A Classicist's Note on Two-, Three-, and Four-Valued
Joseph S. Fulda
The classicist's principal dictum, «A proposition is either true or false, not neither, and not both,» still leaves considerable room for multi-valued logic. To the classicist, two-valued logic is the logic of reality, three-valued logic is the logic of knowledge about reality, and four-valued logic is the logic of beliefs about reality.
The three values are known to be true, known to be false, and unknown; the four values are believed to be true, believed to be false, not believed to be either true or false (note that this is not «believed not to be either true or false» which is a belief within the domain of philosophy of logic and, more generally, a meta-belief which some, the classicist would say, believe to be true, some -- including the classicist -- believe to be false, etc.), both believed to be true and false (note that this is not «believed to be both true and false» -- with the same comment as made above).
The philosophical distance between knowledge and reality is a huge matter treated in countless philosophical papers and treatises, but which we shall not even touch on here. The philosophical distance between belief and knowledge, on the other hand, is smaller: An excellent summary of current thinking on the matter can be found in Sturgeon (1993).
The logical distance between four-valued logic and three-valued logic is bridged by De Morgan's Law: T&F<->~(TVF). The logical distance between three- valued logic and two-valued logic is bridged by (a) the «closed world assumption» which renders the value «unknown» as F, and (b) the definition of conjunction, which renders the gap value, ~(TVF), i.e. T&F, as F.
We now consider some potential challenges to this account. Goldstein (1992) has convincingly argued that the Liar has neither truth value, and has proposed an elegant solution to the paradox that also does not fall afoul of the Strengthened Liar. But his truth value gap does not pose a challenge to the classical account given here, for his solution distinguishes between use and mention and on his account it can easily be argued -- indeed, it is hard to argue otherwise -- that it is not the case that the Liar is a proposition with a third truth value (neither T nor F), but simply that the Liar, like so many other sentences, is just not a proposition at all.
G”del's incompleteness theorem and, more generally, unsolvability, unprovability, and incompleteness results also pose no problem for our account, since each such result is within a system -- or all systems considered individually -- and therefore it is the logic of knowledge -- three-valued logic -- that is appropriate.
Heisenberg's uncertainty principle and, more generally, much of modern quantum physics pose no problem for our account, for it is not so much that certain statements lack a truth value as that either (a) their truth values cannot be known, or, and this is different from the case above, (b) that such statements are either vague (more often) or ambiguous (less often). In the latter case, the sentences are propositional functions and not propositions and, indeed, propositional functions bear no truth value.
What will be seen as troubling to many is the fourth truth value in the logic of beliefs. Certainly it troubled Moore. But the consensus solution to the paradox of the preface holds that the author rationally believes each of the statements in his book to be true, for he has researched them. He also rationally believes that at least one of them is false, knowing his own fallibility. Yet the two beliefs are contradictory. For those not accepting this consensus solution there is also Crimmins' (1992) elegant example. Some, like Goldstein (1993), reject that, too. We may respond that we are speaking about systems of beliefs -- perhaps those implicit in a knowledge base formed by entries from different agents, perhaps those of a philosophical system elaborated on by more than one thinker. We can even say that the fourth value does not ever represent a rational belief choice, but it is still a belief choice that the empirical evidence shows is made with great frequency.
There is a place for multi-valued logic even for adherents of «the three laws of thought.»
This note is dedicated with much appreciation to my most inspiring college teacher, Professor Michael Anshel.
Crimmins, Mark, «I falsely believe that p,» Analysis 52/3 (July 1992): 191.
Goldstein, Laurence, «`This statement is not true' is not true,» Analysis 52/1(January 1992): 1-5.
Goldstein, Laurence, «The fallacy of the simple question,» Analysis 53/3 (July 1993): 178-181.
Sturgeon, Scott, «The Gettier problem,» Analysis 53/3 (July 1993): 156-164.
Joseph S Fulda
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