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Suppose we only have all the standard left and right logical inference rules ($∧L_{1}$, $∧L_{2}$, $∨L$, $→L$, $¬L$, $∨R_{1}$, $∨R_{2}$, $∧R$, $→R$, $¬R$) and on top of that 4 quantifer rules (see the Wikipedia page: https://en.wikipedia.org/wiki/Sequent_calculus)

My professor said that in predicate logic we cannot prove the Left Contraction Rule unlike in Propositional logic (without these 4 extra quantifier rules). For those who are curious he suggested that we should try proving it with induction on the length of the inference.

The user Dan Christensen here suggested a slightly different idea in previous post that I also had came up with: we can't prove the rule with the premise (A ∧ A ⊢ B) and conclusion (A ⊢ B)

P.S. I already posted a similar post but it was deleted due to being unclear (???). I hope explained everything fine this time :(

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    $\begingroup$ you should state what you mean by left contraction rule. $\endgroup$ Commented May 18 at 0:12

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