mod3 - Module 3: Proof Techniques Theme 1: Rule of...

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Module 3: Proof Techniques Theme 1: Rule of Inference Let us consider the following example. Example 1 : Read the following “obvious” statements: All Greeks are philosophers. Socrates is a Greek. Therefore, Socrates is a philosopher. This conclusion seems to be perfectly correct, and quite obvious to us. However, we cannot justify it rigorously since we do not have any rule of inference . When the chain of implications is more complicated, as in the example below, a formal method of inference is very useful. Example 2 : Consider the following hypothesis: 1. It is not sunny this afternoon and it is colder than yesterday. 2. We will go swimming only if it is sunny. 3. If we do not go swimming, then we will take a canoe trip. 4. If we take a canoe trip, then we will be home by sunset. From this hypothesis, we should conclude: We will be home by sunset. We shall come back to it in Example 5. The above conclusions are examples of a syllogisms defined as a “deductive scheme of a formal argument consisting of a major and a minor premise and a conclusion”. Here what encyclope- is to say about syllogisms: Syllogism , a mode of argument that forms the core of the body of Western logical thought. Aristotle defined syllogistic logic, and his formulations were thought to be the final word in logic; they underwent only minor revisions in the subsequent 2,200 years. We shall now discuss rules of inference for propositional logic. They are listed in Table 1. These rules provide justifications for steps that lead logically to a conclusion from a set of hypotheses. Because the emphasis is on correctness of arguments, these rules, when written as a proposition, are tautologies . Recall that a tautology is a proposition that is always true. 1
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Table 1: Rules of Inference Rule of Inference Tautology Name Explanation Addition If the hypothesis is true, then the disjunction is true. Simplification If a conjunction of hypotheses is true, then the conclusion is true. Conjunction If both hypotheses are true, then the conjunction of them is true. Modus ponens If both hypotheses are true, then the conclusion is true. Modus tollens If a hypothesis is not true and an implication is true, then the other proposition cannot be true. Hypothetical syllogism If both implications are true, then the resulting implication is true. Disjunctive syllogism If a disjunction is true, and one proposition is not true, then the other proposition must be true. Let us start with the following proposition (cf. Table 1) The table below shows that it is a tautology. TT T T T TF F F T FT T F T FF T F T This tautology is the basis of the rule of inference called modus ponens or law of detachment that we actually used in Example 1 to infer the above conclusion. Such a rule is often written as follows: .
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This homework help was uploaded on 04/19/2008 for the course CS 182 taught by Professor W.szpankowski during the Fall '08 term at Purdue University-West Lafayette.

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mod3 - Module 3: Proof Techniques Theme 1: Rule of...

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