The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises?
[3] Logicians make precise accounts of logical consequence regarding a given language
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.
[3] The most widely prevailing view on how best to account for logical consequence is to appeal to formality.
Syntactic accounts of logical consequence rely on schemes using inference rules.
This is in contrast to an argument like "Fred is Mike's brother's son.
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.
[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 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.
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 any interpretation of the formal system.
true is a subset of the set of the interpretations that make
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 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.
The accounts discussed above all yield monotonic consequence relations, i.e. ones such that if