What does ⋀ mean in Isabelle?

Question

(This is pretty much a logic question.) ⋀ such as in ⋀y . Py seems to be a kind of placeholder. It occurs after using apply (rule allI) or apply (erule exE. At first I thought it was like a pretend ∀, because of the rule allI which is (!!x. Px) => ∀x. Px (I believe that !! is used to represent ⋀ in the place I got this from). But I didn't seem to be able to apply allI in certain scenarios which appeared to match this rule. What does ⋀ really mean? I'm wondering if ⋀ can mean different things in different situations.

Answer 1

⋀ is an universal quantifier, yes, but it's the metalogical one. I.e. the one which you use to describe the logic you work in, including ∀. Similarly, is the metalogical implication operator. And these are the only two primitives of Isabelle's metalogic; there is no negation, so it's much weaker than classical logic. See http://isabelle.in.tum.de/coursematerial/PSV2009-1/session2/document.pdf or https://www.cl.cam.ac.uk/teaching/1011/L21/5%20-%20Logic.pdf for more.