Logical Entailment Relation

A Logical Entailment Relation is a Logic Relation between two Logic Statements (A, B) that holds iff in every model in which A holds B is also true.

• AKA: ⇒, Logical Consequence.
• Context:
  • It can be expressed as: A ⇒ B, or A |= B
  • It can be detected by an Entailment Decision Task.
• Example(s):
  • [math]\displaystyle{ A ⇒ B }[/math].
  • …
• Counter-Example(s):
  • a Logical Implication Relation, such as [math]\displaystyle{ A }[/math] → [math]\displaystyle{ B }[/math].
• See: Propositional Logic System, Knowledge Representation System.

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References

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Logical_consequence
  • Logical consequence is a fundamental concept in logic. It is the relation that holds between a set of sentences (or propositions) and a sentence (proposition) when the former "entails" the latter. For example, 'Kermit is green' is said to be a logical consequence of 'All frogs are green' and 'Kermit is a frog', because it would be "self-contradictory" to affirm the latter and deny the former. Logical consequence is the relationship between the premises and the conclusion of a valid argument. These explanations and definitions tend to be circular; the provision of a satisfactory account of logical consequence and entailment is an important topic of philosophy of logic.
  • The truth of the above consequence depends on both the truth of the antecedents and the relationship of logical consequence between the antecedents and the consequence. The consequence might NOT be true if not all frogs were green. Logical consequences or inferences by deductive reasoning are a major aspect of epistemology that communicates to the general public hypotheses about causality of risk factors.
  • A formally specified logical consequence relation may be characterized model-theoretically or proof-theoretically (or both).
  • Logical consequence can also be expressed as a function from sets of sentences to sets of sentences (Tarski's preferred formulation), or as a relation between two sets of sentences (multiple-conclusion logic).
  • …
  • A formula A is a syntactic consequence within some formal system FS of a set Г of formulas iff there is a formal proof in formal system FS of A from the set Г.
    • Г |-_FS A
  • Syntactic consequence does not depend on any interpretation of the formal system. [1]
  • A formula A is a semantic consequence of a set of statements Г
    • Г ⇒ A,
  • if and only if no interpretation \mathcal{I} makes all members of Г true and A false. [2]
• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Entailment
  • In logic and mathematics, entailment or logical implication is a logical relation that holds between a set T of propositions and a proposition B when every model (or interpretation or valuation) of T is also a model of B. In symbols,
    • 1. T |= B
    • 2. T ⇒ B
    • 3. T \therefore B
  • which may be read "T implies B", "T entails B", or "B is a (logical) consequence of T". In such an implication, T is called the antecedent, while B is called the consequent.
  • In other words, (1) holds when the class of models of T is a subset of the class of models of B. Without using the language of models, (1) states that the material conditional formed from the conjunction of all the elements of T and B (i.e. the corresponding conditional) is valid. That is, it is valid that
    • (A₁ \land … \land A₁) → B,
  • where the Ai are the elements of T. (If T has infinite cardinality then, provided the language of T has the compactness property, some finite subset of T implies B.) The statement in terms of the material conditional holds only in logics that have the semantic equivalent of the deduction theorem (and, as noted earlier, if T is infinite, then the compactness property is also required if the language disallows conjunctions over infinite sets of formulas). Thus, the original statement in terms of models is more general. The weaker truth function material implication, denoted by '→', should not be confused with the stronger logical implication.

2005

• (Goldrei, 2005) ⇒ Derek Goldrei. (2005). “Propositional and Predicate Calculus: A Model of Argument." Springer.
  • We are about to describe a formal proof system and say what is meant by a formal derivation of a formula. Our aim is that the formal system should match logical consequence. For a set Γ of formulas and a formula ϕ, we write Γ |= ϕ to express that ϕ is a logical consequence of Γ. We can record Γ as a set of assumptions from which ϕ follows. We shall use the similar notation Γ |- ϕ to express that there is a formal derivation of ϕ from Γ.

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