As I see it, the "law" isn't a law, but rather an earnest (in the sense of "a token of something to come; a promise or assurance.").

IOW, like the apriori generally, it simply reflects our intent to use language consistently.

We posit consistent natures for things, proceed as if things have those natures (belief=trust, expectation), and then subsequent experience of them, or of things causally connected to them, either conforms to that posited nature - or not. If experience pans out as expected, well and good, we continue to use the term consistently; if experience answers in the negative, we think up another kind of nature or essence, or modify the original, and continue.

Now it so happens that the "middle-sized furniture of the world" amidst which we've evolved has things that do in fact have consistent natures or essences through time. But we've already seen how that breaks down at the micro-level and macro-level (although we can still draw conclusions that fit the logic-obeying middle-world we live in, in terms of scientific meter readings and dial readings on equipment that's causally connected to the bizarre goings-on).

I just read a book on philosophical logic and it states that one cannot deny the LNC. To do so would be to claim it isn't TRUE or that it is FALSE but not both, thereby affirming the LNC.

This claim by the author seems to rest on self-reference. Applying the denial of LNC to LNC itself results in self-refutation.

Perhaps rejecting the LNC shouldn't apply to the LNC i.e. we ignore the self-reference and continue.

Can anyone clarify? Thanks. — TheMadFool

The negation of "no statement is both true and false" isn't "all statements are both true and false". If at least one statement is both true and false then the law of noncontradiction is false (and only false).

It might be worth looking into dialetheism. Priest et al. only argue that there are true contradictions, not that every contradiction is true.

Who is the author?

As I see it, the "law" isn't a law, but rather an earnest (in the sense of "a token of something to come; a promise or assurance.").

IOW, like the apriori generally, it simply reflects our intent to use language consistently.

We posit consistent natures for things, proceed as if things have those natures (belief=trust, expectation), and then subsequent experience of them, or of things causally connected to them, either conforms to that posited nature - or not. If experience pans out as expected, well and good, we continue to use the term consistently; if experience answers in the negative, we think up another kind of nature or essence, or modify the original, and continue.

Now it so happens that the "middle-sized furniture of the world" amidst which we've evolved has things that do in fact have consistent natures or essences through time. But we've already seen how that breaks down at the micro-level and macro-level (although we can still draw conclusions that fit the logic-obeying middle-world we live in, in terms of scientific meter readings and dial readings on equipment that's causally connected to the bizarre goings-on). — gurugeorge

Finally, someone on this forum who gets it.

The claim is circular and kind of ignorant, quite honestly. Dialetheism has been around as a legitimate theory for decades now, as has paraconsistent logic. The fact that one may deny the LNC doesn't entail one accepts the LNC. It simply means you believe there is at least one proposition which is true and has a true negation. And ironically, according the semantics of standard dialetheic paraconsistent logics, the LNC is a dialetheia, it's both true and false.

Seriously, Graham Priest (and others) has a thorough defense of this view. Work has even been done to develop purely paraconsistent meta theory (see Patrick Girard's work). The book you referred to is simply incorrect.

I think you're barking up the wrong tree. Nothing in the microscopic world has even been suggested to have an inconsistent nature. Quantum mechanics uses the standard mathematical formalism (Zermelo-Frankel set theory + classical logic). There is such a thing as quantum logic, but that formal system does not violate the LNC, it gets rid of the Law of Distribution. The only respected defender of a truly inconsistent quantum theory I know of is Newton da Costa, but even he knows he is almost alone in his view regarding the nature of superpositions.

I think you're barking up the wrong tree. — MindForged

I defer to your greater knowledge of the matter. If things are actually consistent at the micro level as well as at our familiar, evolved level, that's even better.

I hold similar views. Logic, at least in its useful form, must conform with experience. Does this "middle-world" you speak of violate the LNC? It does not and how do we actually go about rejecting the LNC? Please read below.

The negation of "no statement is both true and false" isn't "all statements are both true and false". If at least one statement is both true and false then the law of noncontradiction is false (and only false). — Michael

That's a good point. Rejecting the LNC doesn't require that ALL statements are both true and false. Finding just one statement that is both is a good enough counterexample to the LNC. So, in a sense, rejecting LNC shouldn't be self-referential. Thank you very much.

And ironically, according the semantics of standard dialetheic paraconsistent logics, the LNC is a dialetheia, it's both true and false. — MindForged

Can you explain that a bit. I didn't understand. Thank you.

