Page 187 - Encyclopedia of Philosophy of Language

cept 'neither true nor false,' since this is a phenomenon which occurs in other contexts, as in the case of pre- supposition failure (e.g., 'the present king of France is bald'). In any case, the transition from met- alanguage to metametalanguage does not seem to require any further 'conceptual' change.
6.3 Separating Assertibility and Deniability from Truth and Falsity
By means of the diagonal argument, it is possible to construct a Liar sentence L which says: L is not true (either false, or neither-true-nor-false, or paradoxical, or whatever). Clearly, one cannot truthfully assign the value 'true' to L, since this quickly leads to a contradiction. So, if the semantic theory says anything at all about L, it must say that L is not true (because paradoxical, or whatever). But this means that the theory will include a statement which is equivalent in meaning to the paradoxical sentence L. This is known as the problem of the Strengthened or Extended Liar. (The problem of superheterologicality above is an example of this phenomenon.) In the 1980s, two novel solutions to this problem were proposed. T. Parsons (1984) has suggested that we deny that L is true with- out asserting the paradoxical claim that L is not true. Feferman (1984) has proposed that it should be asserted that L is not true without claiming that what has been asserted (which amounts to L itself) is true. Thus, either deniability is distinguished from the assertibility of the negation, or assertibility is dis- tinguished from truth. Both involve quite radical departures from ordinary practice.
6.4 Accepting Some Contradictions as True
An even more radical departure from ordinary prac- tice was suggested by Graham Priest (1984). He rec- ommends accepting as true the claim: 'the Liar sentence is both true and false.' Priest does not pro- pose the development of a consistent theory (in some metalanguage) about an inconsistent semantic theory (expressed in the object language); such a proposal would be a variant of Kripke's (see Sect. 6.2 above). Instead, Priest rejects the object/metalanguage dis- tinction and knowinglyembraces an inconsistent the- ory about the paradoxical. This necessitates the development of a 'paraconsistent logic' in which, unlike classical logic, not everything follows from a contradiction. Unfortunately, such logic turns out to be quite weak, lacking such rules as modus ponens and the disjunctive syllogism.
6.5 Context-Dependent Type Theory
The semantic paradoxes can be averted and the uni- versality of natural language preserved if a natural language is identified with a transfinite Tarskian hier- archy of formal languages. Unfortunately, there are several obvious objections to such an identification. First, there is nothing in the syntax of natural
language to suggest the existence of Tarskian type restrictions. Second, paradoxical statements like the Liar do not seem to be ungrammatical. Third, when making some claim about all or some sentences of a certain kind, such as 'All of Nixon's utterances about Watergate are false,' the speaker typically has no way of knowing the Tarskian levels of Nixon's relevant statements, and so has no idea of the appropriate level to attach to his own use of 'false.' Fourth, as Kripke (1975) and Prior (1961) have pointed out, the para- doxicality of some statements depends on contingent, empirical facts. Paradoxicality does not seem to be an intrinsic feature of the meaning or logical form of a sentence. For example, the sentence 'the sentence written on the blackboard in Waggener Hall 321 on June 12, 1990 at noon is false' is paradoxical if that very sentence is in fact on that blackboard at that time, a fact which cannot be ascertained simply by inspecting the sentence itself.
All of these objections can be met if the relativity to a Tarskian level is a pragmatic, context-dependent feature of a sentence token. This idea was first pro- posed by Ushenko (1957) and Donnellan (1957), and developed by Charles Parsons (1974), Burge (1979), Gaifman (1988),and Barwiseand Etchemendy(1987). Burge combined the Tarskian hierarchy idea with Kripke's truth-value gap theory, stipulating that sen- tence tokens which are interpreted as containing an inappropriately low level of 'is true' are not to be categorized as ungrammatical or meaningless (as in Russell's or Tarski's type theory). Each level of truth is semantically incomplete: for each level a, some tok- ens are neither truea nor falsea. Each level of truth incorporates all of the semantical information about lower levels. Surge's account of the Strengthened Liar goes as follows. The Liar token L =L is not true is assigned the level 0. So, L = L is not true0. The token L is not in fact true0, since trutho cannot include any evaluation of paradoxical tokens like L (a conse- quence of the diagonal argument). Since L correctly states that it is not true0, L is truei. Truth, can incor- porate the information about L's nontrutho. It is not in fact contradictory to conclude 'L is not true and L is true,' since the interpretation of the predicate 'is true' has shifted between the first and second conjunct, for context-dependent reasons.
Barwise and Etchemendy (1987) have developed a similar account using what is known as 'situation theory,' combining non-well-founded set theory with a realist theory about such entities as properties, relations, and propositions. Gaifman (1988) and Koons (1990) have developed algorithms for assigning Tarski/Burge levels to occurrences of 'is true' in con- crete networks of tokens. A difficulty which remains to be overcome is the development of an account of how the theory itself can be stated with sufficient generality, given the restriction that every occurrence of 'is true' must be assigned to some definite level
Paradoxes, Semantic
165