Jump to content
Objectivism Online Forum

"A priori" in the scientific vs. philosophic case?

Rate this topic


Recommended Posts

I think you are confusing a priori/ a posteriori and analytic/synthetic.

Kant thought it was possible for something to be any combination of these two. In other words he also thought there was such a thing as synthetic a priori knowledge. In fact his entire metaphysics depends on the existence of such.

Link to comment
Share on other sites

  • Replies 76
  • Created
  • Last Reply

Top Posters In This Topic

How would you prove that A is A? Is the axiomatic nature of the proposition enough, or would you do more to prove it? Or, are you saying that it is just "obvious"?

A proof is an argument consisting of several connected statements leading to a conclusion. Axioms by their nature cannot be proved, but I wouldn’t call A is A an axiom. It’s a tautology, and nor is there any need to “prove” a tautology, since it’s a statement that is justified by reference to the meaning of the component terms.

Tautologies are logical rather than factual truths. For A is A, you can substitute anything for A -- table, chair, rainbow, unicorn – and it will always be at least valid, regardless of factual content.

Eddie

Link to comment
Share on other sites

Re previous post:

In formal logic there are two kinds of axioms: logical axioms and non-logical axioms. Logical axioms are logically true, which is to say, roughly, that they're true by their form. The truth (or possibly even falsehood) of non-logical axioms is not ensured by their form alone. For example 'A or not A' could be a logical axiom since it is true no matter the value of A, which is to say that it is true by its form alone. On the other hand, for example, a field axiom such as 'x + y = y + x', is not a logical axiom, since the symbol '+' can stand for a non-commutative operation different from the commutative operation of field addition and the domain might not be that of a field, so 'x + y = y + x' is not logically true, though it is true if the domain is a field and '+' stands for field addition.

But, in a technical sense, it is not true that an axiom (logical or non-logical) cannot be proven. Though it's a trivial fact, it is still a fact that an axiom can be proven simply by entering it on a line of a proof. And in another sense that is not at all trivial, if the independence of a set of axioms has not been proven, then it might not be known whether or not one of them is provable from other axioms.

As to tautologies (in the specific sense of tautologies of sentential logic), there are two different approaches that sentential logic can use (while both approaches, if performed properly, yield equivalent results): 1) We can adopt all tautologies as axioms. This is justified since there is an effective method for checking whether a sentence is a tautology, or 2) We can adopt some proper subset of the set of tautologies to serve as axioms and then prove additional tautologies from our axioms; and if we choose a strong enough set of axioms, then all tautologies will be provable.

And it is not accurate to say that tautologies are "true by the meaning of the component terms." On the contrary, tautologies are always true no matter the assignments of meaning given to the terms (here I'm going along with the poster's own loose, but reasonable, use of 'term'). Granted, though, that the meaning of the sentential connectives themselves are presumed to be fixed (and thus are usually not referred to as 'terms') and that these meanings do account for the logical truth of tautologies.

Edited by LauricAcid
Link to comment
Share on other sites

But, in a technical sense, it is not true that an axiom (logical or non-logical) cannot be proven.

Strictly speaking, yes, but I was trying to relate my comments to the way Ayn Rand uses the term axiom, which as I understand it is a claim about some fundamental aspect of the world, rather than a proposition within the context of a proof.

As for the axiomatic status of “A is A”, or more accurately, A=A, that type of statement can also only be regarded as an axiom within the context of a proof. But Rand doesn’t seem to be using it in that way. She’s using it more in the sense of a self-evident proposition, so in that sense it is a tautology.

But I take your point. It would have been more accurate to say that an axiom is a statement for which no proof is required, and that a tautology is true regardless of the meanings of the component terms (assuming the connectives are fixed).

Eddie

Link to comment
Share on other sites

A proof is an argument consisting of several connected statements leading to a conclusion. Axioms by their nature cannot be proved, but I wouldn’t call A is A an axiom. It’s a tautology, and nor is there any need to “prove” a tautology, since it’s a statement that is justified by reference to the meaning of the component terms.

Tautologies are logical rather than factual truths. For A is A, you can substitute anything for A -- table, chair, rainbow, unicorn – and it will always be at least valid, regardless of factual content.

[...]

