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Transcript
207 Some example proofs: (a) 1. (G H) v (I ~J) 2. G ~H 3. I (J v A) 4. 5. 6. 7. 8. 9. 10. A Pr Pr Pr / A 208 (b) 1. (A B) ~(C D) 2. ~(A v F) 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. ~(D v F) Pr Pr / ~(D v F) 209 (c) 1. B (C E) 2. E ~(J v H) 3. ~S 4. J v S 5. 6. 7. 8. 9. 10. 11. 12. 13. B ~C Pr Pr Pr Pr / B ~C 210 (d) 1. I (B M) 2. M T 3. ~T B 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. ~I Pr Pr Pr / ~I 211 Conditional Proof CONDITIONAL PROOF is a rule of logic that allows you to prove a conditional by using the ASSUMPTION of the conditional's antecedent to get its consequent. -----> simple example: 1. N O 2. N P 3. N 4. O 5. P 6. OP 7. N (O P) Pr Pr / N (O P) Assp (CP) MP 1, 3 MP 2, 3 Conj 4, 5 CP 3-6 ASSUMING the conditional's antecedent is different from having the antecedent as a premise: - the assumption is TEMPORARILY made for the SPECIFIC purpose of getting the conditional's consequent - once the assumption has served this specific purpose, it can't be used in the justifications of further steps 212 So why is assuming the conditional's antecedent legitimate at all? - to prove a conditional, all you have to show is that IF the antecedent is true, then the consequent is true -----> take our simple example again: 1. N O Pr 2. N P Pr / N (O P) 3. N Assp (CP) 4. O MP 1, 3 5. P MP 2, 3 6. OP Conj 4, 5 7. N (O P) CP 3-6 - to prove N (O P) follows, all you have to show is that IF N is true, then (O P) is true (using rules of logic and prior lines of the proof as your resources) - the assumption on line 3. in effect says "If N is true..."; of course, this doesn't mean anything by itself, but it is completed on line 6 with "then (O P)." Then on line 7., the results of lines 3-6 are summarized as N (O P); this summary, which is part of the main proof, discharges the assumption. 213 Some terminology: -----> take our simple example again: 1. N O Pr 2. N P Pr / N (O P) 3. N Assp (CP) 4. O MP 1, 3 5. P MP 2, 3 6. OP Conj 4, 5 7. N (O P) CP 3-6 - lines 3-6 are called a SUBPROOF; the first line of the subproof is the assumed antecedent of the conditional you are proving and the last line is its consequent - the SCOPE of the assumption on line 3. is the subproof (the idea is: the scope of an assumption encompasses the lines where the assumption is operative) - the arrow and vertical line marking the scope of the assumption is called a SCOPE MARKER - note that on line 7., the justification includes the line numbers for the WHOLE SUBPROOF 214 Three restrictions on CP to keep it legitimate: 1. Every assumption must be discharged by the end of the proof. Remember that in using CP assumptions are temporarily made for the specific purpose of proving a conditional. So you CAN'T leave an assumption undischarged--that would be to treat it as an additional premise. 2. Once an assumption has been discharged, neither it nor any step in its scope can be used in the proof again. Remember that the lines in the scope of the assumption are DEPENDENT on the assumption. So you CAN'T use them in justifications AFTER the assumption has been discharged--that would be to treat them as INDEPENDENT of the assumption. 215 3. Assumptions inside the scope of other assumptions must be discharged in the reverse order in which they were made. Numerous uses of CP can be made within a single proof; in fact a use of CP can be made WITHIN another use. When a use of CP is made within another use, the scope of one must be COMPLETELY within the other--scope markers cannot cross. 216 Some example proofs using the rule of Conditional Proof: (a) 1. (A B) (C B) Pr / (A v C) B 217 (b) 1. (A B) v (~A ~B) Pr / A B 218 (c) 1. A D 2. (B D) ~C Pr Pr / A (B ~C) 219 Indirect Proof INDIRECT PROOF is a rule of logic that allows you to prove a negation by using the ASSUMPTION of the OPPOSITE of the negation to get a contradiction, and then inferring the negation. -----> example: 1. N O 2. (N O) P 3. P ~O 4. N 5. O 6. NO 7. P 8. ~O 9. O ~O 10. ~N Pr Pr Pr / ~N Assp (IP) MP 1, 4 Conj 4, 5 MP 2, 6 MP 3, 7 Conj 5, 8 IP 4-9 AGAIN, as with CP, an ASSUMPTION is different from a premise: - with IP the assumption is TEMPORARILY made for the SPECIFIC purpose of getting a contradiction - once the assumption has served this specific purpose, it can't be used in the justifications of further steps 220 How does Indirect Proof work? - you can think of it this way: if you can show that an assumption leads to a contradiction, then you've shown that the assumption leads to an absurd consequence; so we can infer the negation of the assumption 221 - or you can think of it this way: treat the subproof in IP as being like the subproof in CP in the following way: -----> take our example again: 1. N O 2. (N O) P 3. P ~O 4. N 5. O 6. NO 7. P 8. ~O 9. O ~O 10. ~N Pr Pr Pr / ~N Assp (IP) MP 1, 4 Conj 4, 5 MP 2, 6 MP 3, 7 Conj 5, 8 IP 4-9 - the assumption on line 4. in effect says "If N is true..." and this is completed on line 9. with "then O ~O." So we've gotten the conditional N (O ~O). But we know that O ~O is false (it's a contradiction). So we know ~(O ~O). So we can use MT to conclude ~N. (Note that the use of MT ISN'T explicitly stated.) - then on line 10., the results of lines 4-9 are summarized as ~N; this summary, which is part of the main proof, discharges the assumption. 