Does this "middle-world" you speak of violate the LNC? — TheMadFool

It would depend on what is meant by "contradiction." You can make up any number of systems of rule-governed symbol-shuffling in which something called "contradiction" is possible. But what we normally call "contradiction" in the middle-world isn't possible, because things have natures, i.e. they behave consistently within a narrow range of "logical possibilities" (horrible phrase, but what the hey, it's what people use).

A struck match doesn't turn into a gerbil or sprout wings and sing the Hallelujah Chorus, it has a delimited range of possible behaviours under given circumstances. We define "match" as just that sort of thing that has that limited range of behaviours, i.e. we will refuse to call "match" a thing that doesn't behave in that sort of way. If we struck a match and it did turn into a gerbil, or grace us with a rendition of Händel's magnificent oeuvre, we'd no longer consistently call it a match, but given it another name/concept. But that just means it wasn't a match after all, like we thought it was, but this other new, peculiar thing that looks like a match at first, but behaves differently from the things we normally call "matches."

However we could if we wished continue to call it a "match" and expand our definition of "match" to include things that behave like normal matches (i.e. they look like little sticks with black blobs on the end that burst into flame when struck) and things that don't behave like normal matches (i.e. things that look like sticks with black blobs on the end but they turn into gerbils when struck, take a dump in your hand and scamper off). It's really a matter of convenience which path we choose. All that would have happened would be that the world turned out to be a lot weirder than we thought it was, and we'd have to revise huge chunks of what we thought was settled knowledge. :)

(We could go even further and call anything in the universe, or even the entire universe a "match" if we wanted to, but then our system of categorization would be unwieldy and impossible to retain in the mind, because we'd have to constantly qualify the type of match we're talking about. But the job the symbols are doing would be the same - simply tracking changes in experience and nature (and in fact, we'd just cancel the symbol "match" out and use the sub-categories). It's just easier to think of the "ultimate match" as "God/the Absolute/Everything/Existence", etc., and subdivide everything with whatever mixture of symbols and sub-categories and cleanly separate symbols we find convenient.)

Now, in this context, a contradiction would be like using "match" (ordinary) and "match" (expanded ordinary+gerbil) in the same context; but it's not really a law of any sort that we'd be breaking if we did that, and we wouldn't be saying anything informative about the world (the world is just moseying along doing whatever the hell it does, and we can consistently use concise or clumsy symbol systems to get a handle on it - or not).

Another way of putting the above: it's related to what they call "interpretation" in maths. When we say "1+1=2" whether that's true or false depends on what "+" means, what "=" means and what the natures of the objects we're talking about are. Does "+" mean roughly "laying contiguously side-by-side, or side-by-side near each other up to a range of, say, a few inches apart"? Normally we think of the units as smallish solid objects. But what if we're talking about one blob of mercury dropped next to another blob of mercury in a bowl? On opposite sides of a ceramic trench across America?

Similarly, paradoxes and similar mental Chinese finger puzzles depend on semantics ("interpretation"), on the natures of the things we're talking about - you can keep the contradiction or remove it, at will. But the things still behave as they do. In the middle-world, in the real world: either all Cretans are liars or at least one Cretan isn't a liar, either Epimenides is a Cretan or he isn't, either he's lying or he isn't. The symbols alone don't, or rather the sheer syntax of the apparently paradoxical statement laid out before you doesn't, say anything on its own, and can be made to say whatever you want depending on interpretation; the content of the paradox, and whether there's a contradiction or not, depends on the semantics. Without the interpretation, without the semantics - or in Wittgenstein's terms, without a "grammar" or "language game" - there's simply no intrinsic there there, no fact of the matter, no puzzle and therefore no resolution to a puzzle to be found or discovered, all we're dealing with is patterned scribbles on a page, patterned noises from a mouth.

Four arguments on the topic or undermining the topic:

On the topic:

Mapping true and false onto the two horns of the principle of excluded middle is a way of begging the question for this demonstration of the LNC.

LNC is right or wrong.
(1) To be right is equivalent to being true.
(2) To be wrong is equivalent to being false.
(3) The principle of excluded middle is true.
(4) The negation of the principle of excluded middle is wrong
(5) The negation of the principle of excluded middle is false.
(6) Introduce theorem: De Morgan's Law.
(7) LNC is right (from 5, , P or not-P iff not(P and not-P) through 6.)

IE, in order to prove LNC, its negation is assumed by equating 'something is either right and wrong' with the negation of the principle of excluded middle's falsehood (which is itself the LNC).

