The best evidence I've encountered yet that the way we've learned to use binary logic is not our natural way of thinking comes out of psychology, where they have developed a trivalent way to evaluate truth values out of necessity, because people keep using logic this way, naturally:

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 conditionalif A then Cwhich was often evaluated as true whenAandCare true, false whenAis true andCis false but irrelevant (devoid of value) whenAis false (whatever the value ofC). 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.

Source: *Frontiers | The Psychology of Uncertainty and Three-Valued Truth Tables | Psychology*