Truth Tables Logic Mathematics Worksheet

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Truth Tables (Answers on the next page.) 4.1 Construct truth tables for the following sets of formulas and then answer the questions. A 1. 2. B D a. b. c. d. e. A&~A A~A AAA ~AAA ~(AAA) a. b. c. d. e. ~B~D ~(BD) ~(BCB) B~D ~(BA~D) a. 3. S T b. c. ~(ST)A(~S&~T) ~((ST)A~~(SCT)) ~(~(~ST)A~(~SCT)) 4. G H M a. b. c. GA(HAM) ~MA~(G&H) (G&H)AM Questions 1. Which formulas, if any, are tautologies? 2. Which formulas, if any, are contradictions? 3. Which formulas, if any, are contingencies? 4. Which formulas, if any, are logically equivalent to formula a? 5. Which formulas, if any, are logically equivalent to formula b? 6. Which formulas, if any, are logically equivalent to formula c? 7. Which formulas, if any, are entailed by formula a? 8. Which formulas, if any, are entailed by formula b? 9. Which formulas, if any, are entailed by formula c? 10. Which formulas, if any, are entailed by formula d? 11. Is this set of formulas logically consistent? Table Rules p q p&q T T T F F T F T F F F F d. pq T T T F HA(~MA~G) pAq T T F T pCq T F F T ~p F T F T Table Rules Expressed in English & True output iff both inputs are true.  True output iff at least one input is true. A True output except when input is TAF. C True output iff both inputs are the same. ~ Opposite value. Truth Tables Answers (Answers from the previous page.) 4.2 Construct truth tables for the following sets of formulas and then answer the questions. b. c. 2 1 1 d. 1 e. 2 2 1 1. A A&~A A~A AAA ~AAA ~(AAA) All columns. a. 2 1 T F T F FT F F TF T T FT F T TF T T T F T F FT T T TF F F F T T T F F T F a. 3 2 2. B D ~B~D All columns shown. 1 T F T F FT F FT TF F F T FT F TF TF T TF T T F F 1 F F F T T T T F a. b. 1 ~(BD) F TT T F FT T F TT F T FF F c. 2 d. 1 2 1 ~(BCB) B~D F T T T F F T F F T T T F F T F T T FT F F FT T T TF F T TF b. 3 e. 3 2 1 ~(BA~D) T T F FT F F T FT F T T TF F F T TF 6 3 5 4 6 1 5 4 3 2 8 1 2 T T T T F T F T F F F T F T T F T T T F T T F F F T F F T T T F T F F T F F F F F T T T F F F T T T F T c. 7 6 4 5 T T T F F F T T F T F T F T T F 3. S T ~(ST)A(~S&~T) ~((ST)A~~(SCT)) ~(~(~ST)A~(~SCT)) Main columns only. 2 2 T F T F 2 a. 1 F F T T 1 4 T T T T F T T T b. 3 2 F T T T F T T T T F F F T F F F 4. GHM GA(HAM) ~MA~(GH) Useful columns only. T T F F T F T F T F T F T T F T T T F T T F F T T T F T F F F F T T T T T T F F T T F F T T T T F F F F T T T T F F T T 1 c. 2 (GH)AM T F F F T F F F T T T T F T T T 1 1. Which formulas, if any, are tautologies? ................................................ b,c 2. Which formulas, if any, are contradictions? ........................................... a,e 3. Which formulas, if any, are contingencies? ............................................ d 4. Which formulas, if any, are logically equivalent to formula a? ............. e 5. Which formulas, if any, are logically equivalent to formula b? ............. c 6. Which formulas, if any, are logically equivalent to formula c? ............. b 7. Which formulas, if any, are entailed by formula a? ............................... all 8. Which formulas, if any, are entailed by formula b? ............................... c 9. Which formulas, if any, are entailed by formula c? ............................... b 10. Which formulas, if any, are entailed by formula d? ............................... b,c 11. Is this set of formulas logically consistent? ........................................... no T T T T F F F F 4 1 3 2 F F F F T T T T T F T T T F T T F F T T F F T T HA(~MA~G) T T T T F T F T T F T T F T F T 2 d. 3 4 none a c none a,b,d,e b,c b none a none none none b,d none a,d a all a,b none no yes none none all all all all all all all all yes 4.3 Truth Tables for Validity Use the truth table method to show whether the following arguments are valid or invalid. 1. EA~H, ~H "–  E 5. ~((~MAW)A~(WAM)), ~(WAW), ~~M "– ~(~MCW)&(M~M) 2. ~RCW, ~(~R&W) "– ~(W~R) 6. (D~G)&(MAD), ~(~G&M), ~~(DAD), ~DM "– MC~G 3. RA~S, ~(S&R), ~((R~S)A~R) "– ~R 7. ~(~AAB)&(BA~A), ~((~BA(B&~C))A~A) "– ~(AB)AC 4. ~(AC~B), ~(B&A)A~B "– ~(BAA) 8. ~(RAS)(~SAT), ~(~(~TCR)&S), ~~~S "– ~(RS)A~T Answers (note there are often alternative ways to number the connectives) 1. Invalid, counterexample on row 4 2 1 2. Valid, no counterexample 1 2 3 1 2 3 2 1 T F T T FT F T TF T T FT F F TF F F F F T F T TF T T TTF F F F T F TTF E H EA~H, ~H E R W ~RCW, ~(~R&W) ~(W~R) T F T F T F F T F T F T T T TF F T TF F T F T T F T F T F T F T F T F T T F F FT F TF T FT T TF F T T F F T T F F 3. Invalid, counterexample on row 3 2 1 2 1 5 2 1 4 3 T F T F T T FT F F FT T T TF F T TF F T F T FT TF FT TF 3 2 R S RA~S, ~(S&R), ~((R~S)A~R) ~R T F T F T F FT F T FT T T TF F T TF T T F F F TT T T TF F T F F T T F F F F T T F F T T F 4. Invalid, CE on 1 and 4 8. Invalid, CE on 4 T F F T T T T T T T T F T F T T F T F F 5. Valid, no counterexample 6 1 2 5 4 1 3 ignore 1 2 6 F F T F T T T T T F F F T T T T 5 4 M W ~((~MAW)A~(WAM)), ~(WAW), ~~M ~(~MCW)&(M~M) T F T F T T F F T F T F FT TF FT TF T T T F T T F F F F T T F F T F T T F F T F T T T F T F F F F F T T F F T T T T T T F F T F T F T F T F T F F F T T T F F T T F T T F F T F F T T T FT F T TF T T FT F T TF 6. Invalid, counterexamples on rows 1 and 8 (construction shows connective columns only) 2 1 DGM 4 3 3 1 2 ignore 1 1 2 2 (D~G)&(MAD), ~(~G&M), ~~(DAD), ~DM 1 MC~G T T T F T T T F T F F T T F F F T T T T T F T F T F T F T T F F F F T T F F T T T T T T F T F T T T T T F F F F T T T T T T F F T F T F F F F F T F F F T T T T T F T T T T T T T T T T F F T T F F F F T T T T F T F T F T F T T F T F F T F T 7. Valid, no counterexample (construction shows connective columns only) 3 1 2 6 5 4 7 1 4 3 2 6 5 2 1 3 ABC ~(~AAB)&(BA~A), ~((~BA(B&~C))A~A) ~(AB)AC T T T F T T T F T F F T F F F T F F T T T T T F F F F T F F T T T F T T T F F F F F T T T T F F F F F F F F F F F T T T F T F T F F F T T T T F T T T T T T F F T F T F F F F F F F F T F F T T T T T F F F F T F F T T T F T T T F F F F F T T T T F F T T T T F T F T F T T T F T F T F F F T T T T F T T T F T T T F T T T T 4.4 True/False Questions To answer the questions below you should make use of tables. Clarify your thoughts about each sentence by illustrating all relevant information on a table. In general, your strategy should be to attempt to draw a table that would make each claim false. If you succeed, then you know it's false, and if you can't make the sentence false, then the attempt will help you see why it's true. ‡7KHUHDUHDQVZHUVEHORZEXWPRUHLPSRUWDQWO\WKHUHLVDYLGHRRQLISWKHQTQHWWKDWH[SODLQV each sentence. I. Say whether the following statements are true or false. 