As for the axiomatic status of “A is A”, or more accurately, A=A, that type of statement can also only be regarded as an axiom within the context of a proof. But Rand doesn’t seem to be using it in that way. She’s using it more in the sense of a self-evident proposition, so in that sense it is a tautology.

A proof is not an argument. Proof consists of reducing a concept, step-by-step through the hierarchy of knowledge, to the sense-perceptions from which it was formed. Argument is an aid in that process, the more important as the the concept to be proved is the more complex.

A is A is an axiom, and is means "a thing is itself", or "a thing is neither more nor less than what it is", or "a thing has only the properties which it has, and has none of the properties which it does not have". It's not a tautology, since there is no such thing as tautologies. A is A is not justified by reference to the double appearance of the letter A, but by reference to all of the facts of reality from which one forms the concept. Moreover, A is A is not to be "resolved" to "this table is this table", "this other table is this other table", "that table is that table", "this first chair is this first chair", etc. A is not a variable to be "plugged in" or substituted.

Lastly, A is not equivalent to or equal to A: it is A. Things are not equal to themselves: they are themselves. Equation and equivalence are mentally grasped relationships between two measurements. Identity is a property of entities, of A.

Edited by y_feldblum
Link to comment
Share on other sites

All of what you wrote may pertain in a particular philosophy of informal logic, but hardly any of it holds in virtually any professional discussion of formal logic.

Perhaps it is best to agree that there are at least two separate fields, each with their own terminological conventions: Objectivist logic and formal logic. Since the field of formal logic offers rigorous, precise, and fruitful systems for these notions, we benefit from understanding formal logic. Therefore, these contrasts with your stipulations need to be made:

"A proof is not an argument."

'proof' has a precise mathematical definition. 'argument' is an informal synonym of 'proof'.

"Proof consists of reducing a concept, step-by-step through the hierarchy of knowledge, to the sense-perceptions from which it was formed."

The primitive terms of an axiomatic system may or may not be motivated or understood by sense perceptions, but a proof is not allowed to appeal to those motivations or understanding. An explanation of a proof, to make the proof easier to comprehend, may appeal to those motivations or understandings, but the proof itself is independent of them. Though, to be fair, Kleene, for example, does mention Euclid's system as an example of material axiomatics, which is close enough to the notion of axioms rooted in sense perception. But I gather that mathematicians have for a long time recognized that Hilbert's formal axiomatization of Euclidean geometry is needed. If there are mathematicians who reject Hilbert's formalization, then, sincerely, I'd like to know more about that.

"A is A is an axiom, and i[t] means "a thing is itself", or "a thing is neither more nor less than what it is", or "a thing has only the properties which it has, and has none of the properties which it does not have"."

In identity theory, the axiom is of the form: A = A.

And an additional axiom is required to assert that if B = A and A has property P, then B has property P (roughly speaking, without mentioning technicalities of free variables, there is an axiom schema that covers well formed formula "expressing" a property).

As to A not having properties that it doesn't have, that is not expressible in first order logic but is a theorem in second order logic.

"[A is A] It's not a tautology, [...]"

In most usual usage, formal logic agrees that 'A = A' is not a tautology.

"since there is no such thing as tautologies."

Tautologies are well formed formulas of propositional logic that map to the value 'true' for all assignments of the propositional letters. Tautologies exist.

"A is A is not justified by reference to the double appearance of the letter A, but by reference to all of the facts of reality from which one forms the concept."

'A = A' is usually taken as a logical axiom because the symbol '=' is fixed as being mapped to the identity relation in any structure (and the identity relation is usually given this special dispensation in logic). The variable before and after '=' must indeed be the same variable else the formula is not an axiom.

"Moreover, A is A is not to be "resolved" to "this table is this table", "this other table is this other table", "that table is that table", "this first chair is this first chair", etc. A is not a variable to be "plugged in" or substituted."

If it's not a variable, then what is it? The letter 'A' is used in the formulation of the axiom, so the letter 'A' is a linguistic object, since letters are linguistic objects. If it is not a variable, then what linguistic object is it in the axiom?

Also, I don't know the meaning of 'resolved' you have, but each of the above examples is an instantiation of the axiom.

