The Psychology of Uncertainty and Three-Valued Truth Tables

Psychological research on people's understanding of natural language connectives has traditionally used truth table tasks, in which participants evaluate the truth or falsity of a compound sentence given the truth or falsity of its components in the framework of propositional logic. One perplexing result concerned the indicative conditional if A then C which was often evaluated as true when A and C are true, false when A is true and C is false but irrelevant“ (devoid of value) when A is false (whatever the value of C). This was called the “psychological defective table of the conditional.” Here we show that far from being anomalous the “defective” table pattern reveals a coherent semantics for the basic connectives of natural language in a trivalent framework. This was done by establishing participants' truth tables for negation, conjunction, disjunction, conditional, and biconditional, when they were presented with statements that could be certainly true, certainly false, or neither. We review systems of three-valued tables from logic, linguistics, foundations of quantum mechanics, philosophical logic, and artificial intelligence, to see whether one of these systems adequately describes people's interpretations of natural language connectives. We find that de Finetti's (1936/1995) three-valued system is the best approximation to participants' truth tables.

The conditional. Six of the nine systems, (1) -(6) in Appendix B, extend de Finetti's 2x2 table for the conditional by using a 3x3 conditional table identical to de Finetti's ( | !" , Table 1, column 5). System (1) is Fi itself, but these six systems differ from each other by having different conjunctions or disjunctions (see below). The last three systems, (7) -(9) in Appendix B, extend de Finetti's 2x2 table for the conditional by using a 3x3 conditional table that differs from de Finetti's. We call one of them, in (7)  Conjunction and disjunction. The nine systems select their conjunctive and disjunctive connectives among four types that were defined earlier by various researchers working on three-valued logic. They are the following (their truth tables are given in Table A.2): (i) Kleene-Łukasiewicz-Heyting (henceforth KLH), denoted by ∧ ! and ∨ ! (ii) Bochvar (internal), denoted by ∧ ! and ∨ ! (iii) Sobocińsky, denoted by ∧ ! and ∨ ! (iv) McCarthy, denoted by ∧ ! and ∨ ! .  (⋀) and disjunction (⋁).
The constitution of the nine systems in terms of the basic types of connectives (conditional, conjunction, disjunction) is shown in Table A3. We denote the third value by ∅ for all the systems to avoid increasing the number of symbols. Material conditional and material biconditional. Once the systems (denoted generically by X) have been defined and differentiated by their conditional, conjunction, and disjunction connectives, they can be enriched by their internal material conditional connective ⊃ ! C. This connective is an extension of the four logical cases of the 2x2 material conditional ⊃; the way the five empty cells (that is, the other five logical cases) are filled up will again yield different truth tables (see Table A.4). Similarly a material biconditional connective ⇔ ! can be defined (based on the conjunction of ⊃ ! C and its converse C ⊃ ! like in two-valued logic, see Table A.4).
Biconditional. With two exceptions, there is no definition in the nine systems of a biconditional connective based on the conditional (that would extend the "defective" biconditional C|| """ A). However, from a psychological point of view, if people naturally interpret the natural language conditional following the conditional table, they must also interpret the natural biconditional as the conjunction of two conditionals. It is then important to introduce the biconditional in the systems in which it is lacking. This supplements the systems without logical alteration, because the biconditional is defined using two connectives already defined, the conjunction and the conditional following the formula: As far as the Fi system is concerned, Gilio, Over, Pfeifer and Sanfilippo (2017) showed that (using our notations) the following equalities obtain: The different biconditional connectives for the nine systems are presented in Table A.5.
For the systems 2, 3, 4, 7 and 8 (denoted by X) the following relation obtains: The tables for C|| !" A and C|| !"# A do not extend the 2x2 defective biconditional table C|| """ A.