Undermining the topic:

Uselessness of formal logic in terms of proposition evaluation in most contexts

Looking at real arguments in real circumstances; to be true or false is less important than other valuations (useful, right enough, justified, approximately correct, legal, virtuous, probably true, probably false etc...). Informal logic and argument strategies are broader than the strictures of formal logic, even in mathematics. Many interesting philosophical arguments are difficult or impossible to formalise - transcendental deduction and its phenomenological and dialectical derivatives are a prime example in philosophy. If you doubt this, try and take the derivation of the categories in The Critique of Pure Reason and accurately render its propositions and inference rules in first or a higher order predicate logic.**

Differences in formal logic systems

Intuitionism (provability logics, not (P or not-P) is no longer equivalent to the LNC since double negation elimination no longer is allowed) and paraconsistency (LNC not just false, principle of explosion false) are viable logics. The capacity for other logics to be adopted which don't stress the LNC in the same way or don't include it at all to be developed means it must be possible to think about the LNC without it being self evidently true, no?

Formal logic's material implication is an impoverished form of 'following on from' in an argument

Who cares about the LNC when it won't do anything to help you evaluate counterfactual arguments, imperatives and can't even deal with what a semicolon or a colon can do in discursive english. Human inference forms are far richer than the semantic equivalence between deduction and material implication in propositional and predicate logics imply. Even mathematical logic requires a richer notion (see your thread on the incompleteness theorems)...

**: I tried this, it is madness. It's difficult to individuate propositions in the first place, and the steps between them follow in a modal sense rather than a material conditional.

↪gurugeorge I hold similar views. Logic, at least in its useful form, must conform with experience. Does this "middle-world" you speak of violate the LNC? It does not and how do we actually go about rejecting the LNC? Please read below.

I'm responding to a bit more than what you said to me because I feel more can be said.

Logic need not "conform to experience" in order to be useful. Aside from the fact, that human experience is inherently limited and varied, not all humans interpret or understand their experience in the same way. Like take the so-called "Laws of Thought" (this is a terrible name IMO). People often speak of these are obvious, indisputable certitudes that were never reasonably questioned and the only ones questioning it are modern relativists. But this is demonstrably false. I mean, one need look no further than the "cacuskoti"/tetralemma of Indian philosophers, in which propositions are thought to be capable of being only true, only false, both true and false, or else neither true nor false.

Or if someone wants to just dismiss this as nonsense on the spot because something something Eastern philosophers are wacky frauds, let's take a classical Western Philosopher: Aristotle. Aristotle believed that sometimes the Law of the Excluded Middle was not always true. Specifically, that it failed when making contingent statements about the future. In a sense, this lies beyond experience and yet if Aristotle were right about this, then we would want to have a logic that can handle "violations" of the LEM.

This makes an important point broadly about theories, which is just as applicable to logic. Data in theories are soft, that is, the data itself is also fallible and can be overturned by a good enough theory.

A theory which helps us understand why our data is flawed or incomplete or misunderstood. Logical systems (e.g. Classical Logic (Frege's Logic, different from Aristotle's), Intuinistic logic, etc.) are theories about logical consequence (what follows from what). And in this case, you seem to be saying human experience must be data which logical systems have to conform to. But it can just as easily be the other way around: logical systems can overturn what we believe to be true from human experience, just as Non-Euclidean Geometry upturned assumptions we had about the possibilities of different spaces.

/long-winded, sorry

That's a good point. Rejecting the LNC doesn't require that ALL statements are both true and false. Finding just one statement that is both is a good enough counterexample to the LNC. So, in a sense, rejecting LNC shouldn't be self-referential. Thank you very much.

Well, if you're using an explosive logic (i.e. every type of logic besides Paraconsistent logic), rejecting the LNC does require that all proposition are true and false. It's provably so.

And ironically, according the semantics of standard dialetheic paraconsistent logics, the LNC is a dialetheia, it's both true and false.
— MindForged

Can you explain that a bit. I didn't understand. Thank you.

It takes a bit of delving into the semantics of dialetheic logics, but I'll try to state it better than I did. Essentially, if you believe there are contradictions which are also true (known as Dialetheism), you are committed to believing Dialetheism itself is both true and false. It falls right out of the logic. It seems counter-intuitive (in an already counter-intuitive theory of truth), but it is provably the case.

I don't fully understand you. However, your thoughts on ''interpretation'' make sense but doesn't really refute the LNC. If A and B both interpret ''dead'' identically then the statement ''the cat is both dead and not dead'' is a contradiction.

Can you explain what Wittgenstein means by ''language game''?

