Can 'definition' be 'axioms'?

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humy
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Can 'definition' be 'axioms'?

In formal logic, can a 'definition' be (but not necessarily I presume?) or consist of one or more 'axioms'?

And can an 'axiom' itself have a 'truth value'? (albeit in a trivial way I presume)

Obvious Leo
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Re: Can 'definition' be 'axioms'?

Define "truth value".

humy
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Re: Can 'definition' be 'axioms'?

Obvious Leo wrote:Define "truth value".
either 'true' or 'false'.
By the definition of 'proposition' (in formal logic), all propositions are statements that have truth values in the sense that what they mean is either 'true' or 'false'.

https://en.wikipedia.org/wiki/Proposition
https://en.wikipedia.org/wiki/Truth_value

JSS
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Re: Can 'definition' be 'axioms'?

humy wrote:In formal logic, can a 'definition' be (but not necessarily I presume?) or consist of one or more 'axioms'?

And can an 'axiom' itself have a 'truth value'? (albeit in a trivial way I presume)
A definition is an explanation of the meaning of a word. It can be an incorrect definition, the wrong explanation, if the word is taken as belonging to a standard in the language. If the word is taken as a temporary usage for the duration of a document or discussion ("what this word means in this document") and is given by the author of the document, it cannot be incorrect, although perhaps a poor, possible ambiguous definition.

Every axiom, of any kind, is really saying:
IF this is true...

Axioms usually leave of the "IF" because they presume that there would be no argument. All axioms are presumed to be true before the rest of the argument has any meaning. If the axioms cannot be accepted as true, the argument is void, not necessarily false.

So
Yes a definition can be an axiom, and
Yes axioms have truth value.

And declarative definitions are always true (eg. "This is what I mean when I say this word").

humy
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Re: Can 'definition' be 'axioms'?

JSS wrote:
humy wrote:In formal logic, can a 'definition' be (but not necessarily I presume?) or consist of one or more 'axioms'?

And can an 'axiom' itself have a 'truth value'? (albeit in a trivial way I presume)
...

So
Yes a definition can be an axiom, and
Yes axioms have truth value.

And declarative definitions are always true (eg. "This is what I mean when I say this word").
Thanks for that JSS. I personally found that concise and informative and I believe it does answers my question. Very much appreciated (I really needed to know and get my facts completely straight on this for a very specific reason other than a purely academic one).

Obvious Leo
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Re: Can 'definition' be 'axioms'?

I also agree with JSS. Syllogisms in formal logic always take the IF/THEN form. IF we accept that proposition A is true THEN we can conclude that proposition B is also true. In this sense proposition A is accepted as conditionally true, which makes it either an axiom, postulate or theorem, depending on context. It cannot be said to have an absolute truth value in the true/false sense to which you refer.

humy
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Re: Can 'definition' be 'axioms'?

Obvious Leo wrote:I also agree with JSS. Syllogisms in formal logic always take the IF/THEN form. IF we accept that proposition A is true THEN we can conclude that proposition B is also true. In this sense proposition A is accepted as conditionally true, which makes it either an axiom, postulate or theorem, depending on context. It cannot be said to have an absolute truth value in the true/false sense to which you refer.
just curious; what is an "absolute truth value" as opposed to a truth value that isn't "absolute"?
Do you mean a truth value that is totally independent of the context that a statement, providing it doesn't contain both premise and inference, is said in? If so, I think I would have generally always agreed with that. Metaphorically speaking, context is everything here.

