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Chapter 4: Truth Tables and Arguments |
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4.1: Introduction |
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Recall what we said about arguments in Chapter One. An argument is valid if the truth of the premisses guarantees the truth of the conclusion.
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4.2: Valid and Invalid |
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In propositional logic we're interested in arguemnts which are valid because of the propositional structure of the premisses and conclusion. An argument is valid iff, on every row of its tuth-table on which all the premisses are true, so is the conclusion.
Consider the argument form of modus ponens (affirming the antecednt) with the form
p p É q q
This has a truth-table
p q p ( p É q ) q 1 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 0 1 0
There is no row on which all premisses are true and the conclusionis false, so the argument form is valid.
An argument is invalid iff there is at least one row of its truth-table on which all the premisses are true and the conclusion is false. The invalidating row is marked with an arrow. For example:
q p É q p
has the truth-table
p q q ( p É q ) p 1 1 1 1 1 1 0 0 0 1 0 1 1 1 0 ¬ 0 0 0 1 0
The invalidating row provides a truth-tabular counter-example to an argument. We can also describe the counterexample by listing the values of the letters for the invalidating row. In this example p = 0, q = 1.
A good trick is to find the values of the conclusion first, because you're only interested in the rows where it is false. For just those rows you can then check whether the premisses are true. And of course, you only need to continue until you find your first invalidating row. One is enough for invalidity.
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4.3: Translating and Testing |
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Arguments are translated into PL for testing in PC in the same way that we did the translations in Chapter One. The usual practice is to use the translation for testing. We test the explicit form of the argument, the most detailed form of the argument.
General Comments on Translating
1. The six operators we've defined are not abbreviations of words in English. They are representations of the propositional operations that you have identified in the text. You should keep this in mind when translating. EG: The word 'and' in the sentence 'Kiss me and a prince will appear' doesn't act as a conjunction but as an implication. The sentence means 'If you kiss me then a prince will appear.'
2. Beware of ambiguity. EG: does 'Jacjk and Tom are cricketers or footballers' mean 'Both J and T are cricketers or both are footballers' or does it mean 'J is a C or an F, and T is a C or an F'?
3. Sometimes the words 'and ' and 'or' are not the most important indicators of the propositional operations involved. EG: 'Exactly one of Joe, Bill, and Harry came' means the same as 'Exactly one of Joe, Bill, or Harry came'.
4. Treating conditionals as material (purely truth-functional) conditionals is often wrong. EG: 'It's not true that if it rained yesterday, then my car leaked' has the form ~ ( R É L ). This is equivalent to ( R & ~ L ) but the English sentence doesn't mean the same thing as 'It rained yesterday and my car didn't leak.'
5. Statements about necessary conditions can usually be easily recast 'only if' conditionals. EG: 'Oxygen is necessary for life' is equivalent to 'Thee is life only if there is oxygen' Statements about sufficient conditions can usually be easily recast 'if ... then ---' conditionals. EG: 'A spark is sufficient for an explosion' is equivalent to 'If there is a spark then there is an explosion'
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4.4: Curious Cases of Validity |
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1. Any argument which has a tautology for a conclusion will be valid. 2. Any argument which has an inconsistent set of premisses will be a valid argument.
We don't like these results - they seem to approve of 'bad' arguments.
An argument is logically unsound iff it has an inconsistent set of premisses.
(Note that all logically unsound arguments are valid.)
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