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(Also posted in mathstackexchange prior to this).

I stumbled across a set of axioms for first order logic a bit ago. Intrigued, I decided to try to write it all down and organise what I read. After I did that, it seemed to me as though one could quite simply define 'truth' using these axioms and deduction rules.

If β, γ, δ are wfs, x, y are variables, and t is a term, then the following wfs are called logical axioms:

  1. β → (γ → β)
  2. (β → (γ → δ)) → ((β → γ) → (β → δ))
  3. (¬β → ¬γ) → ((¬β → γ) → β)
  4. ((∀x)β) → β[t/x] if t is free for x in β
  5. (∀x)(β → γ) → (β → (∀x)γ) if β contains no free occurrence of x
  6. (∀x)(x = x)
  7. (∀x)(∀y)((x = y) → (β → β[y/x]))

(Then give one's particular set of mathematical axioms, if one intends for them to be thought of as true).

We then define a subclass of wfs which we call the true statements. If α, β are wfs, and x is a variable, then:

All logical axioms are true, as are all mathematical axioms. If α and α → β are true, then β is true. If β is true, then (∀x)β is true. Finally, a statement φ is false if and only if ¬φ is true.

When I showed this to some of my philosophy of mathematics friends, they thought for quite a while before deciding that the above did not suffice to define truth within such a formal system. When pressed as to why not, they weren't entirely sure, though a few possible places where issues could arise were discussed. We couldn't figure out what sort of true statement would not fall under such a definition, however.

My question is therefore - does the above fail to define truth? If so, why does it fail? Are there any amendments that could be made to fix the above? What sort of true statements would not be "true" in the above sense? What have I defined above?

Additionally, any sources for further reading on this topic would be of great, great interest to me, and would be appreciated.

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    Unnecessary, but certainly more convenient :/ – Nethesis May 15 '16 at 10:17
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    Make a little effort to rewrite the symbols? Or do you expect us to copy paste this into a latex editor? – Eliran May 15 '16 at 20:42
  • It's okay, I've had an answer now – Nethesis May 15 '16 at 20:43
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At most, axioms and rules for first-order logic (with equality) may define the concept of "logical truth", assuming the standard semantics and equating validity with logical truth.

What are logical truths ? They are formula which are true irerspective of the domain of discourse, i.e. truths that are not topic-specific, like:

p → p or ¬ (p ∧ ¬ p).

But the philosophical issues related to Logical Truth are many and interesting.

  • are you sue your examples are logical truths? x=x may be true but I'm not sure it's a logical truth. – user20153 May 15 '16 at 20:36
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    @mobileink Do you have any counter example in mind? – Eliran May 15 '16 at 20:41
  • well, I'm thinking you could just map = to "greater than", for example. But more generally my understanding is that logical anything must involve inference, or reasoning, or whatever you want to call it, and x=x does not. the core concept being validity, but that's a property of arguments, not statements. it's amazingly complex, as your ref to the SEP points out. – user20153 May 15 '16 at 20:46
  • fwiw I spent about 3 months working thru Etchemendy's short book on logical consequence. still not sure I get it all, but it's quite striking that he claims the model theoretic account of logical consequence is not in the end any better than the formal deductive one. I've found other papers along this line (not at hand now alas) so I personally have come to reject the idea that the model theoretical account of LC is definitive. – user20153 May 15 '16 at 20:51
  • ps. and obviously equality is a major hairball. how can one infer that one thing is equal to another? does equality of two distinct things even make sense? – user20153 May 15 '16 at 20:55
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the problem is circularity. if you want to define X as Y, then X cannot occur in Y. what you've listed might be treated as conditions of adequacy that must be satisfied by any candidate definition of truth, but not as a definition of truth.

e.g. "all logical axioms are true" does not define true, it just uses an antecedently available (and informal) notion of true to say something about logical axioms.

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