Biconditionals and “biconditional probability” in reasoning

Conditionals play crucial roles in human reasoning. They represent rules, causal relationship, and explicit inference. Psychology of reasoning, from its inception, has been studying how humans use and understand conditionals. One prominent feature of humans’ conditionals is that it does not follow the pattern of the standard model of conditionals. In classical logic, the corresponding connective to conditional “if p then q” is material implication, p ⊃ q = ¬p ∨ q. p ⊃ q is true as far as the antecedent p is false, or the consequent q is true. This property directly leads to the paradoxes of material implication that are counterintuitive. Wason (1966) found that people’s truth table for “if p then q” differs from material implication in the ¬p cases (p is false and q is true, or p is false and q is false). People very often construct a truth table lacking truth values for the ¬p cases. They consider the ¬p cases as irrelevant to the truth of the conditional “if p then q” The truth table with the truth value gaps was later called “defective conditional” pattern (e.g., Evans & Over, 2004). Recently, a new paradigm psychology of reasoning (Over, 2009; Oaksford & Chater, 2007) was proposed. It propounds a Bayesian, probabilistic framework that well describes human reasoning behavior and proves its rationality. The paradigm is based on “the Equation” (Edgington, 1995),