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Show that it’s provable in the Hilbert system that identity behaves as an equivalence relation, i.e., show that for any terms $t_1, t_2$, and $t_3$ the following hold.

(a) $\vdash_H t_1 \approx t_1$

(b) $\vdash_H t_1 \approx t_2 \rightarrow t_2 \approx t_1$

(c) $\vdash_H t_1 \approx t_2 \rightarrow t_2 \approx t_3 \rightarrow t_1 \approx t_3$

relevant axioms

  1. $\alpha \rightarrow \big(\beta \rightarrow\alpha \big)$

  2. $ \big({\alpha \rightarrow \big(\beta \rightarrow\gamma \big)}\big) \rightarrow \big({\big(\alpha \rightarrow\beta \big) \rightarrow \big(\alpha \rightarrow\gamma \big)} \big)$

  3. $\big( \lnot \beta \rightarrow \lnot \alpha \big) \rightarrow \big( (\lnot \beta \rightarrow\alpha) \rightarrow \beta \big)$

Gwen Di
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  • Can you provide the relevant axiom(s) regarding identity in the Hilbert system? Also, what is going on with $(b)$? Should that just be $\vdash_H t_1 \approx t_2 \rightarrow t_2 \approx t_1$? Finally, and most importantly, what did you yourself try? – Bram28 Dec 17 '17 at 21:16
  • @Bram28 Tried my best to fix it. So far I'm actually pretty stuck on how to prove an equivalence relationship in this scenario and can't find much outside assistance. – Gwen Di Dec 17 '17 at 21:22
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    @GwenDi These aren’t enough yet. Equality axioms (which they’re usually called) should at least include the predicate symbol for equality—in your case, $\approx$. For abstract equality theory, there are usually reflexivity axiom and one or two substitution axiom schemas—search for these. (In concrete theories—e. g. ZFC—their count could be greatly reduced. But I suppose it’s not your case.) – arseniiv Dec 17 '17 at 21:35
  • @Bram28 See, this is why I am confused. In the text this class is using Enderton's "Mathematical Introduction to Logic," and in the lecture powerpoint there is no mention of equality axioms. Would there be any other way to do this. Incredibly confused – Gwen Di Dec 17 '17 at 21:40
  • @GwenDi: Somewhere it must define $\approx$; I strongly doubt the symbol is used without any definition. – celtschk Dec 17 '17 at 21:42
  • @GwenDi Also there’s a transitivity proof: questions/583855/how-would-one-prove-transitivity-in-first-order-logic. It’s even about Enderton’s book. :) – arseniiv Dec 17 '17 at 21:45
  • @GwenDi Well, as others have pointed out, you really can't do it with the three axioms you listed, which are all about truth-functional properties, but have nothing to do with equality. But one of the anwers to the post linked in the previous comment does mention the following rule: $\xi = \zeta \to (\varphi(\xi) \to \varphi(\zeta))$. You'll need something like that .. but by itself that one is still note enough to prove the first two theorems. – Bram28 Dec 17 '17 at 21:51
  • You have "tagged" the question as: "first-order logic": thus, we need the axiom for FOL (with equality). – Mauro ALLEGRANZA Dec 18 '17 at 07:16
  • See Mendelson, page 93. – Mauro ALLEGRANZA Dec 18 '17 at 07:40
  • See also the post: equality and its axioms. – Mauro ALLEGRANZA Dec 18 '17 at 08:16

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