The Three Valued Logic as a Type 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, 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 done 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, have been rendered to solve other phenomena. Kleene’s logic originally developed to elucidate partiality in computation, which has been applied to the treatment of presupposition projection to the semantic paradoxes and to the vagueness. So far we have seen how indeterminate sentences occurs 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 the 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 system. 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, 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 original system, the situation is very different. Lukasiewicz symbolises 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 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.

The introduction of the third truth value in logic raises some delicate questions of interpretation. In most of the interpretation, the third value oscillates between an ontic interpretation and an epistemic interpretation. In Lukasiewicz’s original system, ‘possible’ was taken as ‘factually unsettled’. De Finetti argue that for him propositions can only be false or true, but later he takes the third truth value to represent subjective uncertainty about the proposition and keeps the third truth value undecidable by the algorithms whether true or false. The relevant sense of epistemic needs to be qualified, because the excluded middle (A ∨ ¬A) remains undefined rather than true. Many systems of the three-valued logic are inclined to both interpretations. A more unbiased condition on the interpretation of the third truth value is that, depending on the application, some sentences are assigned a special semantic value, other than True or False, to present the fact that such sentences are not arguable in the way True and False sentences are, and that they do not necessarily support the same inferences. In theories of presupposition projection, the third truth value is used to represent cases in which a sentence is concluded inappropriate, for cases of presuppositional failure. The value ½, here, stands for inappropriate or defective. In some theories of Vagueness, the ½ value is used to assign a special semantic status of borderline cases and again the choice of the scheme will depend on how the vagueness of a subsentence is inherited to larger sentences. If someone considers the Bochvar’s treatment of paradoxical sentences, then the third value will come out to be meaningless to separate a class of sentences from true or false values.. The same concern is at stake in Kripke’s theory of truth, where Kleene’s three-valued logic is used to layout sentences such as the Liar or the Truth-Teller, which Kripke defines as ‘ungrounded’.

From the logical point of view, the three-valued logic can be seen as only a first step towards the broader family of many-valued logics. Sometimes, the three-valued logics are viewed as offering only a limited excess of freedom over two-valued logic for that matter, mostly by increasing the space of interpretations for the logical connectives, that is the option of 39 truth-functional tables vs 24 choices in a two-valued logic for a binary connective. This seems to be a significant rise in the possibilities, especially when it comes to the modelling of subtle connectives like the conditionals. But depending on the applications it is sometimes considered desirable to introduce even more truth values. On closer examination, the three-valued logic offers more options. As already mentioned, the elucidation of connectives in three-value settings need not be truth-functional, the third value can be assigned non-truth functionally. And in another way, the definition of logical consequence in a three-valued system leaves various choices open. When logical results are interpreted in terms of the preservation of the designated values, then one basic choice is between the preservation of the value (1), that is the preservation of Truth or strong consequence, and other is the preservation of the non-zero values(1, ½), that is the preservation of nonFalsity or weak consequence.

In the case of three-valued conditional logics, many combinations of the choices have also been considered. For example, it is standard to require both directions of the preservation simultaneously which corresponds to intersecting the logics, in the case of conditionals. And off course there will be another possibility. We can define results in a mixed way: when all premises take the value 1, not all conclusions should take the value 0; or dually, when no premise takes the value 0, some conclusion takes the value 1.

This perspective of logical consequence has been encouraged and taken further in recent years in relation to the treatment of the paradoxes of vagueness, as well as the semantic paradoxes. Such developments have opened new logical perspectives, in particular regarding the interest of nontransitive logics toward a unified treatment of the paradoxes of the vagueness and of the self-referential truth. On the technical side, further recent explorations of the three-valued systems have troubled the proof theory of three-valued logic. The paradigmatic logic of Lukasiewicz, Kleene and De Finetti all share a common base, which stressed on the interpretation of negation, disjunction and conjunction, but they differ in systematic ways on the clarification of the conditional. The number of variants of the systems is too large, but the relation between these logics and the search of integrated proof systems for them has been a fact of continued interest. Some recent interest has also concerned the relation between three-valued logics and modal logics. From early days of many-valued logic, it has been known that normal modal operators are not appropriately expressible by means of n-valued truth-functional connectives, which clearly shows that modal logics are more expensive than the n-valued logics, can also say the three-valued logics. But all these discussions arise a natural question to us, whether many-valued systems and in particular the three-valued systems, can be planted in a two-valued modal system?