Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Many-Valued Logic Daniel Bonevac February 27, 2013 Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Bivalence ◮ Classical logic is bivalent —every sentence is true or false. ◮ Aristotle restricted logic to statements , sentences that can be true or false. ◮ Bivalent systems go beyond that to assume that all statements are actually true or false. ◮ Most logicians throughout history have been willing to make that assumption. Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Truth Value Gaps ◮ Some, however, think there are good reasons to deny certain sentences a truth value. ◮ Many-valued logics revise classical logic by reinterpreting the truth-functional connectives themselves, allowing for additional truth values or truth value gaps. Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux History ◮ Hugh MacColl (1906), Charles Sanders Peirce (1909), Nikolai A. Vasilev (1910), Jan Lukasiewicz (1920), and Emil Post (1921) asked what might happen if bivalence were rejected, and developed the first many-valued logics. ◮ D. A. Bochvar (1939), Kurt G¨ odel, and Stephen C. Kleene (1952) devised alternative systems. Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Presupposition and Denotation Failure ◮ Some sentences have presuppositions: ◮ Has Jones stopped beating his wife? ◮ The present King of France is bald. ◮ Bill Clinton regrets his affair with Margaret Thatcher. ◮ Bill Clinton does not regret his affair with Margaret Thatcher. Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Presupposition and Denotation Failure ◮ The first is a classic example of a complex question, a question with presuppositions. ◮ Every possible answer to it presupposes something—that Jones has been beating his wife. ◮ In general, one sentence presupposes another if its assertion is infelicitous—that is, inappropriate—unless the latter is true. ◮ The other sentences are all inappropriate, since there is no King of France at present and since Bill Clinton never had an affair with Margaret Thatcher. Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Presupposition and Denotation Failure ◮ Are these sentences true or false? ◮ Philosophers and linguists disagree, but, on one view of the matter, they are neither. ◮ Suppose someone were to ask you, “Is the present King of France bald?” It seems you can say neither “Yes, he is” nor “No, he isn’t,” for there is no King of France. ◮ That suggests that (2) is neither true nor false. ◮ Similarly, suppose someone asks, “Does Bill Clinton regret his affair with Margaret Thatcher?” ◮ You can’t answer with “Yes, he does,” or “No, he doesn’t.” Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Vagueness ◮ Some expressions appear to be vague. ◮ Is Word and Object long? Is Austin large? Is turquoise blue? ◮ Long, large, and blue seem to have no very definite boundaries. ◮ Some books are definitely long: War and Peace , for example, or Remembrance of Things Past . ◮ Others, such as Goodnight Moon , are definitely not long. ◮ But many books fall in between. It’s hard to say whether they are long or not. Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Vagueness ◮ Similarly, New York City is definitely large, and Salt Flat, Texas, is definitely not. ◮ But it’s hard to say whether Pittsburgh, Austin, Tucson, and Hartford are large or not. ◮ The sky, on a clear day, is definitely blue; the desert, on a clear day, is definitely not blue. ◮ But it’s not so easy to say whether turquoise is blue or not. Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Vagueness ◮ Now, consider the sentences ◮ Word and Object is long. ◮ Austin is large. ◮ Turquoise is blue. Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Vagueness ◮ Are they true or false? It’s hard to say. ◮ Now, we might adopt two attitudes about that. ◮ We might be epistemicists and say that they are true or false, but that we just have a hard time telling which. ◮ Or, we might say that they are neither true nor false. Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Degrees of Truth ◮ Some expressions, such as fairly, sort of, and in between , suggest that there are intermediate positions between truth and falsehood, or at any rate that there are degrees of truth and falsehood. ◮ Word and Object is fairly long, but that doesn’t imply that it’s long. ◮ Austin is sort of large, but that doesn’t imply that it’s large. ◮ Turquoise is between blue and green, but that doesn’t imply that it’s both blue and green. ◮ Expressions such as fairly, sort of , and in between seem hard to understand in a logic in which every statement is either true or false. Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Indeterminacy ◮ If we think of the world as determining the truth values of sentences, we might think that some sentences escape determination for other reasons. ◮ Works of fiction, for example, leave many features of their imaginative ”worlds” undetermined. ◮ Consider Sir Arthur Conan Doyle’s Sherlock Holmes stories. Holmes was a detective is plainly true, in the world of the stories, at least; Holmes lived with his wife, Edna, in Rio Linda , is plainly false; Holmes lived in London when he was a child seems indeterminate. ◮ The stories provide no information to corroborate or contradict such a claim. Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Indeterminacy ◮ If this problem were confined to fiction, it might not seem very serious. ◮ One could hold that (almost) all sentences about fictional objects are simply false. ◮ But some philosophers hold that, in other kinds of situations as well, there is no fact of the matter. Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Indeterminacy ◮ Aristotle worried that sentences about the future, for example, might lack a truth value. ◮ If it is now true that there will be a sea battle tomorrow, he reasoned, it must already be determined that there will be a sea battle tomorrow. ◮ If it is now false that there will be a sea battle tomorrow, then it must already be determined that there will not be a sea battle tomorrow. Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Indeterminacy ◮ So, if the sentence There will be a sea battle tomorrow is now either true or false, it must already be determined whether there will be a sea battle tomorrow. ◮ In general, if sentences about the future have truth values, the future must already be fixed. Or so it has been argued. Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Indeterminacy ◮ Lukasiewicz and others sought to unify many-valued and modal logic, thinking of intermediate values as something like possibility. ◮ But this was a dead end. James Dugundji (1940) showed that none of the Lewis modal logics S1-S5 have finite characteristic matrices—meaning, among other things, that they cannot be viewed as many-valued systems with a finite number of truth values. Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Verificationism ◮ Some philosophers hold a theory of meaning according to which the meaning of a sentence is its method of verification. ◮ To say that A is true, on this view, is to say that A can be or has been verified. ◮ Some sentences are decidable in the sense that one can verify whether they are true or false. But many sentences are not decidable in that sense. Daniel Bonevac Many-Valued Logic
Rationales for Many Valued Logic The Strong Kleene System K3 Lukasiewicz’s L3 Many-Valued Tableaux Verificationism ◮ Defenders of bivalence treat this as an epistemic problem: The sentences are either true or false, even if we may not be able to tell which. ◮ But those who identify meaning with verification decline to assert that they are either true or false. ◮ To assume that they are is to venture beyond what we know to the realm of metaphysical speculation. Daniel Bonevac Many-Valued Logic
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