Logical consequence (also entailment or logical implication) is a fundamental concept in logic which 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 the following questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises? All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth. 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 L {\displaystyle {\mathcal {L}}} , either by constructing a deductive system for L {\displaystyle {\mathcal {L}}} or by formal intended semantics for language L {\displaystyle {\mathcal {L}}} . 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 (sense experience); and (3) The logical consequence relation has a modal component.
Formal accounts The most widely prevailing view on how best to account for logical consequence is to appeal to formality. This is to say 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. Syntactic accounts of logical consequence rely on schemes using inference rules. For instance, we can express the logical form of a valid argument as:
All X are Y All Y are Z Therefore, all X are Z. This argument is formally valid, because every instance of arguments constructed using this scheme is valid. This is in contrast to an argument like "Fred is Mike's brother's son. Therefore Fred is Mike's nephew." Since this argument depends on the meanings of the words "brother", "son", and "nephew", the statement "Fred is Mike's nephew" is a so-called material consequence of "Fred is Mike's brother's son", not a formal consequence. A formal consequence must be true in all cases, however this is an incomplete definition of formal consequence, since even the argument "P is Q's brother's son, therefore P is Q's nephew" is valid in all cases, but is not a formal argument.
A priori property If it is known that Q {\displaystyle Q} follows logically from P {\displaystyle P} , then no information about the possible interpretations of P {\displaystyle P} or Q {\displaystyle Q} will affect that knowledge. Our knowledge that Q {\displaystyle Q} is a logical consequence of P {\displaystyle P} cannot be influenced by empirical knowledge. Deductively valid arguments can be known to be so without recourse to experience, so they must be knowable a priori. However, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. So the a priori property of logical consequence is considered to be independent of formality.
Proofs and models The two prevailing techniques for providing accounts of logical consequence involve expressing the concept in terms of proofs and via models. The study of the syntactic consequence (of a logic) is called (its) proof theory whereas the study of (its) semantic consequence is called (its) model theory.
Syntactic consequence
A formula A {\displaystyle A} is a syntactic consequence within some formal system F S {\displaystyle {\mathcal {FS}}} of a set Γ {\displaystyle \Gamma } of formulas if there is a formal proof in F S {\displaystyle {\mathcal {FS}}} of A {\displaystyle A} from the set Γ {\displaystyle \Gamma } . This is denoted Γ ⊢ F S A {\displaystyle \Gamma \vdash _{\mathcal {FS}}A} . The turnstile symbol ⊢ {\displaystyle \vdash } was originally introduced by Frege in 1879, but its current use only dates back to Rosser and Kleene (1934–1935). Syntactic consequence does not depend on any interpretation of the formal system.
Semantic consequence
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