Conditionals & Arguments | |
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Conditionals, typically (though not always!) expressed in English as sentences of the form "If .................. , (then) .................. , state necessary and sufficient conditions. In a conditional so expressed, the clause that follows the "if" is the (1) If someone is a mother then they're female. If you know that someone is a mother (not just a parent) that is enough to show that the person is female therefore being a mother is a sufficient condition on being female.
(2) If you study then you'll pass. (2) clearly says that studying is sufficient for passing. It's not so clear that it says passing is necessary for studying and, in fact, it sounds peculiar because passing is something that happens The oddity arises because conditionals in ordinary English, like (2), may express something more than necessary and sufficient conditions: they may express causation and, hence, temporal succession. Let us however ignore this feature of some conditionals (we'll talk about it later when we "go formal") and consider them only insofar as they are expressions of necessary and sufficient conditions. If we strip away the causal and temporal connotations of (2) it becomes more plausible to see passing as a necessary condition on studying, especially if you recognize that once you discount tense (2) is equivalent to (3) If you didn't pass then you couldn't have studied. When one thing is necessary for another thing that means that if you haven't got the first then you haven't got the second. Water, for example, is necessary for plant life: no water, no plants. Similarly, (3) says no pass conclusively shows no study, hence that passing is necessary for studying. In general, for any sentence of the form, “If P then Q,” its (4) For any integers
(5) If Michelle Bachmann is elected then I'll eat my hat. (6) Michelle Bachmann will be elected, therefore I will eat my hat.
(7) If you study then you'll pass. If you pass then you'll graduate. Therefore if you study you'll graduate. (8) If a number is even then it's divisible by 2 without a remainder. 4 is divisible by 2 without a remainder. Therefore, 4 is even.
For any given argument, the conditional that is formed by taking the conjunction (the "and-ing") of its premises as the antecedent and the conclusion of the argument as its consequent is the corresponding conditional to that argument. For example, (9) below is an argument and (10) is its corresponding conditional: (9) All men are mortal. Socrates is a man. Therefore, Socrates is mortal. (10) If all men are mortal and Socrates is a man then Socrates is mortal. |