Philosophy 3349
Topics in Logic

 

type of logic	syntactic changes	semantic changes	proof theory changes
many-valued	none	additional truth values (two main varieties: deal with truth-value gaps by adding a value for "neither true nor false"; deal with truth-value gluts by adding a value for "both true and false". Values for compound sentences determined by truth tables with mn rows, where n is the number of atomic sentences and m is the number of truth values.	good question! Minimally: if you add a value for "neither," excluded middle should no longer be valid (so negation elimination has to go); if you add a value for "both," the law of noncontradiction should go (so negation introduction has to go)
fuzzy	none	additional truth values: treat T as 1 and F as 0; allow any real number in the interval [0, 1] as a truth value. Truth values for compound sentences cannot be determined by truth tables; instead, introduce semantic rules such as F(¬P) = 1 - F(P).	good question! Depends on the details. If logical implication is defined so that P1 . . . Pn ⊨ C iff F(C) >= F(P1 ∧ . . . ∧ Pn), then some classical rules of inference will still be OK (e.g. conjunction elimination and disjunction introduction). Others will no longer work (e.g. conditional elimination).
free	none (or possibly the addition of an Exists(x) predicate, but this is typically defined to mean ∃y(y = x).)	only affects predicate logic, since in propositional logic an interpretation is just an assignment of truth values to atomic sentences. Either allow a null object in the universe (in which case every nondenoting term refers to the same thing), or have a universe with two levels: existing objects (the "inner domain") and nonexistent objects (the "outer domain").	Need to abandon Existential Introduction and Universal Elimination in their unrestricted forms. For instance, we could adopt a modified EI: from F(a) and Exists(a), infer ∃xF(x).
modal	Add necessity and/or possibility operators: □ and/or ◊	We can no longer simply assign truth values to atomic sentences. Now truth-value assignments must be relativized to a world.	No alterations to classical logic are necessary, except that modal logics are essentially free logics (the universe consists of all possible objects, whether they actually exist or not; the "inner domain" is the subset of the universe that exists at the actual world, and the "outer domain" is the rest of the universe). We simply need to add rules for the possibility and necessity operators to the formal system. (For example: from ⊢P, infer ⊢□P.)
temporal, deontic, epistemic, etc.	Add operators for always and/or at some time; obligatory and/or permissible; it is known that; etc.	As with modal logic, truth-value assignments become relativized. For instance, in temporal logic, they are relativized to times.	As with modal logic, we can keep a completely classical logic if we wish, merely adding rules for the new operators. However, some changes may be natural. For instance, one motivation for many-valued logic was "future contingents," the idea that some statements about the future are neither true nor false. Similarly, temporal logic can be many-valued, with sentences about a particular time being neither true nor false at earlier times.
intuitionistic	none	Developing a semantics for intuitionistic logic is tricky! The truth-table approach will not work. Kripke's semantics for intuitionistic logic involves relativizing assignments of truth values (or rather, warranted assertibility values) to atomic sentences to a particular "information state." Information states are rather like times in a system that permits branching time.	unlike any of the other alternative logics discussed here, intuitionistic logic is defined in terms of a change to the proof theory: the abandonment of ¬Elim. (From Barwise and Etchemendy's system, the only change we need to make is the elimination of ¬Elim. However, if we began with a classical system that did not have ⊥Elim (which after all is redundant in B&E's system), we would need to add this rule.)