Logical consequence (alsoentailment orlogical implication) is a fundamentalconcept inlogic which describes the relationship betweenstatements that hold true when one statement logicallyfollows from one or more statements. Avalid logicalargument is one in which theconclusion is entailed by thepremises, because the conclusion is the consequence of the premises. Thephilosophical analysis of logical consequence involves the 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?[1] All ofphilosophical logic is meant to provide accounts of the nature of logical consequence and the nature oflogical truth.[2]
Logical consequence isnecessary andformal, by way of examples that explain withformal proof andmodels of interpretation.[1] A sentence is said to be a logical consequence of a set of sentences, for a givenlanguage,if and only if, using only logic (i.e., without regard to anypersonal interpretations of the sentences) the sentence must be true if every sentence in the set is true.[3]
Logicians make precise accounts of logical consequence regarding a givenlanguage, either by constructing adeductive system for or by formalintended semantics for language. The Polish logicianAlfred Tarski identified three features of an adequate characterization of entailment: (1) The logical consequence relation relies on thelogical form of the sentences: (2) The relation isa priori, i.e., it can be determined with or without regard toempirical evidence (sense experience); and (3) The logical consequence relation has amodal component.[3]
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 orlogical form of the statements without regard to the contents of that form.
Syntactic accounts of logical consequence rely onschemes usinginference rules. For instance, we can express the logical form of a valid argument as:
This argument is formally valid, because everyinstance 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-calledmaterial consequence of "Fred is Mike's brother's son", not a formal consequence. A formal consequence must be truein all cases, however this is an incomplete definition of formal consequence, since even the argument "P isQ's brother's son, thereforeP isQ's nephew" is valid in all cases, but is not aformal argument.[1]
If it is known that follows logically from, then no information about the possible interpretations of or will affect that knowledge. Our knowledge that is a logical consequence of cannot be influenced byempirical knowledge.[1] Deductively valid arguments can be known to be so without recourse to experience, so they must be knowable a priori.[1] 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.[1]
The two prevailing techniques for providing accounts of logical consequence involve expressing the concept in terms ofproofs and viamodels. 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.[4]
A formula is asyntactic consequence[5][6][7][8][9] within someformal system of a set of formulas if there is aformal proof in of from the set. This is denoted. The turnstile symbol was originally introduced by Frege in 1879, but its current use only dates back to Rosser and Kleene (1934–1935).[9]
Syntactic consequence does not depend on anyinterpretation of the formal system.[10]
A formula is asemantic consequence within some formal system of a set of statements if and only if there is no model in which all members of are true and is false.[11] This is denoted. Or, in other words, the set of the interpretations that make all members of true is a subset of the set of the interpretations that make true.
Modal accounts of logical consequence are variations on the following basic idea:
Alternatively (and, most would say, equivalently):
Such accounts are called "modal" because they appeal to the modal notions oflogical necessity andlogical possibility. 'It is necessary that' is often expressed as auniversal quantifier overpossible worlds, so that the accounts above translate as:
Consider the modal account in terms of the argument given as an example above:
The conclusion is a logical consequence of the premises because we can not imagine a possible world where (a) all frogs are green; (b) Kermit is a frog; and (c) Kermit is not green.
Modal-formal accounts of logical consequence combine the modal and formal accounts above, yielding variations on the following basic idea:
The accounts considered above are all "truth-preservational", in that they all assume that the characteristic feature of a good inference is that it never allows one to move from true premises to an untrue conclusion. As an alternative, some have proposed "warrant-preservational" accounts, according to which the characteristic feature of a good inference is that it never allows one to move from justifiably assertible premises to a conclusion that is not justifiably assertible. This is (roughly) the account favored byintuitionists.
The accounts discussed above all yieldmonotonic consequence relations, i.e. ones such that if is a consequence of, then is a consequence of any superset of. It is also possible to specify non-monotonic consequence relations to capture the idea that, e.g., 'Tweety can fly' is a logical consequence of
but not of
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