222 All the terminology (subproof, scope, scope marker) that applies to CP applies to IP as well. Three restrictions on CP apply to IP as well. 1. Every assumption must be discharged by the end of the proof. Remember that in using IP assumptions are temporarily made for the specific purpose of proving a contradiction. So you CAN'T leave an assumption undischarged--that would be to treat it as an additional premise. 2. Once an assumption has been discharged, neither it nor any step in its scope can be used in the proof again. Remember that the lines in the scope of the assumption are DEPENDENT on the assumption. So you CAN'T use them in justifications AFTER the assumption has been discharged--that would be to treat them as INDEPENDENT of the assumption. 223 3. Assumptions inside the scope of other assumptions must be discharged in the reverse order in which they were made. Numerous uses of IP can be made within a single proof; in fact a use of IP can be made WITHIN another use (or within a use of CP). When a use of IP is made within another use, the scope of one must be COMPLETELY within the other--scope markers cannot cross. 224 An example proof using the rule of Indirect Proof: 1. B (A ~B) Pr / ~B 225 An example proof using both Conditional Proof and Indirect Proof: 1. (A v B) ~C 2. D (~F ~G) Pr Pr / (A v D) ~(C F) 226 Theorems Theorems are statement forms that can be proven without any PREMISES; rather theorems are proven with ASSUMPTIONS using CP or IP. - the idea is that theorems follow from the rules of logic ALONE - every theorem is a tautology, and every tautology is a theorem. (REMEMBER: TAUTOLOGIES are statement forms which are true for every substitution instance). -----> example using IP: (a) prove: p v ~p 1. ~(p v ~p) 2. ~p ~~p 3. ~~(p v ~p) 4. p v ~p Assp (IP) DeM 1 IP 1-2 DN 3 227 -----> examples using CP: (b) prove: p (q p) 1. p 2. q 3. pvp 4. p 5. qp 6. p (q p) Assp (CP) Assp (CP) Add 1 Dup 3 CP 2-4 CP 1-5 (c) prove: p v ~p 1. p Assp (CP) 2. pvp Add 1 3, p Dup 2 4. p p CP 1-3 5. ~p v p CE 4 6. p v ~p Comm 5 - in proving a theorem, NEVER assume the theorem ITSELF 228 -----> REVIEW three points: (i) when doing proofs, the arguments are valid; the goal is just to prove that they are valid - the system of 20 logical rules that you've learned (8 rules of inference, 10 replacement rules, CP, and IP) can prove ALL AND ONLY valid arguments - proving ALL valid argument indicates this system of rules is COMPLETE - proving ONLY valid arguments indicates that this system of rules is CONSISTENT (ii) proofs CANNOT be used to demonstrate INVALIDITY - but you can demonstrate invalidity using truth tables 229 (iii) proofs CANNOT be used to demonstrate SOUNDNESS - to demonstrate soundness, you have to show that premises are true: this goes BEYOND LOGIC - think of a premise such as: all whales are mammals--this is a question of biology, not logic 230 Predicate logic REMEMBER this example from the second lecture: Premise 1 All whales are mammals. Premise 2 All mammals are warm blooded animals. Conclusion All whales are warm blooded animals. This is a valid argument. But symbolize this argument in sentential logic. - set up dictionary: W All whales are mammals. M All mammals are warm blooded animals. H All whales are warm blooded animals. - we get: argument form: P1 W P2 M C H P1 p P2 q C r - THAT'S not a valid argument form (you can show that it isn't using a truth table) 231 - in order to understand the validity of: Premise 1 All whales are mammals. Premise 2 All mammals are warm blooded animals. Conclusion All whales are warm blooded animals. we need to represent the logical structure INTERNAL to simple sentences (REMEMBER: a simple sentence is one that does not contain any other sentence as a component--for example, dictionary entries used for symbolization in sentential logic are always simple sentences). PREDICATE LOGIC represents the logical structure internal to simple sentences. The FORM of the argument instance above is: Premise 1 All A's are B's. Premise 2 All B's are C's. Conclusion All A's are C's. - in this argument form, A, B, and C are variables for class terms