the law of identity, it had the same meaning as "=" and can be interpreted as "=". There is no effective equivalent of "=" in the formal rules of Aristotelian logic. (and no law of identity, which as you say came about much later, in the 14th century) – causative Jun 02 '21 at 02:35
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    Consistency of names is not the law of identity. If we reference Socrates twice, time has passed, so his age is different by some microseconds. So he fails Leibniz law of identity. So do we need to take a page out of Hericlitus and not consider him the same as himself? Then we have a problem. But within an ordinary context, we can keep a consistent reference, without invoking 'identity'. – hide_in_plain_sight Jun 02 '21 at 17:00
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    You can form useful statements with ∈ ("element of") or ⊂ ("proper subset") without strict equality. e.g. 1. Socrates ∈ human, 2. human ∈ mortal things. – BurnsBA Jun 02 '21 at 17:12
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    @hide_in_plain_sight: what I explain is the other way around: we cant have name consistency if we don't have identity, I.e. If the object we point to by a name is not guaranteed to stay the same. Also, there is no notion of time passing in Aristotelian logic. I think you are confusing logical identity with ontological identity (ship of Theseus, etc...) – armand Jun 02 '21 at 22:42
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    @burnsba the problem has nothing to do with the = sign. From your propositions 1. and 2. you can't deduce that Socrates is included in mortal things if you can't assume that "Socrates", "human" and "Mortal things" are referring to the same thing between propositions. – armand Jun 02 '21 at 22:47
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    Or even ∈ and ⊂, for that matter – armand Jun 02 '21 at 22:53
  • Socrates was a human. He is a character in the work of Plato. Is the character mortal? People change, so like the ship of Theseus, they may be the same in some contexts, different in others - on 4 legs, 2 legs, then 3 legs, say as the Sphinx's riddle has it, making an identity inference about locomotion not valid. – CriglCragl Jun 03 '21 at 08:53
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    References to the ship of Theseus are irrelevant. Don't confuse ontological identity (what it means to name an object, like the ship of Theseus) and logical identity (the out of time sameness of an concept with itself) – armand Jun 03 '21 at 10:28
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    inb4 "comments are not for extended discussion" :) – msouth Jun 03 '21 at 12:29
  • Note to mods: I am fascinated by this exchange in the comments and find it enlightening. There are some repeated thoughts near the end that can be sent to chat, but I think it would be useful to keep up the original arguments. – Just Some Old Man Jun 04 '21 at 20:57
  • I think it's important to note that OP isn't just asking what happens if we remove the law of identity and never assume that a = a. We can create more nuanced laws to replace the law of identity in this hypothetical logic. – tox123 Jun 05 '21 at 03:44
  • @armand. I can pick an element out of a group in mathematics. That element is merely a representative of an equivalence class. I do not know what element it is. It does not obey Leibniz law. And I can bind that unidentified element with a quantifier. It could still be any element satisfying the requirements under which I chose it. And I can still have a reference. Math works. So, yes I can have rules that work, without having anything, anywhere that obeys Leibniz law. – hide_in_plain_sight Jun 06 '21 at 21:59
  • @armand. Who says there is no notion of time passing? Why assume that? Why not assume that in choosing Socrates, you are inventing a reference that includes all ages of Socrates, the same way that if I choose an integer, and it happens to be 2, that is all forms of 2, the real number 2.0, the fractions 2/1 and 306/153... Why assume identity instead of letting selection be relative to the context. When I point at an electron, I don't choose for it to be a particle or a wave. Why should Socrates be more special than my electron? – hide_in_plain_sight Jun 06 '21 at 22:17
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You can have a form of this in extended algebras with an 'indeterminate' element, to represent things like 0/0. For most of the objects in the algebra, the law of identity applies. But if x = 0/0 and y = 0/0 that doesn't mean x = y. One example used in computer science is the Not-a-Number (NaN) in IEEE 754 Floating point numbers, which are not considered equal to themselves.

Another possible case is number systems designed to cope with uncertain/approximate values. If you have values rounded to finite precision, then x = 0.5 may represent any value 0.45 < x < 0.55. You can still do certain operations and comparisons on such numbers, but others like equality are invalid or unknown. A formula like "0.5 = 0.5" isn't generally true. Again, the common computer programming injunction not to rely on equality tests between floating point numbers is a practical example. A reasoning system designed to check the correctness of numerical algorithms has to implement a logic with a non-standard 'equality' operator.

You can still do logic in these situations, because identity works some of the time, in limited but generally well-defined circumstances. But it's not a universal law.

Nullius in Verba
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  • You say that x = 0/0, y = 0/0 but x != y (so sometimes there isn't equality). I would go as far as saying x = 0/0 is an invalid statement altogether. x = 0/0 is not saying "x is undefined", but rather it is saying "we are failing to define x in this statement" (however an alternative statement could succeed in defining x). And thus if you interpret that statement in that way, then there is no need to define an x != y exception to this scenario. Instead, the scenario x = 0/0 does not exist. (This is just my opinion, and my point might be confusing) – David Callanan Jun 04 '21 at 17:49
  • This answer is incorrect. Imaginary numbers do not somehow disprove the reflexive axiom, because it does not apply to values that aren't real. – forest Jun 04 '21 at 23:32
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I assume by the "law of identity" you mean the thesis that everything is identical with itself. This is usually taken to hold in logic, but it needn't do so. One can have logic with identity or without, that is to say, one can treat the identity predicate as a logical constant, or one can treat it like any other predicate and interpret it. Schrödinger logic is a non-classical logic that dispenses with the law of identity. It is motivated by the idea that one cannot meaningfully speak of the identity of elementary particles.

