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What is True? Does A = A?

1137

Here until I storm off again
Premium Member
How is it that I am mistaken. Axiom without question is a math term first used by Euclid in Geometry. Is there a use prior to Euclid?

You are confused: you seem like most people today not to know the difference between deductive logic and Mathematical-Logic. Aristotle used the term axiom ONLY when referring to geometry. Aristotle did not use axioms in his system whatsoever. You learned that from a math guy or computer science guy.

This is irrelevant, if you choose to ignore how axioms are presently understood then that's your problem.
 

Logikal

Member
Actually logic is the foundation of math, not the equivalent.


Okay , so why is it that you believe axioms always exist? I did not say YOU said that. I am asking if that is true or not.
Secondly, if something is sense verifiable how is it an axiom? Just asking and not suggesting YOU said this.
 

Logikal

Member
How do you know they're true? :D

How human KNOW something is true is changing the topic. How people usually know is sense verification (induction).
To say you have to be aware of x for x to exist is a scientific claim which doesn't make too much sense: there can be life on Jupiter but because YOU don't know you would tend to think the claim "there is life on Jupiter" has no truth value. That is a psychological issue, honestly.

If we discover life there tomorrow are you suggest all of a sudden the claim is true? What about a week ago was the claim NOT TRUE? Granted that the life has aged prior to you becoming aware of it. Of course it would be perfect timing for you discover life as soon as it is created.
 

lovemuffin

τὸν ἄρτον τοῦ ἔρωτος
I wound up asking him if A=A because I was wondering if he believes that logical axioms like that are true regardless of whether or not God or human is observing or not. He seemed to think you can't know that A=A without God. Whatever. Then his more reasonable friend came up to me and we had a really nice conversation. He said God is the precondition of knowledge, and I think there's something to that in a way.

Another question: "What is True?"
The exact question the guy asked me was actually somewhere between "What is True?" and "How do you ever know what's true?"

Forgive me in advance because I'm going to take a little bit to get to the point here. In order to get there I think I need to address 3 questions:

1. Defining "truth" and understanding some of the various theories of truth
2. Discussing the relation between truth and knowledge ("what is true?" in relation to "how do we know?")
3. Addressing the foundations of logical reasoning and the transcendental argument for God (the claim that God is a precondition for knowledge is the transcendental argument)

1. Defining truth

So, as you might guess, the word has been given a number of definitions and uses which are all kind of related but distinct. I'm going to suggest that the most intuitive theory of the nature of truth, and the one generally assumed by your interlocutor, is called the "correspondence theory" of truth, which deals with the truth of propositions, where a proposition P is a sentence like "the sky is blue". In the correspondence theory of truth we might say:

A proposition P is true if and only if the state of affairs described by P exists in reality
That is, the truth of P depends on the correspondence between P and the world.

2. Truth and Knowledge

In and of itself, correspondence theory is silent about epistemology, i.e the question "how do we know the truth?" It draws a distinction between "true" and "known", and while that distinction is useful, in practice of course the question of truth is tightly connected to the question of knowledge. This tension between the two polls also exists in epistemology, where the classical definition of "knowledge" is "justified, true, belief". But just as with the correspondence theory, the pressing issue is that justification really is everything, since apart from a justification we have no means of deciding whether the belief is "true" or not.

In modern usage, when we're using a correspondence theory of truth we are generally also using an epistemology that emphasizes empirical evidence, as in science, because the history of science and philosophy has demonstrated that scientific/empiric methodology is the most appropriate for attaining knowledge about the world.

One of the primary distinctions between correspondence theory and other theories of truth is that most of the others take into account epistemological concerns directly. That is, coherence theory says that the criterion of truth is the coherence of P with the set of other true propositions. The objection of course is that some set of beliefs might be coherent but completely disconnected from reality, but the value of coherence theory is that coherence is really an epistemological justification. We are more justified to believe that P if P is coherent with other things we know. Other theories take into account that we would like to use the word "true" to describe things that aren't easy to reduce to questions about the physical world which can be given decisive scientific answers. Social construction theories of truth for example deal with the kinds of knowledge that arises relative to particular cultures and societies, where the body of knowledge is not really about the objective physical world. Pluralism with regard to truth recognizes that we may need different epistemologies and different criteria for truth in different contexts.

In any case, this is a bit of a sideline, but I think it's useful to understand how truth and knowledge are tightly connected. Mainly, the question put to you was concerned with correspondence theory.

