11.3.2: The Logic of And

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Bradley H. Dowden
California State University Sacramento

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Sentential logic explores not only the patterns of sentences but also the patterns of arguments. In this section we will explore arguments whose validity or invalidity turns crucially on the use of the word and. The truth table for “ and,” or “ &,” or conjunction as it is called by logicians, has four rows for all the possibilities:

Each row of the truth table represents a possible situation or way the world could be. If you were to learn that x = 5 and that y < 7, would it be valid for you to infer that y < 7? Yes. It would also be trivial. The general point of that example of reasoning is that the following is its deductively valid form:

If you were to learn that x = 105, would it be valid for you to infer both that x = 105 and that y = 14? No. The general point is this:

The truth table for conjunction can be used to demonstrate that in the previous argument form there are possibilities in which the premise is true while the conclusion is false. [Just left P be true and Q be false.] There are no such counterexample possibilities for deductively valid forms. In this way, the tables provide a general method of assessing the validity of arguments in sentential logic.