"Lastly, A is not equivalent to or equal to A: it is A."

The letter 'A' is equal to the letter 'A'. And the values of the letter 'A' are each equal to themselves.

"Things are not equal to themselves: they are themselves. Equation and equivalence are mentally grasped relationships between two measurements. Identity is a property of entities, of A."

Equality and equivalence are different relations from each other. And neither is confined to measurement.

Identity is usually stated as a property held by certain ordered pairs. But one could take identity to be a unary predicate, and formal logic has no problem with fixing a functional constant, say, 'I', to stand for 'A = A' so that I(A) is true in identity theory. But that cannot supplant the binary predicate, since there would be no way to express that B is not A.

Edited by LauricAcid
Link to comment
Share on other sites

I wrote:

'argument' is an informal synonym of 'proof'.

More accurately I should say:

'argument' is an informal synonym of 'purported proof'. And an argument is a proof if it is not just a purported proof but rather is indeed a correct argument (a correct argument being a proof).

Link to comment
Share on other sites

A is A is not justified by reference to the double appearance of the letter A, but by reference to all of the facts of reality from which one forms the concept.

And what are these facts of reality? They clearly cannot be such facts as: "a thing is itself", or "a thing is neither more nor less than what it is", or "a thing has only the properties which it has, and has none of the properties which it does not have". These are merely synonyms for A is A, and to appeal to these facts would be to beg the question.

You also seem to have ruled out appeals to objects, such as “this table”, so I’m not sure which “facts of reality” you have in mind. You may wish to clarify what you mean here.

Eddie

Link to comment
Share on other sites

Everything that exists. (This answer also explains why A is A is an axiom of metaphysics.)

Is the statement “everything that exists” a fact of reality, or is it that all existing things are facts of reality? Previously, you said “Moreover, A is A is not to be "resolved" to "this table is this table"…” So you clearly believe that some facts of reality have nothing to do with A is A.

In that case your claim that A is A is justified by reference to all of the facts of reality cannot be the case. Why don’t you present an argument – ie a series of connected statements – that demonstrates your understanding of this subject? Show how A is A is justified by reference to “everything that exists”.

Eddie

Link to comment
Share on other sites

The question was what facts of reality support the proposition A is A.

The answer given was: 'everything that exists'. But the expression 'everything that exists' is a not a proposition since it has only a noun phrase but nothing predicated of it. So the expression 'everything that exists' cannot assert or even describe a fact of reality.

I surmise that the idea is that it is a fact of each existent that it exists, but to be an existent is to have an identity, which is to say that every exisistent is itself and nothing else. So no existent can violate the law of A is A. So everything that exists is evidence, in its existence itself, of the law.

However, why one would assert that the propostion A is A is not instantiated (if that's what 'resolved' means) by existents or why the proposition is not true by its form is beyond me.

Edited by LauricAcid
Link to comment
Share on other sites

Everything that exists - get ready for this! - is what it is. And there you have the law of identity, or A is A.

It is an axiom of metaphysics partly because it is a concept which subsumes everything. Similar axiomatic concepts, or concepts which are corollaries of axioms, include the law of existence and the law of causality.

I say that A is A is not a mathematical expression in one variable in to which one is supposed to plug concrete objects, thus yielding an infinite set of mathematical expressions in no variables, such: {"this table is this table", "this other table is this other table", "this table over there is this table over there", and so on indefinitely}. Contrariwise, A is A is a means of expressing a concept with indefinitely many referents, and, in its capacity as a concept, serves to reduce an indefinite number of units down to one.

Link to comment
Share on other sites

Everything that exists - get ready for this! - is what it is. And there you have the law of identity, or A is A.
And don't you agree that every existent, A, has the property that A is identical with itself? Or do your other remarks imply that the letter 'A' does not refer to existents nor to indefinitely many of them?

And don't you agree that for every existent to be what it is requires that every existent is different from every other existent?

[...] A is A is a means of expressing a concept with indefinitely many referents, and, in its capacity as a concept, serves to reduce an indefinite number of units down to one.

What are the referents of the concept?

To what does 'its' refer? The expression 'A is A' or the concept it expresses?

What are the units? The referents?