2a. "Truth" as a transcendental

One final note before (3). I've been using "true" as a word that describes a property of some particular proposition P. The other word we encounter is "Truth" with a capital-T, which is supposed to represent the kind of overarching unity of truth. "Truth" is somehow more than just the collection of truths. This idea owns in large part to certain foundational laws of classical logic which I'll address in (3). Essentially, the law of non-contradiction seems to require that all truths be compatible, as with the coherence theory of truth. Because A and not-A can't both be true. This intuition about truth becomes important to the transcendental argument for God that was put to you.

3. Foundations of Logic and the TAG

"God is a precondition of knowledge" is an argument of presuppositional apologetics, sometimes called the transcendental argument for God. I believe the first version of it owes to Immanuel Kant. In order to understand the argument we have to talk about the foundational rules of classical logic. There is more than one formulation of classical logic but I'll just deal with the most common ones, which go back to Aristotle more or less:

1. The law of identity (A=A)
2. The law of non-contradiction (not (A and not-A) -- A and not-A can't both be true)
3. The law of the excluded middle (a proposition is either true or false, there is no middle ground)
We call them first principles of logic precisely because they are taken as assumptions, as principles which are self-evidently true. Or, if we're being more cautious, we simply say that we assume them because we can't get anywhere with classical logic without assuming them, and we can't figure out how to derive them from any more fundamental principles. The important part, from the perspective of TAG, is precisely that they are presuppositions, i.e things we hold to be true despite a lack of demonstration within the logical systems we build on top of them.

The transcendental argument, then, asserts that in assuming the truth of the foundational laws, we already implicitly assume the existence of God. The argument hinges on the concept of God as a kind of ultimate ground of everything. It's similar in that way to the cosmological argument I referenced before. The cosmological argument recognizes a possible infinite regression of causes if everything in reality requires a cause, and declares the sui generis uncaused first cause to be God. The transcendental argument recognizes a possible infinite regression of justification in logic and epistemology, and declares the un-justified sui generis laws of logic to depend on that same God.

Another way of expressing the transcendental argument is more abductive. We recognize that the reason logical reasoning is useful is because for some reason the world has a kind of logical structure. It is intelligible in this way, and demonstrates patterns that are amenable to our thinking. The transcendental argument asserts that without God we have no reason to believe that we should have expected the world to be like that.

The problem with TAG

As I said in my first response, I think the main issue is that the "God" of the transcendental argument is basically content-less. It just gives a name to a kind of property that is supposed to be necessary (the property of being the terminus of an otherwise infinite regress) but that property, even if we agree it must exist, doesn't imply any other property that is usually attributed to God. In the case of the cosmological argument, we might accept that the principle of sufficient reason (everything must have a cause) breaks down, and that there must be some existence which is just a brute fact, or is self-existent, but we have no reason to call that thing "God" rather than just "universe". The same applies to the TAG.

In fact, as an abductive argument the opposite of TAG might be stronger than TAG. The success scientific methodology in understanding the world, including the intelligibility and ordered-ness of the world would rather seem to imply that it's unlikely that the world was created by an omnipotent and anthropomorphic will, which, considering human will, we might expect would have led to a much less "logical" reality than the one we find. Beyond that, there is of course the fact that nowhere does the logic or methodology of science require any consideration of a deity or of the value of sacred texts or traditions. It's probably worth noting that many classical theologies end up de-anthropomorphizing their conceptions of God for reasons related to the first, but as a whole the intelligible order of the natural world would seem to do more to call into question traditional religious dogmas than to confirm them.

What is true? How do we know?

Going back to the discussion about truth and knowledge, I tend to think the pluralist critique of the idea of a single, overarching and absolute truth is reasonable. Because truth and knowledge are so intricately connected, the idea of an absolute and objective truth entirely disconnected from "how do we know?" isn't useful at all. It's an abstraction that can't really exist, at least for us, because we have no access to it. We might reject this idea of truth as excessively metaphysical in the same way Buddhism is skeptical of metaphysics. The idea of absolute certainty about knowledge is likewise less than useful.

