Difference between Entailment and Implicaiton

[friend]

What's the difference between the logical concepts of entailment and implicaiton? I know what implication is between two propositions; every case is allowed except a true premise and a false conclusion. But I'm not quite sure what entailment is.

As I understand it, entailment occurs when a conjunction of statements is not inconsistent with another statement. But doesn't this also mean that the conjunction of statements implies the other? Can anyone give me a case when the relationship is entailment but not implicaiton? Any clarification would be appreciated. Thank you.

 

[SW VandeCarr]

What's the difference between the logical concepts of entailment and implicaiton? I know what implication is between two propositions; every case is allowed except a true premise and a false conclusion. But I'm not quite sure what entailment is.

As I understand it, entailment occurs when a conjunction of statements is not inconsistent with another statement. But doesn't this also mean that the conjunction of statements implies the other? Can anyone give me a case when the relationship is entailment but not implicaiton? Any clarification would be appreciated. Thank you.

They can correspond. That is, if P entails Q, P may also be said to imply Q. However the first is statement based on a proof that P necessarily entails Q. The second is simply based on truth tables. So if P is true and Q is true, then P implies Q under both strict and material implication. However the two statements don't necessarily have anything to do with each other.

For example, P (S is a man) and Q (S likes ice cream). If both statements are true, we can say the P implies Q, but P doesn't necessarily entail Q. For that, you would need a third statement, that all men like ice cream.

EDIT: I can't think of a case where P entails Q would not also indicate P implies Q under material implication.

 

[Preno]

Implication (\rightarrow) is a logical connective, just like \wedge or \vee. Entailment (\vDash) is a relationship between formulas.

\varphi \rightarrow \psi is a formula, a mathematical object. It makes no more sense to assert that \varphi \rightarrow \psi than it makes to assert 2 or \mathbb{R}. \Gamma \vDash \varphi is a mathematical statement about the relationship between the set of formulas \Gamma and the formula \varphi, which may be true or false. In the context of classical propositional logic, it says that \varphi is true in every interpretation in which each formula of \Gamma is true.

The two are related by the deduction theorem: \Gamma, \varphi \vDash \psi iff \Gamma \vDash \varphi \rightarrow \psi.

 

[friend]

Did you mean: \Gamma, \varphi \vDash \psi iff \Gamma \vDash (\varphi \rightarrow \psi)

Or did you mean: \Gamma, \varphi \vDash \psi iff (\Gamma \vDash \varphi )\rightarrow \psi

 

[Preno]

The former. The latter makes no more sense than (2 < 3) \times 2 (as opposed to 2 < 3 \times 2). Implication connects formulas, \Gamma \vDash \varphi is not a formula.