Nothing in the microscopic world has even been suggested to have an inconsistent nature. — MindForged

Well, we do have a problem here in defining "consistent behavior" in the microscopic world, in which nothing repeats. Everything is evolving.

Thanks for the informative post.

Well, if you're using an explosive logic (i.e. every type of logic besides Paraconsistent logic), rejecting the LNC does require that all proposition are true and false. It's provably so. — MindForged

Can you show me how that is implied? How do you prove it?

Another way of putting the above: it's related to what they call "interpretation" in maths. When we say "1+1=2" whether that's true or false depends on what "+" means, what "=" means and what the natures of the objects we're talking about are. — gurugeorge

Yes. but the subtleness if what you are suggesting will not be grasped on one reading. It is the essence of understanding the fundamental issue that arises when one attempts to replace that which we observe with symbols. One cannot understand movement with stagnant symbols.

Can you show me how that is implied? How do you prove it?

Here's the simplest way. Take 2 arbitrary assertions and assign them to the letters "P" and "Q".

"P" = "It's Monday"
"Q" = "I'm green"

1) P AND ~P
[Assume a Contradiction is the case]

2) P (because if an AND statement is true, each of the components must be true)
[Conjunction Elimination from #1]

3) P OR Q (Only 1 part of a Disjunction needs to be true for the entire statement to be true, and we just proved that P is true)
[Disjunction Introduction from #2]

4) ~P (Remember, if an AND statement is true, both components must be true. In this case, the other component is the negation of P)
[Conjunction Elimination from #1]

5) Q (In an OR statement, if one component is not true, we know the other component must be true. And we just proved above that ~P is true)
[From 3 & 4 via Disjunctive Syllogism]

So if it's both Monday and it's not Monday, I am therefore green.
That's trivial. It let's you prove any and every proposition is true if you assert a contradiction in a logic that is not Paraconsistent.

Aaaaaaaahhhh!

I get it. Assume P and ~P...one single contradiction

Take any other proposition Q

1. P & ~P.....premise
2 Q..............premise
3 P...............1 simp
4 P v ~Q......3 add
5 ~P.............1 simp
6 ~Q............4, 5 disjunct elim
7 Q & ~Q.......2, 6 conjunction

Yea Explosion is wild. That's why dialetheists adopt a Paraconsistent Logic, most often dropping either Disjunction introduction or disjunctive syllogism. By challenging these, you might also lose something like Excluded Middle or something else as well, but you can then formalize logical theories involving contradictions. Perhaps you want to solve the Liar Paradox or else you want to resurrect Frege's Logicism. After all, Godel's Incompleteness Theorems do suggest that you can build a complete mathematics on an inconsistent foundation, but that's only available when using Paraconsistent Logics.

There are a number of reasons you might want to avoid reasoning explosively besides Dialetheism, these are just some interesting reasons you might consider going to paraconsistency.

I don't fully understand you. However, your thoughts on ''interpretation'' make sense but doesn't really refute the LNC. If A and B both interpret ''dead'' identically then the statement ''the cat is both dead and not dead'' is a contradiction. — TheMadFool

I don't think you can refute the LNC, because it's not a "law," it's not a thing for refuting; it's a reflection of our commitment to speak consistently (e.g. to interpret "dead" identically for A and B). What would be the sense in refuting our own commitment? How do you refute a commitment? It doesn't make sense.

Can you explain what Wittgenstein means by ''language game''? — TheMadFool

It's an analogy between language and games, which usually have rules. He's saying that in many cases (not all, but most) we use words in a way that's analogous to the way we play games, which involve patterned interactions with the world, and sometimes the use of objects in patterned ways. Chess, for example, has rules about how to move the pieces, and moving the pieces in a different way simply isn't playing Chess. Similarly, with language, we have rules about "moving words around," and shuffling the words around in a different way makes us unintelligible. (Note that it's possible to invent a new game by moving the pieces differently; similarly it's possible for language to branch out by using words differently. Note also that it's not just words we shuffle around - some language-games involve physical interactions with the world in patterned ways as well, alongside the word use, even if only implicitly.)

So usually, the meaning of a word is its use, or its place in the language game (which is defined more or less by the "grammar", the pattern of use, the rules and criteria of proper usage). Now reference (or interpretation, as discussed above) is one kind of use. We do use language in the "game" of referring to things - a lot. But (Wittgenstein thinks) a lot of philosophical trouble and misunderstanding has been caused by philosophers trying to make all language seem like it's fundamentally about referring to things (which is what Wittgenstein did in the Tractatus). But it's more accurate, and gives a bigger, better picture, to see reference as a (large) sub-game in the welter of language games, for which the more general understanding is meaning-as-use.