In practice, the "how?" is more important, and the best way we've found to arrive at truth is also the best way to make our knowledge more reliable. That is, by intelligently applying criteria that are relevant to the kind of question we are asking, by being humble and aware of our own limitations and cognitive biases, by being self-critical and skeptical, and by looking for objective and repeatable ways of testing hypotheses where those tests are possible.
 

lovemuffin

τὸν ἄρτον τοῦ ἔρωτος
In deductive logic the law of identity is not an axiom. Axiom is a Math term. Aristotles's logic begins with NO AXIOMS. The law of identity is sense verifiable --That is how we KNOW it to be true.

It's true that axiom is a technical term in formal symbolic logic or in math, but it's also the case that the word has a descriptive meaning that very much applies to principles like the law of non-contradiction. The word axiom comes from the Greek αξιος meaning "worthy". Descriptively it means a premise that is taken to be self-evident. "Self-evidence" is basically what you are referring to when you say there is some evidence for the law of identity. But the kind of evidence that distinguishes the law of identity from other kinds of logically valid conclusions is essentially what makes it "axiomatic" in the descriptive sense of the word. I'm not sure whether it's anachronistic to use the word axiom this way, but it's quite common and I don't think there's anything wrong with it. You can substitute the word "foundational" for "axiomatic" if you want.
 

lovemuffin

τὸν ἄρτον τοῦ ἔρωτος
'm not sure whether it's anachronistic to use the word axiom this way, but it's quite common and I don't think there's anything wrong with it. You can substitute the word "foundational" for "axiomatic" if you want.

My Greek isn't quite good enough yet to read it easily, but Aristotle uses axiom in the Posterior Analytics, and not just in reference to mathematics, and not just as a choice of the translator. The greek word is in the text.

http://www.hs-augsburg.de/~harsch/graeca/Chronologia/S_ante04/Aristoteles/ari_a211.html
http://classics.mit.edu/Aristotle/posterior.1.i.html

I call an immediate basic truth of syllogism a 'thesis' when, though it is not susceptible of proof by the teacher, yet ignorance of it does not constitute a total bar to progress on the part of the pupil: one which the pupil must know if he is to learn anything whatever is an axiom. I call it an axiom because there are such truths and we give them the name of axioms par excellence. (I,2)

The text is a bit obscure until you realize axiom means "worthy", i.e they are worthy premises par excellence because they are worthy of being considered self-evident.
 

Logikal

Member
It's true that axiom is a technical term in formal symbolic logic or in math, but it's also the case that the word has a descriptive meaning that very much applies to principles like the law of non-contradiction. The word axiom comes from the Greek αξιος meaning "worthy". Descriptively it means a premise that is taken to be self-evident. "Self-evidence" is basically what you are referring to when you say there is some evidence for the law of identity. But the kind of evidence that distinguishes the law of identity from other kinds of logically valid conclusions is essentially what makes it "axiomatic" in the descriptive sense of the word. I'm not sure whether it's anachronistic to use the word axiom this way, but it's quite common and I don't think there's anything wrong with it. You can substitute the word "foundational" for "axiomatic" if you want.

You are mistaken! The so called logical laws are SENSE VERIFIABLE!! You cannot show any of them to be false WITHOUT CHANGING THE CONTEXT of the said laws. Axiom has always been a MATH term. Never a logical term. Euclid was the first known writing to use such a term. When Aristotle used the term AXIOM he was talking specifically about GEOMETRY which is of course MATH. Can you demonstrate Aristotle using the term AXIOM without talking about MATH-- That is the question?
 

Logikal

Member
Forgive me in advance because I'm going to take a little bit to get to the point here. In order to get there I think I need to address 3 questions:

1. Defining "truth" and understanding some of the various theories of truth
2. Discussing the relation between truth and knowledge ("what is true?" in relation to "how do we know?")
3. Addressing the foundations of logical reasoning and the transcendental argument for God (the claim that God is a precondition for knowledge is the transcendental argument)

1. Defining truth

So, as you might guess, the word has been given a number of definitions and uses which are all kind of related but distinct. I'm going to suggest that the most intuitive theory of the nature of truth, and the one generally assumed by your interlocutor, is called the "correspondence theory" of truth, which deals with the truth of propositions, where a proposition P is a sentence like "the sky is blue". In the correspondence theory of truth we might say:

A proposition P is true if and only if the state of affairs described by P exists in reality
That is, the truth of P depends on the correspondence between P and the world.