Appendix B. The origin of three-valued logics and the nine "extended" systems
In a range of natural circumstances, people cannot readily classify sentences as true or false. Aristotle made this point in his discussion of future-contingent sentences such as (S1) there will be a naval battle tomorrow. He argued that declaring this sentence definitely true or false now can only be done at the risk of fatalism. Another circumstance pointed out by Frege (1892/1952) concerns sentences that contain non-referring singular terms, such as (S2) Ulysses was set ashore at Ithaca while sound asleep. Declaring the sentence true or false requires that Ulysses have a reference, failing which the sentence is neither true nor false. However, these insights did not lead to the elaboration of logical systems. Rescher (1969) identifies MacColl (1837MacColl ( -1909, C. S. Pierce (1839( -1914( ) and N.A. Vasiliev (1880( -1940 as the first authors to have proposed systems of propositional logic where propositions can have more than two values. Reflecting on Aristotle's case of future-contingent sentences, Łukasiewicz (1920/1967) proposed a third truth value, and he did formulate a full threevalued system that challenged standard two-valued logic. It notably rejects the logical principles of the excluded middle and non-contradiction. The three values are true (denoted by 1), false (denoted by 0), and a third value often denoted by ½, which represents what is possible. In 1922, Łukasiewicz extended his three-valued logic to a many-valued logic, in which a degree of possibility is given a number between 0 and 1. Other circumstances than contingency or lack of reference can result in the attribution of a third truth value. In 1938, Bochvar's investigation of the semantic paradoxes resulted in a three-valued logic, in which the third value indicates that the sentence is undefined or meaningless. For instance, a version of the Liar paradox (S3) this sentence is a lie can be neither true nor false, since assuming that it is true implies that it is a lie and assuming that it is false implies that it is true. In 1938, Kleene's work on recursive functions in mathematical logic resulted in the elaboration of a three-valued logic in which the third value refers to what is non-decidable. Non-decidability can be thought of as a radical type of uncertainty in mathematics. There is a distant relation between Kleene's work and ours, but our notion of uncertainty does not come from a lack of mathematic information, but rather from limited visual information. More important for our purpose is the investigation of conditionals in logic and linguistics. The question of the interpretation of a conditional sentence whose antecedent is false led de Finetti (1935de Finetti ( , 1936 to the definition of the conditional event, which is void in such a case, that is, it cannot be classified as true or false. (It can be declared true or false only in case its antecedent is true). As can be seen, different authors pursued quite different objectives, and accordingly the conception of the third value varied widely. One dividing line separates the third value considered on a non-epistemic par with truth and falsity, as in Łukasiewicz's logic, and the third value as an epistemic absence of truth or falsity, as in de Finetti's logic. In de Finetti, the third value, "void", is superimposed on bivalent logic and is given an epistemic interpretation, which we operationalized in our experiment as uncertainty caused by a lack of visual information. Ultimately in de Finetti, the third value becomes the full range of subjective probabilities. For introductions to many-valued logic, see Haack (1974) andRescher (1969), and for discussions of the interpretation of the third value, see Cobreros, Egré, Ripley and van Rooij (2014), Gottwald (2015), and Rescher (1962).
The nine "extended" systems (1) to (9) Six systems, (1) -(6), out of the nine formally adopt de Finetti's conditional event table  The 3x3 de Finetti conditional event table (C| !" A, Table 1, column 5) has been constructed by several authors. Most of these authors failed to attribute the table to de Finetti; they actually rediscovered the table, giving various interpretations of the third value depending on their research field. Here we consider six three-valued systems that are comprehensive, starting with de Finetti's original Level 1 system Fi (with five additional versions of it numbered (a) to (e)), followed by five systems coming from formal logic or philosophical logic.
(1) The Fi system As laid out by de Finetti, the Fi system includes the involutive negation ¬ ! , the conjunction ∧ ! and the disjunction ∨ ! as well as the material conditional ⊃ ! . (There are two additional connectives that allow to fall back on a two-valued system: a Thesis connective T(A) which means "A is true", and a Hypothesis connective H(A) which means "A is not null", X = C| !" A; T X = C ∧ A and X = T(X)| !" H(X) (see Table A.1, columns 4 and 5). Fi has been used by several authors (Baioletti & Capotorti, 1994, 1996Coletti & Scozzafava, 2002;Milne, 1997;Mura, 2009;Rothschild, 2014). Some authors have defined an expansion of the Fi system that maintains tautologies of bi-valued logic (such as "¬ ! A ∨ ! A", see Mura for a discussion). In this approach Baioletti & Capotorti (1994, 1996 equipped Fi with two additional negations called "left" (denoted by ¬ ! ) which yields F when ∅ is negated and "right" (denoted by ¬ ! A) which yields T when ∅ is negated 1 (see Table A.1, columns 4 and 5). We present below the different authors who have concurred to propose a system formally identical to Fi system, but motivated by different objectives and offering various interpretations of the third value. The following five systems are logicians' creations.