Logical consequence is a fundamental concept in logic. It is the relation that holds between a set of sentences (or propositions) and a sentence (proposition) when the former "entails" the latter. For example, 'Kermit is green' is said to be a logical consequence of 'All frogs are green' and 'Kermit is a frog', because it would be "selfcontradictory" to affirm the latter and deny the former. Logical consequence is the relationship between the premises and the conclusion of a valid argument. These explanations and definitions tend to be circular; the provision of a satisfactory account of logical consequence and entailment is an important topic of philosophy of logic.
The truth of the above consequence depends on both the truth of the antecedents and the relationship of logical consequence between the antecedents and the consequence. The consequence might NOT be true if not all frogs were green. Logical consequences or inferences by deductive reasoning are a major aspect of epistemology that communicates to the general public hypotheses about causality of risk factors.
A formally specified logical consequence relation may be characterized modeltheoretically or prooftheoretically (or both).
Logical consequence can also be expressed as a function from sets of sentences to sets of sentences (Tarski's preferred formulation), or as a relation between two sets of sentences (multipleconclusion logic).
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This section describes some common accounts of logical consequence.
Γ will represent an arbitrary set of premises and A an arbitrary conclusion. Γ/A will denote the logical argument having Γ as its (set of) premises and A as its conclusion. Γ A will mean that A is a logical consequence of Γ.
Formal accounts of logical consequence are variations on the following basic idea:
Two common variations on this basic idea are:
Let us again consider the argument:
Formal account (1) says that the conclusion is a logical consequence of the premises because no matter how we uniformly replace the nonlogical terms (frog, green, Kermit) in the argument, we do not get true premises and a false conclusion. Consider for example:
We can make up arguments of this form all day, but we will never come up with one that has true premises and a false conclusion. The argument is deductively valid by virtue of its logical form, which might be characterized with the following template (in which F, G, and a are meaningless placeholders):
Formal account (2) says that the conclusion of the "Kermit" argument is a logical consequence of the premises because no matter how we interpret the nonlogical terms (frog, green, Kermit) in the argument, we do not get true premises and a false conclusion. Suppose, for example, we interpret frog to mean plumber, green to mean shy, and Kermit to mean Madonna (the singer). Then the argument has two false premises (for not all plumbers are shy, and Madonna is not a plumber) and a false conclusion (for Madonna is not shy). We can come up with as many interpretations of frog, green, and Kermit as we like, but this will never result in an argument with true premises and a false conclusion.
A formula A is a syntactic consequence^{[1]}^{[2]}^{[3]}^{[4]} within some formal system FS of a set Г of formulas if there is a formal proof in formal system FS of A from the set Г.
Syntactic consequence does not depend on any interpretation of the formal system.^{[5]}
A formula A is a semantic consequence of a set of statements Г
if and only if no interpretation makes all members of Г true and A false.^{[6]}
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 necessity and (im)possibility. 'It is necessary that' is often cashed out 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.
Modalformal accounts of logical consequence combine the modal and formal accounts above, yielding variations on the following basic idea:
Most logicians would probably agree that logical consequence, as we intuitively understand it, has both a modal and a formal aspect, and that some version of the modal/formal account is therefore closest to being correct.
The accounts considered above are all "truthpreservational," 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 "warrantpreservational" 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.
The accounts discussed above all yield monotonic consequence relations, i.e. ones such that if A is a consequence of Γ, then A is a consequence of any superset of Γ. It is also possible to specify nonmonotonic consequence relations to capture the idea that, e.g., 'Tweety can fly' is a logical consequence of
but not of
For more on this, see the article on nonmonotonic logic.