1. The negation of a contradiction is a tautology. 2. The negation of a contingency is a contradiction. 3. A conjunction which has a tautology as one of its conjuncts is always true. 4. A conditional whose antecedent is a contradiction is always true. 5. A disjunction which has a contradiction as one of its disjuncts is always false. 6. The conjunction of two contingencies can be a contradiction. 7. Every argument that has a contradiction for a conclusion is a valid argument. 8. No valid argument has a contradiction for a conclusion. 9. Every argument that has a tautology for a conclusion is a valid argument. 10. No argument which has tautologies for all of its premises is invalid. 11. Tautologies only entail other tautologies. 12. Tautologies are only entailed by other tautologies. 13. Contradictions only entail other contradictions. 14. Contradictions are only entailed by other contradictions. 15. All contradictions are logically equivalent to each other. 16. All tautologies are logically equivalent to each other. 17. All contingencies are logically equivalent to each other. 18. No two contingencies are logically equivalent. 19. If two formulas entail each other then they must be logically equivalent. 20. If two formulas, p and q, both entail a third formula, then p and q must be logically equivalent. 21. If all the members of a set of formulas are tautologies then the set must be consistent. 22. If a set of formulas contains a tautology then the set must be consistent. 23. If a set of formulas contains a contradiction then the set must be inconsistent. 24. The following formulas: p~p, pAp, and pCp are all tautologies. 25. Every tautology contains at least one repeated sentence letter. 26. No contingencies contain any repeated sentence letters. 27. Every simple sentence is a contingency. 28. (DA(ZAN))&(NAD) is a contradiction. 29. No disjunction is logically equivalent to any conditional. 30. No tautology has an ampersand as its main connective. $QVZHUV6HH9LGHRIRUH[SODQDWLRQV 1. True 7. False 2. False 8. False 3. False 9. True 4. True 10. False 5. False 11. True [a entails b = a "– b] 6. True [p&~p!] 12. False [a is entailed by b = b "– a] 13. False 14. True 15. True 16. True 17. False 18. False 19. True 20. False 21. True 22. False 23. True 24. True 25. True 26. False 27. True 28. False [all Œ's contain tildes] 29. False >H[aSq=pAq] 30. False >H[ SAp)&(pAp)] 4.5 Truth Trees for Validity Tree Method to determine whether an argument is Valid or Invalid 1. Setup the counterexample for the argument: Premises true and conclusion false. 