Thread: Establish this logical equivalence, where x does not occur as a free variable in A.

1. Establish this logical equivalence, where x does not occur as a free variable in A.

Establish this logical equivalence, where x does not occur as a free variable in A. Assume that the domain is nonempty.

∀x(A → P(x)) ≡ A → ∀xP(x)

How in the heck am I supposed to do this? Am I supposed to prove (translating the logical statement into English in bold) for every value of A there is a value that makes A true is equivalent to there is some value for A that makes A true?

2. Re: Establish this logical equivalence, where x does not occur as a free variable in

You are supposed to start with the definition of when a formula is true in an interpretation, i.e., the definition of the relation ⊧. You need to show that ∀x(A → P(x)) is logically equivalent to A → ∀xP(x). From left to right this means that for every interpretation I, if I ⊧ ∀x(A → P(x)), then I ⊧ A → ∀xP(x). So, assume I ⊧ ∀x(A → P(x)) and apply the definition of ⊧.

3. Re: Establish this logical equivalence, where x does not occur as a free variable in

Originally Posted by emakarov
You are supposed to start with the definition of when a formula is true in an interpretation, i.e., the definition of the relation ⊧.
I'm sorry, I don't recognize the symbol, ⊧. Can you explain what that is please?

4. Re: Establish this logical equivalence, where x does not occur as a free variable in

The point still stands. You should start with the definition of the logical equivalence, i.e., the symbol ≡. This leads to the definition of when a formula is true in an interpretation. This relationship is usually denoted by ⊧. But even if it is not denoted this way in your textbook, you need to start with these two definitions.

5. Re: Establish this logical equivalence, where x does not occur as a free variable in

Originally Posted by emakarov
The point still stands. You should start with the definition of the logical equivalence, i.e., the symbol ≡. This leads to the definition of when a formula is true in an interpretation. This relationship is usually denoted by ⊧. But even if it is not denoted this way in your textbook, you need to start with these two definitions.
Alright so I say, " every value of x is in the domain A if and only if there is some value x in the domain A" ?

6. Re: Establish this logical equivalence, where x does not occur as a free variable in

Two formulas A and B are called logically equivalent (this is sometimes denoted by A ≡ B) if for every interpretation I, A is true in I iff B is true in I. Since this statement starts with "for every interpretation I", you should start your proof by "Fix an arbitrary interpretation I". Then you assume that the left-hand side is true in I and prove that the right-hand side is true also, and vice versa. In doing this, you need to use the definition of what it means for a formula to be true in an interpretation. The definition is somewhat long, so attention to technical details is needed. For example, going left-to-right you assume that ∀x(A → P(x)) is true in I and some variable assignment (or, environment) μ (see the link above). This means that A → P(x) is true in I and every μ' that differs from μ only in the value of x. Then you apply the part of the definition that deals with implications, and so on.

Note that textbooks about mathematical logic duffer a great deal in the details of many definitions, even though these definitions end up equivalent. That's why I said that unless you learn the definitions of logical equivalence and the truth of a formula in an interpretation, and unless you share them here if necessary, we can't help you.

Informally, ∀x(A → P(x)) is true in I if for every x, the truth of A in I implies the truth of P(x) in I. But since the truth of A in I does not depend on x (which formally is a lemma that needs to be proved), this is equivalent to the fact that the truth of A in I implies that P(x) is true in I for every x.