
Logical consequence

Logical consequence (also entailment) 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 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 formula_1, either by constructing a deductive system for formula_1 or by formal intended semantics for language formula_1. 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:
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"&apos;s brother's son, therefore "P" is "Q"&apos;s nephew" is valid in all cases, but is not a "formal" argument.
A priori property of logical consequence.
If it is known that formula_4 follows logically from formula_5, then no information about the possible interpretations of formula_5 or formula_4 will affect that knowledge. Our knowledge that formula_4 is a logical consequence of formula_5 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 formula_10 is a syntactic consequence within some formal system formula_11 of a set formula_12 of formulas if there is a formal proof in formula_11 of formula_10 from the set formula_12.
Syntactic consequence does not depend on any interpretation of the formal system.
Semantic consequence.
A formula formula_10 is a semantic consequence within some formal system formula_11 of a set of statements formula_12
if and only if there is no model formula_21 in which all members of formula_12 are true and formula_10 is false. Or, in other words, the set of the interpretations that make all members of formula_12 true is a subset of the set of the interpretations that make formula_10 true.
Modal accounts.
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 of logical necessity and logical possibility. 'It is necessary that' is often expressed as a universal quantifier over possible 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't imagine a possible world where (a) all frogs are green; (b) Kermit is a frog; and (c) Kermit is not green.
Modal-formal accounts.
Modal-formal accounts of logical consequence combine the modal and formal accounts above, yielding variations on the following basic idea:
Warrant-based accounts.
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 by intuitionists such as Michael Dummett.
Non-monotonic logical consequence.
The accounts discussed above all yield monotonic consequence relations, i.e. ones such that if formula_10 is a consequence of formula_12, then formula_10 is a consequence of any superset of formula_12. 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