Bumble
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Free logic is a generalisation of classical logic which permits discussion of empty terms: terms which do not refer to a thing that exists.

From a quick scan of the version described in the Stanford article, it looks like (if you define an identity symbol) it is typical to define it in the usual way (so that everything is self-identical) but I imagine you could have a version of free logic in which 'non-existent things' are not self-identical. For example, in this hypothetical logic the sentence 'my pet unicorn is identical to my pet unicorn' would fail to be true because my pet unicorn does not exist.

(Classical logic could handle this with a definite description, so that the logical structure is more complicated but the truth-value comes out as false.)

However, as mentioned in the other answers, this would make reasoning about non-self-identical 'objects' very difficult.

dbmag9
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  1. A=B
  2. A=C
  • With the law of identity, logically, B is equal to C.
  • Without the law of identity, A is not necessarily the same A in step (2). Then, logically, B is not necessarily equal to C.

Now, take your conclusions.

RodolfoAP
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  1. All stars emit their own light
  2. Brad Pitt is a star
    Ergo,
  3. Brad Pitt emits their own light

Fallacy of Equivocation.

Logic establishes inferential connections (between premises and the conclusion) So, in the simplest sense, A is linked to B and B to C and then from that A is linked to C. If ~A = A or ~B = B or ~C = C then, the B in A is linked to B is not necessarily the B in B is linked to C and we're not in Kansas anymore. The same goes for C. We can't do logic without the law of identity (A = A).

Agent Smith
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The law of identity is best understood as metalogical. If we didn't assume the law of identity, then logical laws, for example the modus ponens, (A → B) ∧ A ⊢ B, would be useless because the second A couldn't be assumed to refer to the same thing as the first A, and same thing for B. No law of identity, no deductive logic.

Without the law of identity, it would be confusing to use the same symbol more that once. Thus, the modus ponens would become (A → B) ∧ C ⊬ D, an implication which is obviously not true. A ⊢ ¬¬A would become A ⊬ ¬¬B, again not true. Even A ⊢ A ∨ B would become false: A ⊬ C ∨ B.

∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘

It should be said that it is perfectly logical to use expressions which contradict the law of identity. An expression such as for example x ≠ x is perfectly logical, logical in the precise sense that we have no difficulty assigning a truth value to it, namely False, just as we have to say that 0 = 1 is false.

This is on a part with the fact we can assert expressions that contradict the law of contradiction. Thus, we are able to logically handle contradictions, for example A ∧ ¬A, without any difficulty, and this precisely because the law of contradiction says contradictions are false.

∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘

Another point deserve mention. Many programmers and computer scientists will argue that the law of identity is falsified by computers. One answer here even gives an over-sophisticate example of that argument (Not a Number, NaN).

But this argument is fallacious. Computer programs are sequences of instructions, which, clearly enough, implies that they are to be understood as unfolding over time, so to speak. Thus, there is no reason to understand a variable A on line 1089 to be referring to the same thing as the variable A on line 1088. Indeed, one of the most important program statement in computing is that assignment statement which can be used in particular to increment a value, for example: A := A + 1. Obviously, A := A + 1 means that if initially A is 1, then it will be 2 once the assignment statement will have been run. Thus, the same name A will inevitably refer to two different values. However, the law of identity of course does not mean that things cannot change.

Another example is the notion of random number, which by design of the random number generator will most likely be different each time it appears in a statement. So, arguing that the law of identity does not hold because there are example of computer program variables that are not equal to themselves is a red herring. We might just as well argue that we don't bathe in the same river twice or that we ourselves are never identical to what we were just a fraction of a second ago.

Speakpigeon
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  • "We might just as well argue that we don't bathe in the same river twice or that we ourselves are never identical to what we were just a fraction of a second ago." Famous philosophers have done that, though. Namely Heraclitus, & Leibniz. But you know better? – CriglCragl Jun 03 '21 at 08:59
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    @CriglCragl You think the law of identity means that every thing is identical to what it was a fraction of a second ago?! You think that saying a thing is identical to itself can possibly mean that it remains identical to what it was initially? – Speakpigeon Jun 03 '21 at 13:29
  • Predicate logic is like mathematics. A toy world, for a purpose. Don't reify it, and believe it is the only possible useful overlay to simplify phenomena. It depends for it's use on correct assumptions: we tacitly encode 'similar enough' as identical. – CriglCragl Jun 03 '21 at 17:32
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    @CriglCragl Predicate logic?! We are not talking about predicate logic. We are talking about logic and logic is definitely not "like" mathematics. – Speakpigeon Jun 03 '21 at 18:44
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I am in possession of a certain A for which A != A. Logic doesn't have a real problem with it. You have to qualify a few proofs as quite a lot of them have unstated assumptions but it's not a problem.

There doesn't seem to be a good way to answer how many of this A I have because the following are theorems on any such A:

  • A ≠ A
  • A ∉ {A}
  • ∃a (a ≠ a) ∃b (b ≠ b) ↛ a ≡ b

That is, "A does not equal itself", "A is not found in the set containing itself", and "A is not unique".

Yet there is logic on a system containing objects with this property. It has arithmetic, symbolic algebra, and a workable proof system. This is all set down in a book that sits upon my shelf and as sat there since lower division college, and should a computer programmer come by here I would be disappointed if they could not give A its right name.

There's a secondary law that's true in the algebra that I don't know how to state in formal logic that still holds. If you have ∃A ... A ... A ... than the two a are the same by the definition of bound variables as used to define the operation ∃. Note that this continues to be true even though (A ≠ A) is a theorem for this particular A. I don't know any name given to it, but it is somehow both stronger (true in more cases) and weaker (less can be concluded from it) than the classical law of identity.

Joshua
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