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Difference between Entailment and Implicaiton

  1. Dec 24, 2011 #1
    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. jcsd
  3. Dec 26, 2011 #2
    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
  4. Dec 28, 2011 #3
    Implication ([itex]\rightarrow[/itex]) is a logical connective, just like [itex]\wedge[/itex] or [itex]\vee[/itex]. Entailment ([itex]\vDash[/itex]) is a relationship between formulas.

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

    The two are related by the deduction theorem: [itex]\Gamma, \varphi \vDash \psi[/itex] iff [itex]\Gamma \vDash \varphi \rightarrow \psi[/itex].
  5. Dec 28, 2011 #4
    Did you mean: [itex]\Gamma, \varphi \vDash \psi[/itex] iff [itex]\Gamma \vDash (\varphi \rightarrow \psi)[/itex]

    Or did you mean: [itex]\Gamma, \varphi \vDash \psi[/itex] iff [itex](\Gamma \vDash \varphi )\rightarrow \psi[/itex]
  6. Dec 28, 2011 #5
    The former. The latter makes no more sense than [itex](2 < 3) \times 2[/itex] (as opposed to [itex]2 < 3 \times 2[/itex]). Implication connects formulas, [itex]\Gamma \vDash \varphi[/itex] is not a formula.
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