# Difference between Entailment and Implicaiton

1. Dec 24, 2011

### 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.

2. Dec 26, 2011

### SW VandeCarr

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.

Last edited: Dec 26, 2011
3. Dec 28, 2011

### 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 $\Gamma \vDash \varphi \rightarrow \psi$.

4. Dec 28, 2011

### friend

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

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

5. Dec 28, 2011

### 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.