Logical consequence
A fundamental concept in logic describing valid inference.
Logical consequence (also entailment or logical implication) is a fundamental concept in logic that describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves questions about in what sense a conclusion follows from its premises and what it means for a conclusion to be a consequence of premises.
- field
- Logic
- known_for
- Fundamental concept in logic describing the relationship between statements that hold true when one statement logically follows from one or more statements
- key_features
- Necessary and formal; relies on logical form; a priori; has a modal component
Lore & Background
Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to be a logical consequence of a set of sentences, for a given language, if and only if, using only logic (i.e., without regard to any personal interpretations of the sentences) the sentence must be true if every sentence in the set is true. Logicians make precise accounts of logical consequence regarding a given language, either by constructing a deductive system for that language or by formal intended semantics for that language.
Reader's Guide
The Polish logician Alfred Tarski identified three features of an adequate characterization of entailment: (1) The logical consequence relation relies on the logical form of the sentences; (2) The relation is a priori, i.e., it can be determined with or without regard to empirical evidence; and (3) The logical consequence relation has a modal component. The most widely prevailing view on how best to account for logical consequence is to appeal to formality, meaning that whether statements follow from one another logically depends on the structure or logical form of the statements without regard to the contents of that form. The two prevailing techniques for providing accounts of logical consequence involve expressing the concept in terms of proofs (syntactic consequence) and via models (semantic consequence). Modal accounts of logical consequence appeal to the modal notions of logical necessity and logical possibility, stating that a conclusion is a logical consequence of premises if and only if it is necessary that if all premises are true, then the conclusion is true, or equivalently, it is impossible for all premises to be true and the conclusion false.
Did You Know?
- Logical consequence is also known as entailment or logical implication.
- Alfred Tarski identified three features of an adequate characterization of entailment: reliance on logical form, a priori nature, and a modal component.
- Semantic consequence holds if there is no model in which all premises are true and the conclusion is false.
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