2. Apply tree rules to main connectives, numering steps as you go. It s est to apply stac ing rules rst. When applying a rule to a formula, add the result to the end of every open ranch elow the formula you re wor ing on. 3. Chec for contradictions after each step. If any ranch contains a formula and it s negation, then close the ranch with an X. 4. When there are no more rules to apply: All ranches closed valid Even one ranch open invalid Use the tree method to show whether the following arguments are valid or invalid. Simple Trees (Answers on 4.6) 1. M→N, M&(N→P) |– P 2. M&R, (MN)→P |– N→P 3. A→B, A&(B→C) |– C 4 TS, (S&T)→M |– MT 5. ST, (TS)→M |– MS 6. B→~A, DB, A |– ~(D→B) 7. E→~F, GE, F |– ~(G→E) 8. E→~F, G→H, G |– F&H 9. Q R, ~P→Q |– P R 10. Q R, ~(PQ) |– P R 11. ~(E&F), ~(FD) |– E→F 12. ~(A&B), ~CB |– A→B More Interesting Trees (Answers on 4.7) 13. ~(RW)S, R&(S→T), W→~S |– S&H 14. (S→T)&(T→W), W→M, ~S→~H |– ~HM 15. B E, ~(E D), R (S&B) |– R→~D 16. ~(MN), S~T, ~T→(NZ) |– (MZ)~S 17. ~(RS)→~(EF), (RS)→(E&F) |– ~(E&~F) 18. ~(R→W), ~(ST)&M |– (M&R)K 19. S~(W&J), B&~B, M→J |– (D&J)(M&S) 20. R→(S T), ~TS |– ~(R(S&T)) 21. (TR)S, (TR)→~S |– (SR)&T 22. (H→M)→(M→R), M, H→(R→M) |– H&M 23. C→(~B→D), ~DC, D |– C D 24. (~NF)&~(D→E), D→(NE) |– (RE)&F These three have videos on ifpthenq.net. 25. M→(S→R), R, ~RM |– MR 26. (A&B)&C, ~(CD) |– A→~B 27. A→(M→B), ~(M→E), ~E→C |– (AB)&C Example: R→S, ER, S  R→S  ER  SB  ~(EB) B |– EB Counterexample: Premises True, Conclusion False ~E ~B ~R S E X R X E X R S B X All ranches closed Valid ~S ~B X Tree Rules ~~p p ~~ ~(p&q) p&q p q & ~p pq ∨ q ~(pq) ~p ~q q ~(p→q) p ~q p p→q → ~p ~q p q ~(p q) p q ~p ~q p ~q ~p q 6LPSOH7UHHVIRU9DOLGLW\‡$QVZHUV 1 O MAN M M&(NAP) M R N ~P Valid P  M  P ~(TS)  F Valid G  10 ~E E  ~E  ~F  ~G H  ~F E ~G ~H  Open  P ~R Open ~H  11 ~P ~Q O ~(EF) M ~(FD) N ~(EAF) ~F ~D E ~F Valid ~Q ~R Q R  P ~R P ~R ~P R  12 N~(AB) O ~CB M~(AAB) A ~B ~A  ~F  Open  Open Invalid ~E Invalid ~P R Q P~~P Q ~~P Q P Open P   ~P R  ~Q ~R  N QCR M ~(PQ) O~(PCR) ~A  M QCR O ~PAQ N~(PCR)  ~F ~B  Q R Invalid H  ~A B 9 ~F B  G ~(F&H) O ~G ~F D  M EA~F N GAH 8 M ~~(GAE) N GAE ~E   Open O EA~F P GE 7 ~B M ~T ~S ~T Valid ~D T  P ~(S&T) A M ~~(DAB) N DAB Valid S B  O BA~A P DB 6 ~M ~S Invalid Valid C  ~A  O (TS)AM M ~(MS) S ~S P  N S T 5 ~M ~T  ~B ~M ~N  O (S&T)AM M~(MT) ~C A N BAC Valid P ~(MvN) N T S T O AAB M A&(BAC) 3 O (MN)AP N ~(NAP) ~M N  4 M M&R 2 ~P M N N AP ~N 4.6 Invalid ~B ~C Open B  0RUH,QWHUHVWLQJ7UHHVIRU9DOLGLW\‡$QVZHUV 13. Invalid R N SAT ~ST P ~(RW)S Q ~(RW)S  ~R ~R ~W  ~W  ~S  16. Invalid P ~(MCN) Q SC~T R ~TA(NZ) M ~((MZ)~S) N ~(MZ) O ~~S ~M ~Z S ~S ~TW ~TW ~W M ~W M ~W  ~H Open ~~S ~H  R~R R S&B ~(S&B) S   B  18. Valid M ~(RAW) N ~(ST)&M P ~(M&R)Q R ~W O ~(ST) M ~S ~T Q ~(M&R) ~K ~F R  Z  1RWH$VVRRQDVDOO branches close, you’re done!  S 11 E&F V~(RS)E&F T ~(RS)U ~R E ~R E ~S F ~S F NZ 19. Valid S~(W&J) M B&~B MAJ ~((D&J)(M&S)) B ~B     S ~T S en en Op Op 22. Invalid (multiple open) 23. Valid 24. Invalid (one open) ~T 25. Valid  26. Valid (no branches!) 