know that "it is raining" and that "Alicia is wet" then we can be confident that the assertion "If it is raining, Alicia gets wet" is true. However, if it is raining (P is true) and Alicia is not wet (Q is false) then clearly the assertion "If it is raining, Alicia gets wet" is false.

The next two cases are somewhat trickier. Imagine that it is not raining (P is false) and Alicia still gets wet (Q is true), then, psychologically, one may feel uncertain whether the statement if P then Q is true or false. You might want to say, "Well, the statement might be true" or "We don't know whether it's true or not". However, in the context of the logic, we are dealing with a world in which everything is either true or false. Hence, the logicians' choice has been to maintain that the assertion is true, when P is false and Q is true; something else may have made Alicia wet— someone may have thrown a bucket of water over her— so we have no grounds for saying that "If it is raining, Alicia gets wet" is false; therefore, it is true. Again, when P and Q are false—when "it is not raining" and "Alicia is not wet"—the assertion is also considered to be true. This then is the logician's conception of the meaning of if. then (see Table 16.1 for the truth table).

Furthermore, logicians distinguish this treatment of if. then (called material implication in logic) from if and only if, the biconditional (or in logic material equivalence). The biconditional (notated as has a similar truth table to the conditional except for the P is false and Q is true case; it characterises this case as making the assertion false. The reason for this is that the biconditional rules out other states of affairs (like the bucket of water); that is, P ^ Q is read as "if and only if P is true, then Q is true".

We shall see later that people often deviate from these logical interpretations in their reasoning. Using these truth tables it is possible to define valid and invalid inferences (see later section). For example, the inference of Q from the true premises If P then Q and P is a valid inference, called modus ponens, and the inference of not-P from the true premises If P then Q and not-Q is a valid inference, called modus tollens. However, although most people make the modus ponens inference readily, a lot fewer people are willing to make the modus tollens one. As we shall see, these deviations from the dictates of the logic become the evidential bread-and-butter of theories of reasoning (outlined in the next section). The importance of the logical analysis presented here is that it allows us to characterise the abstract structure of reasoning problems and gives us a criterion for determining whether a certain conclusion is valid or invalid, correct or in error.

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