Edited by LauricAcid
Link to comment
Share on other sites

Everything is, identically, what it is. But are you still not clear on the purpose and use of the phrase A is A?

There referents of the law of identity are, as I said, everything. "Its" refers to the law of identity, aka A is A.

Were you actually not clear on my position on any of these mundane question?

Link to comment
Share on other sites

'A is A' is perfectly clear to me. It's an axiom and it must be true. But not the way you handle it.

It was important to be very clear as to 'its' since we need to not conflate an expression with what it expresses. So your answer does the job in that regard.

However, you say the referents of the law are everything. But if the letter 'A' in the expression stating the law is not a variable, then it must have a referent. The letter 'A' and the law are not the same thing. And if the referent of 'A' is something called 'everything', then the referent of the letter 'A' is the same thing as the referent of the law, in which case that thing that is called 'everything' is identical with the concept that is expressed by the expression 'A is A'. So that thing that is called 'everything' is the law that everything is itself, which is absurd.

Ah, but one might say, "Everything is not a thing." Fine, but then whatever is meant to be conveyed by the word 'everything' can't be a referent. And by the argument in the previous paragraph, 'A' must be a variable.

But if you let 'A' be a variable ranging over things in a universe, even allowing that the universe is not a thing, then 'A is A' stands as expressing an incontestable truth that is a prerequiste for all rational thought.

Edited by LauricAcid
Link to comment
Share on other sites

Contrariwise, A is A is a means of expressing a concept with indefinitely many referents, and, in its capacity as a concept, serves to reduce an indefinite number of units down to one.

As I understand it, A is A in Objectivist parlance expresses the concept of identity, and a concept ultimately refers to or means all its referents in reality. In the case of “identity”, that means all tables, chairs, trees, mountains etc. The validity of the concept is also justified by “reducing” it to the observation of specific things, or existents.

In that case, “A is A” ultimately refers to each and every existing thing, in which case it seems quite acceptable to say “A is A” can be expressed as “this table is this table”.

Eddie

Link to comment
Share on other sites

A is not the concept. A is A is the concept. The former does not refer to anything at all. The latter refers to everything that exists.

The treatment of A as a variable which ranges over and can take on all the values in the set of everything that exists is absurd. A is not a variable, and does not range over anything, and does not take on values. A is A is another name for the law of identity. It's shorter than saying "it is what it is". That's it. The law of identity does not depend on a knowledge of algebra, set theory, or any other math.

Link to comment
Share on other sites

It would be fine to say that 'A is A' is just an informal handle for, say, one of these:

Every entity is itself.

All things are themselves.

Every thing is what it is.

Etc.

But you're expressing that in a language. And so the examples above have pronouns: 'itself', 'themselves', 'it', etc. Those pronouns have referents. Or the referents of the pronouns are left undetermined but the pronoun is understood to stand for different things in some range of things.

'If something is crisp, then Jack likes it' provides that we can refute the sentence if we can plug in something, say a certain crisp apple, as a referent for 'it' and show that Jack in fact doesn't like that apple. One doesn't have to know mathematics to understand that.

But you say that taking 'A' as a variable is absurd. If it were absurd, then you could prove that by showing a contradiction. But you can't. You might show a contradiction with some other premises you hold, but the reasoning you've posted so far shows lack of recognition of certain basic implications of the fact that you are expressing laws with sentences that have words that refer.

Edited by LauricAcid
Link to comment
Share on other sites

I cant even fathom what meaning 'A is A' (or 'everything that exists, exists"/'all things are themselves' etc) could have other than a quantificational once. Obviously you dont need to know maths/logic to understand 'A is A', but that doesnt affect the logical form of the statement.

Link to comment
Share on other sites

This thread made my brain bleed.

I thought an Axiomatic concept was an irreducible primary that couldn't be argued against without some reference to it agreeing with its fundamental premise.

And,

In the "Internet Encyclopedia of Philosophy" A Priori is defined as; "A proposition is knowable a priori if it is knowable independently of experience."

I cant think of a thing I know independently of experience. Whether its knowing that particular proposition without experiencing it or knowing ANY proposition with no experience of anything.

Link to comment
Share on other sites

I agree that the entire concept of "A priori" (when defined like that, as Kant did) is comically wrong.