2. Truth and Knowledge

In and of itself, correspondence theory is silent about epistemology, i.e the question "how do we know the truth?" It draws a distinction between "true" and "known", and while that distinction is useful, in practice of course the question of truth is tightly connected to the question of knowledge. This tension between the two polls also exists in epistemology, where the classical definition of "knowledge" is "justified, true, belief". But just as with the correspondence theory, the pressing issue is that justification really is everything, since apart from a justification we have no means of deciding whether the belief is "true" or not.

In modern usage, when we're using a correspondence theory of truth we are generally also using an epistemology that emphasizes empirical evidence, as in science, because the history of science and philosophy has demonstrated that scientific/empiric methodology is the most appropriate for attaining knowledge about the world.

One of the primary distinctions between correspondence theory and other theories of truth is that most of the others take into account epistemological concerns directly. That is, coherence theory says that the criterion of truth is the coherence of P with the set of other true propositions. The objection of course is that some set of beliefs might be coherent but completely disconnected from reality, but the value of coherence theory is that coherence is really an epistemological justification. We are more justified to believe that P if P is coherent with other things we know. Other theories take into account that we would like to use the word "true" to describe things that aren't easy to reduce to questions about the physical world which can be given decisive scientific answers. Social construction theories of truth for example deal with the kinds of knowledge that arises relative to particular cultures and societies, where the body of knowledge is not really about the objective physical world. Pluralism with regard to truth recognizes that we may need different epistemologies and different criteria for truth in different contexts.

In any case, this is a bit of a sideline, but I think it's useful to understand how truth and knowledge are tightly connected. Mainly, the question put to you was concerned with correspondence theory.

2a. "Truth" as a transcendental

One final note before (3). I've been using "true" as a word that describes a property of some particular proposition P. The other word we encounter is "Truth" with a capital-T, which is supposed to represent the kind of overarching unity of truth. "Truth" is somehow more than just the collection of truths. This idea owns in large part to certain foundational laws of classical logic which I'll address in (3). Essentially, the law of non-contradiction seems to require that all truths be compatible, as with the coherence theory of truth. Because A and not-A can't both be true. This intuition about truth becomes important to the transcendental argument for God that was put to you.

3. Foundations of Logic and the TAG

"God is a precondition of knowledge" is an argument of presuppositional apologetics, sometimes called the transcendental argument for God. I believe the first version of it owes to Immanuel Kant. In order to understand the argument we have to talk about the foundational rules of classical logic. There is more than one formulation of classical logic but I'll just deal with the most common ones, which go back to Aristotle more or less:

1. The law of identity (A=A)
2. The law of non-contradiction (not (A and not-A) -- A and not-A can't both be true)
3. The law of the excluded middle (a proposition is either true or false, there is no middle ground)
We call them first principles of logic precisely because they are taken as assumptions, as principles which are self-evidently true. Or, if we're being more cautious, we simply say that we assume them because we can't get anywhere with classical logic without assuming them, and we can't figure out how to derive them from any more fundamental principles. The important part, from the perspective of TAG, is precisely that they are presuppositions, i.e things we hold to be true despite a lack of demonstration within the logical systems we build on top of them.

The transcendental argument, then, asserts that in assuming the truth of the foundational laws, we already implicitly assume the existence of God. The argument hinges on the concept of God as a kind of ultimate ground of everything. It's similar in that way to the cosmological argument I referenced before. The cosmological argument recognizes a possible infinite regression of causes if everything in reality requires a cause, and declares the sui generis uncaused first cause to be God. The transcendental argument recognizes a possible infinite regression of justification in logic and epistemology, and declares the un-justified sui generis laws of logic to depend on that same God.

Another way of expressing the transcendental argument is more abductive. We recognize that the reason logical reasoning is useful is because for some reason the world has a kind of logical structure. It is intelligible in this way, and demonstrates patterns that are amenable to our thinking. The transcendental argument asserts that without God we have no reason to believe that we should have expected the world to be like that.

The problem with TAG

As I said in my first response, I think the main issue is that the "God" of the transcendental argument is basically content-less. It just gives a name to a kind of property that is supposed to be necessary (the property of being the terminus of an otherwise infinite regress) but that property, even if we agree it must exist, doesn't imply any other property that is usually attributed to God. In the case of the cosmological argument, we might accept that the principle of sufficient reason (everything must have a cause) breaks down, and that there must be some existence which is just a brute fact, or is self-existent, but we have no reason to call that thing "God" rather than just "universe". The same applies to the TAG.

