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Deduction, Proofs, and Inference Rules

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Let’s Review What we Know Take a look at your handout and see if you have any questions You should know how to translate all of these fairly simple sentences into their logical components (be able to go from English to logic symbolization)

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Formal Proof of Validity: Translate the Following If Anderson was nominated, the she went to Boston. If she went to Boston, then she campaigned there. If she campaigned there, she met Douglas. Anderson did not meet Douglas. Either Anderson was nominated or someone more eligible was selected. Therefore someone more eligible was selected.

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1. Translate A B B C C D ~D A v E Therefore E ∩ ∩ ∩

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2. Establish Validity It might seem obvious this argument is valid, but we want to prove it We could use truth tables, but this would require us to make a table with 32 rows since there are 5 different simple statements involved So now what? Prove validity by deducing its conclusion from its premises using already-known, elementary valid arguments

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2. Establish Validity (still) We’ll use three of these basic rules of inference (there are a total of 9) in this example: 1. Hypothetical Syllogism (H.S.) If p then q And if q then r Therefore: if p then r 2. modus tollens (M.T.) If p then q ~q Therefore ~p 3. Disjunctive Syllogism (D.S.) p v q ~p Therefore q

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Validity Established Looking at the argument we want to prove valid, we see that the conclusion can be deduced from the five premises of the original argument by four elementary valid arguments (2 H.S. + 1 M.T. + 1 D.S.) This proves that our original argument is valid

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3. Write the Proof 1. Write the premises and the statements that we deduce from them in a single column – to the right of this column, for each statement, its “justification” is written (basically the reason why we include that statement in the proof) 2. List all the premises first, then the logic (e.g. inference rules) used to get at the conclusion (which will be listed last)

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3. What it Looks Like 1. A B 2. B C 3. C D 4. ~D 5. A v E 6. A C 1,2 H.S. 7. A D 6,3 H.S. 8. ~A 7,4 M.T. 9. E 5,8 D.S.∩ ∩ ∩ ∩ ∩ The justification for each statement (the right most column) consists of the numbers of the preceding statements from which that line is inferred, together with the abbreviation for the rule of inference used to get it

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Definitions A formal proof that shows an argument is valid is a sequence of statements, each of which is either a premise of that argument or follows from preceding statements of the sequence by an elementary valid argument (i.e. our inference rules), such that the last statement in the sequence is the conclusion of the argument whose validity is being proved An elementary valid argument is any argument that is a substitution instance of an elementary valid argument form (e.g. our inference rules) We don’t have time to prove the validity of each one of these statements, so take our word for it that they are valid

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More Complex Substitutions (A B) [C ≡ (D v E)] A B Therefore C ≡ (D v E) This sequence above is an elementary valid argument because it is a substitution instance of the elementary valid argument form modus ponens (M.P.), another one of our inference rules. See if you can see it: modus ponens (M.P.): If p then q And p Therefore q∩

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The Nine Rules of Inference (Pt. 1) 1. Modus Ponens (M.P.) If p then q p Therefore q 2. Modus Tollens (M.T.) If p then q ~q Therefore ~p

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The Nine Rules of Inference (Pt. 2) 3. Hypothetical Syllogism (H.S.) If p then q And if q then r Therefore: if p then r 4. Disjunctive Syllogism (D.S.) p v q ~p Therefore q

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The Nine Rules of Inference (Pt. 3) 5. Constructive Dilemma (C.D.) (p q) (r s) p v r Therefore: (q v s) 6. Absorption (Abs.) p q Therefore: p (p q) ∩ ∩ ∩ ∩

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The Nine Rules of Inference (Pt. 4) 7. Simplification (Simp.) p q Therefore: p 8. Conjunction (Conj.) p q Therefore: (p q)

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The Nine Rules of Inference (Pt. 5) 9. Addition (Add.) p Therefore: (p v q) These nine rules of inference correspond to elementary argument forms whose validity is easily established by truth tables. With their air, formal proofs of validity can be constructed for a wide range of more complicated arguments.

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Example: Prove the following given the premises (using inference rules) 1. W X 2. (W Y) (Z v X) 3. (W X) Y 4. ~Z Therefore X ∩∩ ∩ ∩

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Solution: (Strategy Hint: see what you can ‘create’ from the premises using the inference rules we know. Keep in mind what you’re looking for: this will keep you on track) 1. W X 2. (W Y) (Z v X) 3. (W X) Y 4. ~Z 5. W (W X) 1 Abs. 6. W Y 5,3 H.S. 7. Z v X 2,6 M.P. 8. X 7,4 D.S. ∩ ∩ ∩ ∩ ∩ ∩ Line 5: look at line 1. Use our absorption rule Line 6: A little harder: look at line 5 then 3: it follows the H.S. pattern: W (W X) (W X) Y Therefore: W Y Line 7: a fairly simple M.P. form from lines 2 and 6 Line 8: use D.S. from lines 7 and 4 ∩ ∩ ∩

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Example 2 1. I J 2. J K 3. L M 4. I v L Therefore: K v M ∩ ∩ ∩

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Solution 1. I J 2. J K 3. L M 4. I v L 5. I K 1,2 H.S. 6. (I K) (L M) 5,3 Conj. 7. K v M 6,4 C.D. ∩ ∩ ∩ ∩ ∩ ∩

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