Logical consequence
A fundamental concept in logic describing valid inference.
Logical consequence—also called entailment or logical implication—is a core idea in logic. It describes the relationship where one statement necessarily follows from one or more other statements. In a valid logical argument, the conclusion is entailed by the premises, meaning the conclusion is the consequence of those premises. Philosophers analyzing logical consequence ask: In what sense does a conclusion follow from its premises? And what does it mean for a conclusion to be a consequence of premises? The entire field of philosophical logic aims to explain the nature of logical consequence and logical truth.
Logical consequence is both necessary and formal. This is illustrated through formal proofs and models of interpretation. For a given language, a sentence is a logical consequence of a set of sentences if and only if, using only logic (ignoring personal interpretations), the sentence must be true whenever every sentence in the set is true. Logicians make this precise for a specific language by either building a deductive system for it or by defining formal semantics for it. The Polish logician Alfred Tarski identified three features of an adequate account of entailment: (1) it depends on the logical form of sentences; (2) it is a priori, meaning it can be determined without empirical evidence; and (3) it has a modal component.
The most common approach to explaining logical consequence is to appeal to formality—whether statements follow from each other depends on their structure or logical form, not their content. Syntactic accounts use inference rules. For example, the argument "All X are Y; All Y are Z; Therefore, all X are Z" is formally valid because every instance of that scheme is valid. This contrasts with "Fred is Mike's brother's son; therefore Fred is Mike's nephew," which depends on the meanings of "brother," "son," and "nephew." That is a material consequence, not a formal one. A formal consequence must hold in all cases, but this definition is incomplete: even "P is Q's brother's son, therefore P is Q's nephew" is valid in all cases, yet it is not a formal argument.
If we know that Q follows logically from P, no information about possible interpretations of P or Q changes that knowledge. This knowledge cannot be influenced by empirical experience. Deductively valid arguments are knowable without recourse to experience—they are a priori. However, formality alone does not guarantee that logical consequence is free from empirical influence, so the a priori property is considered separate from formality.
The two main techniques for accounting for logical consequence involve proofs and models. The study of syntactic consequence is called proof theory; the study of semantic consequence is called model theory.
A formula A is a syntactic consequence within a formal system FS of a set Γ of formulas if there is a formal proof in FS of A from Γ. This is written as Γ ⊢_FS A. The turnstile symbol ⊢ was introduced by Frege in 1879, but its current use dates to Rosser and Kleene (1934–1935). Syntactic consequence does not depend on any interpretation of the formal system.
A formula A is a semantic consequence within a formal system FS of a set of statements Γ if, under every interpretation of the language, whenever all statements in Γ are true, A is also true.
- 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, also known as entailment or logical implication, is a fundamental concept in logic describing the relationship where one statement necessarily follows from one or more other statements. A valid argument is one where the conclusion is entailed by the premises. The concept is characterized as both necessary and formal: a sentence is a logical consequence of a set of sentences if, using only logic and disregarding personal interpretations, the sentence must be true whenever every sentence in the set is true. The Polish logician Alfred Tarski identified three essential features of an adequate characterization of entailment: it relies on the logical form of sentences, it is a priori (determinable without empirical evidence), and it has a modal component. The most widely prevailing view is that logical consequence depends on the structure or logical form of statements, not their content. For example, the argument "All X are Y; all Y are Z; therefore, all X are Z" is formally valid because every instance of that scheme is valid, unlike an argument relying on the meanings of specific words like "brother" and "nephew." Logicians provide precise accounts either through syntactic consequence, using formal proofs within a deductive system (denoted by the turnstile symbol), or through semantic consequence, using models where no interpretation makes all premises true and the conclusion false. Modal accounts hold that a conclusion is a logical consequence of premises if it is impossible for all premises to be true and the conclusion false, or equivalently, if there is no possible world where that occurs. Modal-formal accounts combine these, stating that a conclusion follows if no argument with the same logical form can have true premises and a false conclusion.
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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