Relevance logic
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Relevance logic, also known as relevant logic, is a group of systems of non-classical logic which attempt to resolve the paradoxes of material implication.
The classical definition of implication, material implication, is truth-functional: if we know the truth value of A and B, then we know the truth value of whether A implies B. A implies B if A is false or if B is true. What this means, is that if A is false, then A implies anything. For example, the following is true according to material implication: "If the Earth has two moons, then JFK was never assassinated". Since the antecedent is false, the implication is true, regardless of the truth or falsehood of the consequent. However, most would say that this implication is false, since the two statements have nothing to do with each other. Relevance logic attempts to capture formally this intuitive idea, that the premises must be relevant to the conclusion for the implication to be true. But as a result, the relevant implication is not truth-functional - knowing the truth or falsehood of the antecedent and consequent is insufficient to know whether or not the implication is true.
One attempt to form a notion of implication which better reflects our naive ideas of the meaning of "implies" is strict implication. Strict implication interprets "A implies B" as meaning "necessarily, A implies B", or "in all possible worlds, A implies B". Whereas material implication would consider "If the Earth has two moons, then JFK was never assassinated", strict implication would suggest that statement is false, since (arguably) there is some possible world in which the Earth had two moons, yet the JFK assassination still happened. But, consider another statement "If 1+1=3, then JFK was never assassinated". Strict implication would consider this true, since the same statement (interpreted according to material implication) is true in all possible worlds, since there is no possible world in which 1+1=3 was true. But relevant implication would reject that implication as false, since the premise is irrelevant to the conclusion. Thus may relevant and strict implication be distinguished.