# Why can't we derive the Left Contraction Rule in predicate logic?

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 :(

• you should state what you mean by left contraction rule. Commented May 18 at 0:12