27. Invalid (one open)  ~R  21. Invalid M (TR)S O (TR)A~S Q ~((SR)&T) N TRS R O S&T S T ~R Q SCT S ~S T ~T ~M  20. Invalid P RA(SCT) ~TS S ~~(R(S&T)) M N R(S&T)  ~E D  M P ~~(RS)~(EF) Q ~E R RS  Open   17. Valid O ~(RS)A~(EF) S (RS)A(E&F) M ~~(E&~F) N E&~F E ~F  S~S ~T ~~T N E ¾LE ~D D ~D       B ~B E ~E ~ST M~M ~N N ~~T 15. Valid O BCE P ~(ECD) Q RC(S&B) M ~RA~D) R N ~~D D 14. Valid M (SAT)&(TAW) WAM R S ~SA~H  N ~(~HM) P SAT Q TAW O ~~H ~M H O~(RW)S M R&(SAT) R WA~S S ~S&H) ~W 4.7 T ~(TR)~S ~(TR)~S ~R R SCT ~T S S ~S ~T  Open T  ~T S  Open R   P ~(TR)~S ~T  ~R  T ~(SR)~T ~S Open R ~(SR)~T ~(SR)~T  ~S Open ~R ~S  ~R ~R Open  S 1RWH$IWHUVWHSLWZDVSRVVLEOHWRFORVHWKHEUDQFKHV ending in ~(TR) because of the TR above. 4.8 Truth Trees for Other Purposes I. Entailment and Logical Equivalence: Use the tree method to answer the following questions for each pair of formulas. 1. Does a entail b? Method: 2. Does b entail a? 1. Setup two trees to test for validity, one with "a" as 3. Are they logically equivalent? premise and "b" as conclusion, and vice versa. 2. Complete trees using standard test for validity. ‡D5AS b. ~SAa5 5HDGWKHDQVZHUV ‡DHQWDLOVELIIWKHDUJIURPDWRELVYDOLG ‡D7AW b. TA(ZAW) ‡EHQWDLOVDLIIWKHDUJIURPEWRDLVYDOLG *3. a. (B&C)AD b. (BAD)&(CAD) ‡DDQGEDUHHTXLYDOHQWLIIWKH\HQWDLOHDFKRWKHU *4. a. BA(C&D) b. (BAC)&(BAD) Example: a. AA(BAC) b. ~CA~A *5. a. ~MAN b. ~MC(NS) P AA(BAC) Q ~CA~A ‡D ,' $E,(D&A) ~(~CA~A) M M ~(AA(BAC)) *7. a. (EAG)&(FAG) b. (EF)AG ~C&~~A N A&~(BAC) N ~C A *8. a. AA(BAC) b. (A&B)AC ~~A ~(BAC) O O ‡D$A(BAC) b. ~CA~(B&A) A P B&~C *10. a. ~(S~S) b. A&(BC) B ~A BAC ~C Q  ~B Open Answer ‡$QVZHUVQH[WSDJH * Answers below. C  ~~C  ~A  1. 1st tree shows that "a" does not entail "b." 2. 2nd tree shows that "b" does entail "a." 3. They are not logically equivalent. Answer II. Tautologies, Contradictions and Contingencies: Use the tree method to show for each formula whether it's a tautology, contradiction or contingency. ‡ $ % A(AB) Method: 1. Setup two trees; one for the positive formula, and ‡ $B)A(A&B) another for the negated formula. ‡ a55 A5 5 2. Complete both trees (if necessary). *4. ~((DAE)C(~DE)) 5HDGWKHDQVZHUV *5. ((BAD)AB)AB ‡,IWKHSRVLWLYHWUHHFORVHVRQDOOEUDQFKHVWKH ‡a :AT)(TAW)) formula cannot be true, thus it's a contradiction. ‡,IWKHQHJDWHGWUHHFORVHVRQDOOEUDQFKHVWKH a 5 6 a 6 5 formula cannot be false, thus it's a tautology. *8. (W&(TZ))A~(W&(TZ)) ‡,IERWKWUHHVKDYHDWOHDVWRQHRSHQEUDQFKWKH formula is a contingency. Example: (AB)A(A&B) ~((AB)A(A&B)) Answers (AB)A(A&B) AB 6HFWLRQ, ~(A&B) 6HFWLRQ,, 3. b entails a 4. contradiction ~(AB)A&B A B 4. equivalent 5. tautology ~A A 5. neither entails 7. contingency ~B B ~A ~B ~A ~B 7. equivalent Open Open 8. contingency Open Open   8. equivalent 1st tree shows that the formula is not a contradiction. 10. a entails b 2nd tree shows that it's not a tautology. Therefore it is a contingency. 