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I'm writing an article about logic for absolute dummies, so I want to make everything crystal clear; now I'm going to discuss predicate logic. After Googling, I found there are mainly 2 slightly different kinds of axiom systems for it, both are present here. let's simply call the first system in the link FOL1, and the alternative system FOL2.

Here's my question:

  1. Why does $x$ have to be bound in $A$ in the 5th axiom $(A\to∀xA)$ in FOL1? Can this axiom be replaced by $A∣-∀xA$ ?

  2. Why is there an extra quantification in axiom $4$ in FOL1 than in FOL2? This axiom in FOL1 seems easier to explain.

TShiong
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盛安安
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  • Well, for #1 you don't want to be able to infer that $A$ is true of everything just because it's true of something... – Malice Vidrine Jul 27 '16 at 16:17
  • How do I interpret a free variable? Can it be seen as being implicitly quantified by ∃ ? – 盛安安 Jul 27 '16 at 16:27
  • Free variables are really their own thing. An expression containing free variables isn't really true or false; it's a little bit like a function mapping things (or tuples of things) to truth values. Closed expressions, on the other hand, have a definite truth value. – Malice Vidrine Jul 27 '16 at 18:13
  • Why is A∣-∀xA right in FOL2 then? – 盛安安 Jul 27 '16 at 18:59
  • There are many "flavours" of FOL axiom systems... The basic issue is that we want an ax syst to be sound and complete for validity. Thus, you have to consider that $P(x) \to \forall x P(x)$ is not valid. If you discard Ax.5 and use instead the addiotional inf rule called Gen (basically: $A \vdash \forall x A$) you have to suitably restrict the Deduction Th, in order to avoid to prove in your proof system the invalid $P(x) \to \forall x P(x)$. – Mauro ALLEGRANZA Jul 28 '16 at 06:12
  • @Bram28: please don't add "predicate logic" if "first order logic" is already present; see this thread on meta where the two are being proposed as duplicates https://math.meta.stackexchange.com/questions/28849/is-tagpredicate-logic-a-synonym-for-tagfirst-order-logic . – Carl Mummert Aug 13 '18 at 16:04
  • @CarlMummert OK, thanks for letting me know! I'd be all for that merger, sure. – Bram28 Aug 13 '18 at 16:07
  • Note that these are just minor variants of the same Hilbert-style systems. There are numerous others kinds of systems. For "absolute dummies", you almost surely do not want to use Hilbert-style. Rather, take a look at Fitch-style systems such as this one. – user21820 Jun 21 '22 at 17:27
  • i question whether you are really writing this article for absoute dummies. – RyRy the Fly Guy Jul 31 '23 at 04:31

1 Answers1

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Question 1:

Variables must be bound because we don't work with open sentences in proof systems, yet it is possible to assign a truth value to open sentences.

The truth value of an open sentence $A$ with an free variable $x$ is true if and only if the value of its closure $\forall x A$ is true.

Question 2:

The difference between axiom 4 in FOL1 and FOL2 is that the rule in FOL1 $\forall x(A\rightarrow B) \rightarrow (\forall x A \rightarrow \forall x B)$ allows $x$ to be free in $A$. The rule in FOL2 $\forall x (A\rightarrow B)\rightarrow (A\rightarrow \forall x B)$ on the other hand explicity states that $x$ must not be free in $A$, therefore FOL1 rule is broader than FOL2's.

Henrique Inonhe
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  • isn't X=X a theorem in FOL? – 盛安安 Jul 27 '16 at 19:30
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    Yes it is, but what's the deal with it? – Henrique Inonhe Jul 27 '16 at 19:46
  • Variables must be bound because we don't work with open sentences in proof systems, yet it is possible to assign a truth value to open sentences. – 盛安安 Jul 28 '16 at 05:10
  • What does "work" mean? Isn't assigning value working? – 盛安安 Jul 28 '16 at 05:11
  • Sorry if I wasn't clear, what I meant is that even though we can assign a truth value to an open sentence, it is unusual to admit open sentences in a proof system, therefore only closed sentences should "appear" within a formal deduction in a proof system. The reason for that is mainly due to the fact that to deal with open sentences in a proof system we would need aditional inference rules. So, to sum it up, the "class" of open sentences exist merely as a way to formally define how to "build" closed sentences from them. – Henrique Inonhe Jul 28 '16 at 13:14
  • Why is X=X (without a quantifier) a theorem then? – 盛安安 Jul 29 '16 at 03:19
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    But we work with open formulae (not sentences) in proof system; see e.g. the following instance of the quantifier axiom: $\forall x \ (x=x) \to (y=y)$. – Mauro ALLEGRANZA Jul 29 '16 at 08:51
  • In some conventions, the truth value of an open formula is the same as the truth value of its universal closure. In other conventions, the truth values do not need to be the same, and every formula has its own truth value, using variable assignment functions. – Carl Mummert Aug 13 '18 at 16:01