Many Valued Logic: Family of Nonclassical Logic

In the 1920s and 1930s, Three valued logic that belongs to the family of nonclassical logic was introduced. The idea prompted by the logicians was that it is not necessary that all the sentences need to be true or false, but some sentences can be indeterminate in truth value. Łukasiewicz(1920) wrote in one of his papers that ‘Three valued logic is a system of non-Aristotelian logic since it assumes that in addition to true and false propositions, there also are propositions that can neither be true nor false and hence, that there exists a third logical value’.

Other logicians had identified other reasons to think of sentences as neither true nor false. Logicians thought that sentences can be indeterminate concern sentences that involve Vagueness. Apart from possibility, reference failure, and vagueness, the development of three-valued logic can be correlated with at least three other phenomena of interest in which the topic of indeterminacy plays an important role.

In 1935, De Finetti proposed a three-valued treatment of indicative conditional sentences in relation to probability, aimed to model the cases in which the preceding of the conditionals is false, and leaving the conditional undefined. In his table, the conditional will be true when the antecedent and the resultant both are true and false when the resultant is false and the antecedent is true. His overall intuition was that a conditional should be undefined in all other cases, especially when the antecedent comes out to be false.

Bochvar published an article, in 1937, in which he proposed a three-valued calculus which he applied to Russell’s paradox and Grelling’s paradox and did an analysis to build certain paradoxical sentences as meaningless. When all arguments are defined, his logic coincides with classical logic, but whenever one of its components is undefined it assigns undefined value to the sentence. Later on, Kleene designed three-valued tables for the logical connectives, to represent cases in which the truth of a sentence might not be decided.

Until today all those areas are active fields of research. Also, all those frameworks are originally developed to approach one class of phenomena and have been rendered to solve other phenomena. Kleene’s logic was originally developed to elucidate partiality in computation, which has been applied to the treatment of presupposition projection to semantic paradoxes and to vagueness.

So far we have seen how indeterminate sentences occur which results in the third value of logic, but a question arises what does this third truth value really stands for?

Unlike Lukasiewicz, Frege did not propose indeterminacy as a separate truth value, but rather as the lack of a truth value. This difference was later on reflected by the difference between the two kinds of three-valued systems. In the first, the third truth value is admitted on a part with the values true and false. And on the other hand, systems of partial two-valued logics based on truth-value gaps, such as supervaluationist logics, in which a sentence fails to receive a value of True or False. In the supervaluationist framework, a sentence can either be semantically defined or undefined. If it is undefined, one can consider all the possible ways of assigning it a classical truth value. It is then called true if it takes the same classical value ‘True’ under all ways of making it defined, false if it takes the same value False under all ways of making it defined, and otherwise, it is neither true nor false.

For example, if Pranav is a case of a bald man, ‘Pranav is bald’ will be undefined, to begin with, and ‘Pranav is bald’ fails to be either true or false, since the sentence can either be True or False depending on the precisification. On the other hand, whether Pranav is bald or Pranav is not bald will be true, since the disjunction is classically True under all classical assignments of value to ‘Pranav is bald.

But in the case of Lukasiewicz's original system, the situation is very different. Lukasiewicz symbolizes True with 1, False with 0, and the third value with ½ which stands for ‘possible’. In his method, disjunction follows truth-function and corresponds to the maximum of the values of each disjunct, and similarly, the value of the negation is 1 minus the value of a negated sentence. This implies that the law of excluded middle (A ∨ ¬A) fails to take the value 1 under all assignments (when A gets the value ½). One can think that any trivalent truth-functional system is one, in which the third truth value needs to stand as a primitive notion, but it is not so. Truth value is generally the set of all classical values a sentence can take under all precisifications. For example, one can superimpose a truth function for the connectives on a supervaluationist having an understanding of the values 1, 0, and ½ as assigned to the sentences. A related outlook on the three-valued logics is to view all the three values as a subset of values within a four-valued logic, where the values true, false, both false and true, and neither false and true, can be seen as resulting from a relational rather than a functional two-valued semantics. True is assigned to the sentence that is related to 1 only, False is related to 0 only, Both to a sentence that is related to both 1 and 0, and Neither is related to neither of 1 or 0.