(This is why Wittgenstein didn't actually repudiate the Tractatus, but thought of it as a less comprehensive explanation of logic and philosophy than he'd originally believed it was - it's more like a study in a particular narrow area. It's like an extended, elaborate version of the ultra-simple "moving blocks around on command" language-games he used in the Philosophical Investigations. But this is also why Wittgenstein didn't entirely repudiate philosophical theory either; it's just that he thought that whereas philosophers previously had believed they were making discoveries about a deep, hidden structure to language, what they were actually doing was creating artificial, simplified language games that give insight into use, just as he does with the simple language game examples in the PI.)

I don't think you can refute the LNC, because it's not a "law," it's not a thing for refuting; it's a reflection of our commitment to speak consistently (e.g. to interpret "dead" identically for A and B). What would be the sense in refuting our own commitment? How do you refute a commitment? It doesn't make sense.

I don't really understand this. The LNC is an axiom in reasoning, there's no reason why it cannot be subject to refutation. That we wish to remain consistent does not entail that we can remain consistent. It's not [merely] a commitment.

Like take Tarski's undefinability theorem. Any language which contains its own truth predicate must be either inconsistent or incomplete. Natural languages most assuredly do contain their own truth predicate (even Tarski had to admit this was clearly the case). So natural languages are inconsistent because they can produce Liar-type sentences; this was why Tarski advocated for a hierarchy of formal languages, and why Wittgenstein eventually just suggested we avoid making Liar paradoxes.

If a Liar-type sentence holds, then the LNC does not hold because there is a counter-example. It's no different than when Intuitionists say that the Law of the Excluded Middle is not a tautology. A Dialetheist would say that not all contradictions are truth and false, but that some of them are. Not because they are violating some commitment, but because they believed it is entailed by features in our language or semantics.

That we wish to remain consistent does not entail that we can remain consistent. It's not [merely] a commitment. — MindForged

You can refute an example of inconsistency, but how do you "refute" the very commitment to remain consistent that defines reason?

(Not trying to be flip here, this is really how I see it. The LNC is on a different level from things that use the LNC. The form of it makes it look like an object-language statement - which could be consistent or inconsistent - but I think it's really a statement of intent.)

You can refute an example of inconsistency, but how do you "refute" the very commitment to remain consistent that defines reason?

Didn't I answer that? We are committed to *trying* to remain consistent, but that doesn't mean we can actually meet that commitment. Nor, do I think the commitment (surely this is normative) to remain consistent is what defines reason. We can be perfectly reasonable in holding contradictory beliefs or separate beliefs that contradict each other. If we have good reason to hold these conflicting beliefs, but no way to resolve the contradiction (if resolution is possible), then the reasonable thing to do is to maintain the contradiction until you are able to resolve it.

(Not trying to be flip here, this is really how I see it. The LNC is on a different level from things that use the LNC. The form of it makes it look like an object-language statement - which could be consistent or inconsistent - but I think it's really a statement of intent.)

It's an axiom and it's generally reasonable, even to the Dialetheist. Even granting that we ought to remain consistent (as possible), what we intend to do might not be identical to what we are able to do in practice.

I don't think you can refute the LNC, because it's not a "law," it's not a thing for refuting; it's a reflection of our commitment to speak consistently (e.g. to interpret "dead" identically for A and B). What would be the sense in refuting our own commitment? How do you refute a commitment? It doesn't make sense. — gurugeorge

Accepted.

But this is also why Wittgenstein didn't entirely repudiate philosophical theory either; it's just that he thought that whereas philosophers previously had believed they were making discoveries about a deep, hidden structure to language, what they were actually doing was creating artificial, simplified language games that give insight into use, just as he does with the simple language game examples in the PI.) — gurugeorge

So Wittgenstein isn't that injurious to philosophy as I supposed. One member, I think it was Banno, said that everything is a game. I wonder if philosophical truths are more about the game rather than anything substantive. The question itself is part of the game I suppose?

It's an axiom — MindForged

No it's not, that's the thing. An axiom would be something presupposed as true, or assumed as true, or necessarily implied as true. But the LNC is not a presupposition or assumption or a necessary implication of anything, it just has the form of such, which is what's misleading. But because it's not a truth claim, it's not for refuting.

