Modal accounts of logical consequence are variations on the following basic idea: :\Gamma \vdash A is true if and only if it is
necessary that if all of the elements of \Gamma are true, then A is true. Alternatively (and, most would say, equivalently): :\Gamma \vdash A is true if and only if it is
impossible for all of the elements of \Gamma to be true and A false. 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: :\Gamma \vdash A is true if and only if there is no possible world at which all of the elements of \Gamma are true and A is false (untrue). Consider the modal account in terms of the argument given as an example above: :All frogs are green. :Kermit is a frog. :Therefore, Kermit is green. 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 Modal-formal accounts of logical consequence combine the modal and formal accounts above, yielding variations on the following basic idea: :\Gamma \vdash A if and only if it is impossible for an argument with the same logical form as \Gamma/A to have true premises and a false conclusion.
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. Non-monotonic logical consequence The accounts discussed above all yield
monotonic consequence relations, i.e. ones such that if A is a consequence of \Gamma, then A is a consequence of any superset of \Gamma. 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 :{Birds can typically fly, Tweety is a bird} but not of :{Birds can typically fly, Tweety is a bird, Tweety is a penguin}. ==See also==