In fact, as an abductive argument the opposite of TAG might be stronger than TAG. The success scientific methodology in understanding the world, including the intelligibility and ordered-ness of the world would rather seem to imply that it's unlikely that the world was created by an omnipotent and anthropomorphic will, which, considering human will, we might expect would have led to a much less "logical" reality than the one we find. Beyond that, there is of course the fact that nowhere does the logic or methodology of science require any consideration of a deity or of the value of sacred texts or traditions. It's probably worth noting that many classical theologies end up de-anthropomorphizing their conceptions of God for reasons related to the first, but as a whole the intelligible order of the natural world would seem to do more to call into question traditional religious dogmas than to confirm them.

What is true? How do we know?

Going back to the discussion about truth and knowledge, I tend to think the pluralist critique of the idea of a single, overarching and absolute truth is reasonable. Because truth and knowledge are so intricately connected, the idea of an absolute and objective truth entirely disconnected from "how do we know?" isn't useful at all. It's an abstraction that can't really exist, at least for us, because we have no access to it. We might reject this idea of truth as excessively metaphysical in the same way Buddhism is skeptical of metaphysics. The idea of absolute certainty about knowledge is likewise less than useful.

In practice, the "how?" is more important, and the best way we've found to arrive at truth is also the best way to make our knowledge more reliable. That is, by intelligently applying criteria that are relevant to the kind of question we are asking, by being humble and aware of our own limitations and cognitive biases, by being self-critical and skeptical, and by looking for objective and repeatable ways of testing hypotheses where those tests are possible.

This response is bit quick due to time but I can't go in detail like I would like to at this moment. I would disagree with the three common "logical laws" as axioms or "assumptions" I reserve those comments only to mathematicians after George Boole.
I would say that the three logical laws are sense verifiable. The issue is that some people do not understand the concepts fully because they take the laws TOO literally. The law of non-contradiction for instance is taken the most literally I find. People who understand the concept understand the definition is way MORE than the short cut definition given. The more accurate definition is that nothing can be and not be at the same time and at the same place and in the same respect. This I would think is sense verifiable. There is a reason you cannot find counter examples. This would give reason to believe the proposition as true at least on the science level. The other laws are similar. It is not Philosophers (who specialize in logic or Epistemology) that come straight out and say "these are assumptions".

The truth correspondence theory is exactly what science does. This is practical. But my point is sort of on the meta physical level and not just on the practical side alone. The objective truth is not the same as correspondence theory. All literal meaningful declarative statements are sense verifiable. Maybe right this moment but potentially there could be a possibility for some form of detection at some point now or at some time in the future. If I say "there is life on Pluto" this claim I may not be aware of the truth of the claim but this does not negate the claim HAS a TRUTH VALUE. The same way many people say "O.J. did commit murder" this claim's truth value may be between God and O.J. but still has an independent truth value. If there is no God the same claim holds an independent truth value. Because you and I may not be SURE which truth value applies is an excuse to doubt things that are non-practical and does not affect an objective truth value. Objective truths are specific truths and are not emotional or subjective. For the Practical / emotion thinking most humans are programmed to do love for God requires something DIFFERENT. There is more than practical clearly with nearly all religious theories.
 
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raw_thought

Well-Known Member
The correspondence theory of truth (from now on referred to as CTT) has many self contradictions. One for example is,
The CTT defines truth as a proposition that corresponds to a reality. If the CTT is true,by its own definition it must correspond to a reality. What is that reality? Another CTT...infinite regress, reminiscent of Plato's third man argument.
 

Logikal

Member
The correspondence theory of truth (from now on referred to as CTT) has many self contradictions. One for example is,
The CTT defines truth as a proposition that corresponds to a reality. If the CTT is true,by its own definition it must correspond to a reality. What is that reality? Another CTT...infinite regress, reminiscent of Plato's third man argument.

That is because the correspondence theory of truth is a science approach but tries to hide that it is science. Science is meant to be practical. Millions of human like zombies have been programed to be "practical" and many people believe the infamous saying "if it is not practical, it makes no sense" or they will say "if it is not practical it is useless". The minority, as it turns out, are the few millions of people with higher education and would NEVER say such a thing: after all there are many UNPRACTICAL degrees one can earn in a University! The majority in this light do not have higher education.
 

lovemuffin

τὸν ἄρτον τοῦ ἔρωτος
You are mistaken!

I don't believe so. Can you demonstrate that I am?

