Non-truth-functional

A concept in logic and philosophy that describes statements, propositions, or logical systems where the truth value (true or false) of a compound statement cannot be determined solely from the truth values of its constituent parts. In truth-functional logic, the truth value of a complex proposition is a function of, and is determined by, the truth values of its simpler components. non-truth-functional contexts, however, introduce elements that go beyond the truth-values of the individual components. These often involve factors like necessity, possibility, belief, intention, or context, that are not reducible to simple true/false assessments. This means the overall truth of a statement isn't predictable solely from knowing the truth/falsehood of its parts. These can be more difficult to analyze. The concept is critical in understanding modalities, intensional contexts, and the limitations of purely extensional logical systems. This type of language often requires more complex semantic analysis, for example when looking at human's thoughts on objects or situations.

Non-truth-functional meaning with examples

• The sentence 'John believes that it is raining' is non-truth-functional because even if it *is* raining (a true statement), the overall sentence could still be false if John *doesn't* believe it. The truth value of the overall statement depends on John's belief, a non-truth-functional element.
• Consider 'It is necessary that 2 + 2 = 4'. The truth value is absolute, yet '2 + 2 = 4' is true, but the overall statement hinges on necessity, not just the truth of the component. The concept of 'necessity' contributes to the truth value of the whole.
• In the conditional statement 'If the cat is black, then the sky is blue,' the truth value hinges on the relationship between the two propositions. However, a counterfactual conditional such as 'If the cat were black, the sky would be green' is non-truth-functional. The outcome depends upon the unrealized connection.
• The imperative sentence 'Close the door!' is non-truth-functional. It does not assert a fact, but it issues a command. It is neither true nor false; it's fulfilled or not fulfilled, making it a type of language that cannot be evaluated for its truth value as if its components held meaning.
• Statements involving attitudes like 'Mary hopes that it will rain tomorrow' are non-truth-functional. Whether Mary hopes it, or not, it does not determine the truth of the component proposition. The proposition 'it will rain tomorrow' can be true or false independently of Mary's hopes, and the whole will remain non-truth-functional.