For example "A = A" (which is the root of the others, which are just "corollaries" IMHO, although even saying that could be misleading) looks like you're making a truth claim about reality, like this is an assumed fact, or a discovery about reality or the world. But it's actually just setting out the rules of the game: "We will use "A" consistently."

What on earth would it mean to say that "a thing is identical with itself"? Is that an informative statement?

So Wittgenstein isn't that injurious to philosophy as I supposed. One member, I think it was Banno, said that everything is a game. I wonder if philosophical truths are more about the game rather than anything substantive. The question itself is part of the game I suppose? — TheMadFool

Yeah I think that's the idea - philosophical truths "limn grammar", it's like we're reminding ourselves of the rules of the game, the criteria for language use and thought, but it seems like we're making substantive claims. That's the main cause of the confusion in philosophy, as Wittgenstein sees it.

(Stepping backwards a bit, I'm still not absolutely totally convinced of that - I still think it's possible that there may be such a thing as philosophical discoveries about the world - but I do tend to argue for the Wittgensteinian position at the moment, to see how it stands up, as I've been exploring it for the past few years after a long period of hostility to W., and latterly being convinced by him - this is in the context of a 40 year obsession with philosophy :) - and I do think he's right about the cause of a lot of the apparently fruitless to and fro in philosophy.

(I'm actually coming round to the idea that modern philosophy was a mistake, and that the few philosophical discoveries that there are to be made were already made by the classical philosophers, that modern philosophy broke philosophy, and we're only just starting to rectify that. Going back to Aristotle is increasingly fashionable - and I actually think that Wittgenstein was starting to reinvent the Aristotelian wheel in On Certainty, which is particularly ironic in view of the fact that he lamented he hadn't read any Aristotle. Perhaps it's good that he didn't - so at least we know that those ideas can be dug up by alternative routes! Like a triangulation or something.)

No it's not, that's the thing. An axiom would be something presupposed as true, or assumed as true, or necessarily implied as true. But the LNC is not a presupposition or assumption or a necessary implication of anything, it just has the form of such, which is what's misleading. But because it's not a truth claim, it's not for refuting.

Um, what? It literally is an axiom. That's how it's introduced within a logic, it's presupposed as true in all models. It is a truth claim. Specifically, it is the claim (assumption) that either a proposition "P" is true or the negation of "P" is true, but not both. So if one wants to attempt to refute it, one merely (if possible) has to show that some propositions is true and its negation is true as well.

For example "A = A" (which is the root of the others, which are just "corollaries" IMHO, although even saying that could be misleading) looks like you're making a truth claim about reality, like this is an assumed fact, or a discovery about reality or the world. But it's actually just setting out the rules of the game: "We will use "A" consistently."

Well I didn't say these had to be assertions about reality. These can be understood purely formally and syntactically. And if you read some previous stuff in the thread, it's not clear identity is universally applicable.

What on earth would it mean to say that "a thing is identical with itself"? Is that an informative statement

I dislike that rendering because it doesn't tell you much. A better definition is that identity is a reflexive, transitive and symmetric relation.

What on earth would it mean to say that "a thing is identical with itself"? Is that an informative statement? — gurugeorge

It could perhaps be referring to the proposition 'x=x' where x is a variable symbol. Some axiomatisations of first-order predicate logic contain an axiom schema which is of that form.

What the schema does really is specify a key property of the '=' symbol, specifically the property named 'reflexivity'. The other schema that's needed to complete the specification of the meaning of '=' is the axiom schema of substitution:

(a=b) -> (F -> F')

where F is a proposition and F' is a proposition obtained from F by replacing one or more of the occurrences of a in F by b. Essentially it says that any property that is true of a is also true of b.

A better definition is that identity is a reflexive, transitive and symmetric relation. — MindForged

That is a necessary but not sufficient condition for identity, assuming that identity means the same thing as '='. All equivalence relations are reflexive, transitive and symmetric. For example 'is the same age as' is an equivalence relation. But Arjun being the same age as Helga does not make them the same person.

The properties of reflexivity, symmetry and transitivity of '=' can be deduced from the above two axiom schemas, but they narrow it down more than just those three properties do.

Fair enough.

Well I didn't say these had to be assertions about reality. These can be understood purely formally and syntactically. — MindForged

Then why talk of "axioms" and "truth?" Purely formally/syntactically you can make up any old rules for moving symbols around in patterned ways, neither truth nor the assumption of truth have anything to do with it surely?

When you're doing logic, you're using these rules and axioms to derive other truths. That's what logical consequence is, no? Like, if I have "A & B" then I can deduce that "A".