The Psychology of Uncertainty and Three-Valued Truth Tables

Abstract

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.

Correlation Detection with and without the Theories of Conditionals: A model update of Hattori & Oaksford (2007)

“…22 (McGee, 1989; Kaufmann, 2009; Gilio and Sanfilippo, 2014; Sanfilippo et al, 2018). Baratgin, Politzer, Over and Takahashi (2018) have confirmed that the abstract (non-causal) biconditionals has a well corresponding truth table to pARIs. In the subjective probability theory developed by De Finetti, conditional probabilities are atomic and that they correspond to conditional events (de Finetti, 1964).…”

Nested conditionals and genericity in the de Finetti semantics

“…Of the various philosophers who have considered a trivalent treatment of indicative conditionals along these lines, several—notably Adams, Mackie, and Ducrot—were particularly interested in expounding the connection between conditionals and supposition that is suggested by the Ramsey test. Indeed, recent psychological studies have supported the trivalent theory (e.g., Baratgin, Over, & Politzer, 2013; Baratgin, Politzer, Over, & Takahashi, 2018; Politzer, Jamet, & Baratgin, 2020; Politzer, Over, & Baratgin, 2010), and it has been taken up in psychological work as the main representative of a “suppositional” theory of conditionals, supplanting the “no truth‐value” theory popularized in philosophical work under the influence of Adams, Edgington, and Bennett. In addition, several more recent formal studies have expanded on de Finetti's observation that his theory enforces the equation between probabilities of conditionals and conditional probabilities, showing that it circumvents a variety of triviality results (Lassiter, 2020; Mura, 2009; Paneni & Scozzafava, 2003; Rothschild, 2014).…”

De Finettian Logics of Indicative Conditionals Part II: Proof Theory and Algebraic Semantics

“…Informally, one can see the jumps , † , and ‡ as corresponding to the operations of extending a given labeled tree to another labeled tree, where the sequents that are added result from applying the rules of CC/TTm 'upside down', that is, going from a sequent to all its possible premises according to the CC/TTm rules. 6 Now we construct a search tree for every sequent, that is, a labeled tree where the above jumps are systematically applied as many times as possible. For every sequent define (for a limit ordinal ):…”