Obvious Leo
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Re: Can 'definition' be 'axioms'?

humy wrote:
Obvious Leo wrote:I also agree with JSS. Syllogisms in formal logic always take the IF/THEN form. IF we accept that proposition A is true THEN we can conclude that proposition B is also true. In this sense proposition A is accepted as conditionally true, which makes it either an axiom, postulate or theorem, depending on context. It cannot be said to have an absolute truth value in the true/false sense to which you refer.
just curious; what is an "absolute truth value" as opposed to a truth value that isn't "absolute"?
That's why I asked you to define what you understood by a truth value, although you left me none the wiser.
humy wrote: Obvious Leo wrote:
Define "truth value".

either 'true' or 'false'.
By the definition of 'proposition' (in formal logic), all propositions are statements that have truth values in the sense that what they mean is either 'true' or 'false'.
It seems to me that all you're saying is that true statements are true and false statements are false, which is hardly breaking news but not particularly helpful in formal logic which is not designed to make the distinction. Logic can only be used to draw valid conclusions from a premise but a valid conclusion cannot be used to validate the premise itself. Advocates of the god hypothesis are routinely guilty of falling for this logical fallacy.

JSS
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Re: Can 'definition' be 'axioms'?

humy wrote:what is an "absolute truth value" as opposed to a truth value that isn't "absolute"?
The ONLY "absolute truths" are those which logically have absolutely no alternative. It is the lack of any alternative that proves a truth.

Declared definitions are void of any possible alternative simply because the author is expressing his intent, nothing else. No one can argue with the author's intent, whether a good or bad intent. It is what it is. Most truths are an issue of "most probably true to the point that I am not going to argue against it". Those are not really absolute, but considered [sufficient] truth.

Most theories in physics are not absolute. Generally it takes a good philosopher to form an absolute truth. No empirical evidence can contradict an absolute truth (invariably formed through declared definitions and proper logic = "Definitional Logic"). Empirical evidence can never provide absolute truth, thus science promoters quite often proclaim the absolute truth that there is no such thing as absolute truth.

Obvious Leo
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Re: Can 'definition' be 'axioms'?

JSS wrote:Most theories in physics are not absolute.
Physics cannot make truth statements because it puts the cart before the horse, thus ignoring a salutary warning delivered by Leibniz when physics was but an infant. Instead of interpreting empirical evidence in the light of metaphysical first principles physics attempts to derive its metaphysical first principles from the way it interprets empirical evidence. Both Leibniz and Kant clearly pointed out that this is not logically kosher. This is why physics makes no sense.

humy
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Re: Can 'definition' be 'axioms'?

Obvious Leo wrote:
It seems to me that all you're saying is that true statements are true and false statements are false, ...
So what would your definition of truth value be then?
I would like to see it to see if it is any better.

Obvious Leo
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Re: Can 'definition' be 'axioms'?

humy wrote:So what would your definition of truth value be then?
I have no such definition of a "truth value". I agree with JSS that there are statements of definition which are true as self-referential rules of language but other than that I don't really acknowledge the notion of a truth value. Perhaps you could offer an example or two to illustrate exactly what you're asking.

humy
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Re: Can 'definition' be 'axioms'?

Obvious Leo wrote: ...there are statements of definition which are true as self-referential rules of language . ...
That could be said to be directly behind what I mean by truth value (I think).
And I don't imagine giving examples would help much!
I guess truly and literally defining what 'true' or 'false' exactly means, just like with many basic words such as 'is' or 'if' etc, which I guess would require assuming absolutely no prior knowledge by the reader whatsoever, would be extremely difficult if not totally impossible. I would guess the best one can do in practice is define it according to certain other terms, words and related concepts, and then just hope the reader already knows all about those those other things. Giving some examples, although that may and probability will aid understanding, wouldn't ever be what a true definition comprises of.

JSS
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Re: Can 'definition' be 'axioms'?

"True" means "in perfect alignment/compliance with ..."

Hobbes' Choice
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Re: Can 'definition' be 'axioms'?

No, definition might be axiomatic.
A definition might be an axiom.
Or definitions might be axioms.

As long as you demand that the definition of axiom is axiomatic, then any definition can be axiomatic. But it is unlikely that all definitions are axiomatic. If they were not one would argue over them.
In other words: if we can all agree what the definition of axiom is, then we have demonstrated the fact that definitions can be axioms.

I don't think there is any thing further to be said.

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