The only things that could be considered "A priori" (using this definition) is the sum total of of the knowledge of a person born into a sensory deprivation chamber. I think it is obvious that that sum equals 0.

Kant on the other hand claims that math, geometry, and all sorts of other things can be known "A priori".

Link to comment
Share on other sites

I've read Kant, and I don't really know how he comes to that conclusion either. Part of the problem is that he has many very different (and very contradictory) definitions of A priori. He says that all things which must be true are A Priori, but also that A Priori is knowledge gained without any experience. He freely switches back and forth between these definitions without making note of the fact.

I think that he somehow equivocated the fact that since a triangle must have three sides, it is then somehow known without any link to experience.

Link to comment
Share on other sites

  • 2 months later...

Punk defines the a posteriori as "something that has been directly empirically observed". But why "directly"?

As far as I know all philosophers who write about a posteriori knowledge equate it with empirical knowledge. And even believers in the a priori believe that both of Punk's examples are a posteriori (empirical).

However, it is true that many of those who believe that some knowledge must be a priori support their position by defining "empirical knowledge" too narrowly. For example, many confine to knowledge from sensation or perception, thereby excluding from introspection or reflection.

Yes, Punk, the distinction goes back before Kant. As BurgerLau pointed out, it was originally a distinction between arguments. Such was the medieval usage. By the time of Hobbes (1600s), it was being applied to statements (sentences, propositions, or the like) or knowledge of them.

What all empiricists, including, in the broad sense of that term, Rand, believe in is the doctrine known as "conceptual empiricism", which is the belief that all concepts are empirical. However, the doctrine known as "propositional empiricism", which is the belief that all knowledge is empirical, is much rarer. It seems to be held by J.S.Mill and Quine, and is held by Rand and Peikoff, as I understand them.

Those who hold the first doctrine but not the second seem to me to be in an indefensible position: they say that all true statements contain only empirical concepts, but some true statements---namely, those derived from empirical concepts---are not known empirically. Why? Because they are conceptual truths. However, I say that they are known empirically because the concepts are empirical. I believe that Rand and Peikoff would agree with this.

Believers in the a priori could object that, even if the truths allegedly known a priori, such as "All bachelors are unmarried", could be known a posteriori, they can still be known a priori as well, because no experience is needed to know them--once you understand the definition. But if you put enough information in the definition then, by this criterion, any truth could be known a priori.

I should also point out, Punk, that there is another usage of "a priori", which is more like your usage. In this sense it doesn't mean non-empirical. In this sense relying on one's general background knowledge is often called "a priori", even though the knowledge was gained through experience. However, Kant repudiated this usage, or something like it. Most philosopher's who talk of a priori knowledge use Kant's concept or something close to it.

Link to comment
Share on other sites

What all empiricists, including, in the broad sense of that term, Rand, believe in is the doctrine known as "conceptual empiricism", which is the belief that all concepts are empirical.  However, the doctrine known as "propositional empiricism", which is the belief that all knowledge is empirical, is much rarer.  It seems to be held by J.S.Mill and Quine, and is held by Rand and Peikoff, as I understand them.

I'm not sure whether I'd agree that "all concepts were empirical' for AR. Although she obviously disagreed with distinctions like a priori/a posterori and rejected innate knowledge, she did distinguish between axiomatic concepts, and things where were "implicit in experience", and more explicit empirical knowledge. I dont think AR would have considered 'existence exists' to be a statement of the type as 'Paris is the capital of France', because the former is a foundational statement which is presupposed by, and implicit in, all further knowledge. Although she didnt write much about mathematics, I cant imagine her agreeing with Mill's "truths of arithmetic are derived from experience" approach, and she would have opposed Quine on pretty much everything.

I cant even fathom what meaning 'A is A' (or 'everything that exists, exists"/'all things are themselves' etc) could have other than a quantificational once. Obviously you dont need to know maths/logic to understand 'A is A', but that doesnt affect the logical form of the statement.

I can't remember what was going through my head when I wrote this, but I now realise that its nonsense.

Edited by Hal
Link to comment
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Guest
Reply to this topic...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

Loading...
  • Recently Browsing   0 members

    • No registered users viewing this page.
×
×
  • Create New...