When Aristotle used the term AXIOM he was talking specifically about GEOMETRY which is of course MATH. Can you demonstrate Aristotle using the term AXIOM without talking about MATH-- That is the question?

Asked and answered:


Beyond that, I'll cite the Stanford Encyclopedia of Philosophy:

Aristotle says that PNC [Principle of Non-Contradiction] is one of the common axioms, axioms common to all the special sciences. It has no specific subject matter, but applies to everything that is. It is a first principle and also the firmest principle of all.

http://plato.stanford.edu/entries/aristotle-noncontradiction/
 

lovemuffin

τὸν ἄρτον τοῦ ἔρωτος
The correspondence theory of truth (from now on referred to as CTT) has many self contradictions. One for example is,
The CTT defines truth as a proposition that corresponds to a reality. If the CTT is true,by its own definition it must correspond to a reality. What is that reality? Another CTT...infinite regress, reminiscent of Plato's third man argument.

Your argument doesn't demonstrate the inconsistency of CTT (or that it is contradictory) but rather that it might be said to be incomplete. I tried to discuss this briefly in my longer post. That is, the proposition "Is the CTT true?" might not be a truth-apt proposition under CTT, because "CTT" doesn't have the kind of ontological status that would allow it to be truth-apt within its own rules, if we're being strict about them. One of the reasons I mentioned alternative theories of truth and the challenges of epistemology is because I agree that CTT is not the beginning and end of the question. I also agree that CTT is mostly useful in empirical science. In any case, the reason I focused on it is because I think the argument being put to the OP about God as a precondition of knowledge -- and the relation of that argument to questions of "truth" -- is easiest to talk about in the context of CTT. It's the right place to start in my opinion.

It's interesting to me that the argument that CTT entails a contradiction depends on the assumption of a transcendental "Truth" as I mentioned in (2a). It becomes contradictory when you make it so that any theory of truth must be absolute. But a kind of pluralism would allow us to evaluate "true" as a predicate with different criteria in different contexts. CTT is "true" in the sense of being perhaps the best theory of truth for the domain it covers, which is certainly a very important domain! But I would agree with you that it's not everything.

There is an interesting blog from a philosopher from last year that speaks in part about some of these epistemological issues and the problem of reducing truth and epistemology to an overly simplistic kind of scientism: http://www.philosophersbeard.org/2014/04/why-i-am-not-atheist.html (see from the heading: "the epistemological problem"...)
 

Logikal

Member
I don't believe so. Can you demonstrate that I am?



Asked and answered:



Beyond that, I'll cite the Stanford Encyclopedia of Philosophy:


I am not sure what You would accept as proof. You did not address that axioms are not required at all to learn deductive logic. That is where math came in. And why people think logic is about assumptions and one must start with assumptions. This is simply FALSE. Can you provide any sources before BOOLE where axioms are stressed? You did not address that.
 

lovemuffin

τὸν ἄρτον τοῦ ἔρωτος
I never claimed that axioms were a requirement to learning logic. I claimed that the nature of the first principles of logic were important to understand in order to make sense of the transcendental argument for the existence of God. That said, I would certainly claim that even classical logic (over against formal symbolic logic or math) depends upon assertions taken as self-evident. I've already provided a source prior to Boole where this is stressed: Aristotle's Posterior Analytics. He also develops the same idea in the Metaphysics.

I would also claim that as soon as one wants to learn about the epistemological foundations of logic in philosophy that one is going to encounter those first principles, even though you can learn logic without them. That's why they are important to TAG; TAG is an argument about the epistemological foundations of logic. I'm not primarily concerned with whether or not we call those first principles axioms, but I think I've amply demonstrated that your insistence that "axiom" is only a mathematical term is false. It is both a technical term in mathematics as well as a descriptive term in Aristotle's development of logic, which I think is certainly the most important classical system of logic.

As far as my request to you to provide an argument to support your claim, I would accept any academic source or classical philosophical text which supports the assertions you've made, namely

a) it is inappropriate to use the term axiom to refer to the law of identity
b) the law of identity is not axiomatic in the sense of "self-evidently true", but is "sense verifiable". (please explain what "sense verifiable" means. induction?)
c) 50 years ago you would never say deductive logic has axioms (I believe I've disproven this in citing Aristotle)
d) Aristotle used the term axiom only in relation to mathematics (I believe I've disproven this also in citing Posterior Analytics)
 
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