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Are the premises of an inference rule in the sequent calculus supposed to be understood as being necessary (necessarily present) to introduce the formula in the conclusion?

It seems weird to me that I could use $\wedge L$ to introduce $A\wedge B$ without having either $A$ nor $B$ in the assumptions. For example, this would be a valid proof with Ax at the top and then $\wedge L$.

$$ P \vdash P \over A \wedge B, P \vdash P $$

I know I could use WL but that is not my question. Similarly, can the Axiom rule be used somewhere in a proof that is not at a leaf of the proof tree? For example, is it valid to have a proof which is two consecutive uses of the Axiom rule?

IllogicalUser
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    The rule above is not correct. In the lower sequent you must have also P on the left. – Mauro ALLEGRANZA May 24 '25 at 06:16
  • No, as @MauroALLEGRANZA pointed out above, that's not how rules work in the sequent calculus. I think the quickest way to get how the system actually works these days is by going through the Logitext tutorial and using it to work through some proofs interactively before doing it by hand. – Z. A. K. May 24 '25 at 06:19
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    @MauroALLEGRANZA Sorry, I fixed the typo. I did mean to have P on the left. – IllogicalUser May 26 '25 at 12:39

1 Answers1

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ETA: The below answer refers to a previous version of the question which had $P$ removed from the antecedents of the conclusion.

Inference rules $\dfrac{\phi}{\psi}$, where $\phi$ is the premise and $\psi$ is the conclusion, are read as: From $\phi$ we can prove $\psi$.

Sequents $\Gamma \vdash \Sigma$, with $\Gamma$ called the antecedent and $\Sigma$ the succcedent, are read as: If all of the statements in $\Gamma$ are true, then also at least one of the statements in $\Sigma$ is true.

So a sequent rule $\dfrac{\Gamma \vdash \Sigma}{\Delta \vdash \Pi}$ reads as: If from $\Gamma$ it follows that $\Sigma$, then from $\Delta$ it follows that $\Pi$.

"It seems weird to me that I could use ∧L to introduce A∧B without having either A nor B in the assumptions."

Indeed, you can't. You can use $\land L$ to introduce $A \land B$ as an assumption if you already have both $A$ and $B$ in the assumptions. The rule doesn't allow you to drop an arbitrary assumption $P$ and replace it by a different one. What the rule says is that you may replace antecedents $A, B$ with $A \land B$. So your $\dfrac{P \vdash P}{A \land B \vdash P}$ is not a correct rule application. From "If $P$ is true, $P$ is true" we can not infer that "If $A \land B$ is true, $P$ is true".

An application of $\land L$ following the identity axiom would be as follows:

$\dfrac{A, B \vdash A}{A \land B \vdash A}$

Read: Given that, when $A$ is true and $B$ is true, it follows that $A$ is true, we also have that when $A \land B$ is true, $A$ is true.

What you're showing in your revised version of the question is a correct application not of the inference rule $\land L$, but of the structural rule called weakening on left ($WL$), which allows introducing arbitrary antecedents. Intuitively, requiring more conditions won't render an already established conclusion invalid. Here, since we showed that P holds under condition P, we can still obtain P if we additionally assume $A \land B$ to be true. It is crucial here that we may only introduce $A \land B$ as an additional (unneeded) antecedent, not replace $P$ with it; the succedent $P$ still depends on the truth of $P$ as an antecedent. But to answer the (intended) first question of yours: The premises antecedents of a rule sequent in the sequent calculus are not, in general, necessary to be present in order for the validity of the formulas in the conclusion succedent to hold.

Natalie Clarius
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  • You say that the rule says "I may replace antecedents $A,B$ with $A\wedge B$". Do I need to have $A$ and $B$, and do I need to replace them if I do have them? – IllogicalUser May 26 '25 at 12:41
  • Yes, if you want to use $\land L$ then you must have had both $A$ and $B$ as assumptions. The rule means: If you can prove $P$ assuming $A$ and $B$ individually, then you can prove $P$ from the conjunction of $A$ and $B$. – Natalie Clarius May 26 '25 at 16:30
  • Strictly speaking, if you apply the inference rule you need to remove the used assumptions in the conclusion. However, there is also the structural rule of weakening, which allows us to introduce arbitrary assumptions. Intuitively, requiring more conditions won't render an already established conclusion invalid. The derivation you're showing in your now revised question is an application of the weakening rule. The $\land L$ rule doesn't occur in your derivation. – Natalie Clarius May 26 '25 at 16:32