4.9 7UXWK7UHHVIRU2WKHU3XUSRVHV‡$QVZHUV O RAS M ~(~SA~R) ~S R ~S N ~~R R ~R ~~S S  P TAW M ~(TA(ZAW)) N T&~(ZAW) N ~SA~R M ~(RAS)   ~T 1 DHQWDLOVE EHQWDLOVD 7KH\DUHHTXLYDOHQW M (ID)&A N ~(I(D&A)) O ID I  P ~(ID) ~I ~D ~D ~A   ~Z W  2SHQ  2 DHQWDLOVE EGRHVQRWHQWDLOD 7KH\DUHQRWHTXLYDOHQW Q AA(BAC) M ~(~CA~(B&A)) N ~C&~~(B&A) N D&A I D O ZAW ~T W  M I(D&A) O ~((ID)&A) A ~I P ~(D&A) T ~W T O ~(ZAW) Z ~W ~R  N TA(ZAW) M ~(TAW) ~A Q ~(ID)  ~I ~D  ~C D A 2SHQ Q ~CA~(B&A) M ~(AA(BAC)) N A&~(BAC) A ~(BAC) O P B&~C B ~C O ~~(B&A) P B&A B A ~A   R BAC ~A  ~B 6 DHQWDLOVE C  EGRHVQRWHQWDLOD 7KH\DUHQRWHTXLYDOHQW R ~(B&A) ~~C  ~B  ~A   9 DHQWDLOVE EHQWDLOVD 7KH\DUHHTXLYDOHQW M (A&B)A(AB) M ~((A&B)A(AB)) M (AB)A(A&B) N A&B ~(AB) O O N ~(A&B)AB O N ~(AB)A&B A ~A ~B A 2SHQ 2SHQ 2SHQ B ~A ~B B 2SHQ ~A ~B A B 2SHQ 2SHQ  M ~((AB)A(A&B)) N AB ~(A&B) O A B ~A ~B ~A ~B  2SHQ 2SHQ  1 VWWUHHVKRZVLW VQRWDFRQWUDGLFWLRQ 2 VWWUHHVKRZVWKDWLW VQRWDFRQWUDGLFWLRQ QGWUHHVKRZVWKDWLWLVDWDXWRORJ\ QGWUHHVKRZVWKDWLW VQRWDWDXWRORJ\ 1RWHWKDWWKHQGWUHHPDNHVWKHVWLUUHOHYDQW 7KHUHIRUHLWLVDFRQWLQJHQF\ M ((~RR)AR)&R M ~[((~RR)AR)&R] M ~((WAT)(TAW)) M ~~((WAT)(TAW)  N (~RR)AR N ~(WAT  N (WAT)(TAW) O  R ~(TAW) ~R N ~((~RR)AR) W 2SHQ O WAT P TAW O ~RR ~T ~(~RR) R ~R O T ~~R ~R  2SHQ ~R 2SHQ R  3 VWWUHHVKRZVWKDWLW VQRWDFRQWUDGLFWLRQ QGWUHHVKRZVWKDWLW VQRWDWDXWRORJ\ 7KHUHIRUHLWLVDFRQWLQJHQF\ ~W ~W 2SHQ T ~T 2SHQ 2SHQ W 2SHQ  6 VWWUHHVKRZVWKDWLW VDFRQWUDGLFWLRQ QGWUHHVKRZVWKDWLWLVQRWDWDXWRORJ\ 1RWHWKDWWKHVWWUHHPDNHVWKHQGXQQHFHVVDU\ 4.10 Test an Argument P1, P2, ... Pn "– C Test a Single Formula p Test a Pair of Formulas p "– q Test a Set of Formulas p, q, ... r Tree Method Set up the tree by listing the premises and negating the conclusion (this is the counterexample). If the tree closes on all branches, then the argument is valid. Table Method Validity An argument is valid if it has no counterexample. This is shown on the table method, if there is NO row on which all the premises are True and the conclusion is False. To Show: Invalidity X p q r ~c p q r ~c Open If the tree for the positive formula closes on all branches, then p is a contradiction. If it's open on at least one branch, then it's not a contradiction. If the tree described above is open on at least one branch, then the argument is invalid. Column under the main connective is all False. If the tree for the negated formula closes on all branches, then p is a tautology. If it's open on at least one branch, then p is not a tautology. An argument is invalid if it does have a counterexample. On the table method this means one or more rows on which all the premises are True and the conclusion is False. Contradiction Œ (Logical Falsehood) Column under the main connective is all True. Two trees: one to check for contradic~p p tion and one for tautology. If both trees have at least one open branch, Open Open then the formula is a contingency. Proof Method Derive the conclusion from the premises. NA Construct a proof for ~p without any assumptions. You cannot show invalidity on the proof method. X Construct a proof for p without any assumptions. X p ~q X X p ~q X q ~p p q r Open p q r X NA NA Construct two proofs: one from p to q, and one from q to p. Construct a proof with p as the only assumption and q as the conclusion. NA ~p p Tautology (Logical Truth) Column under the main connective is any combination of True and False. Set up the tree with p true and q negated. If the tree closes on all branches, then p does entail q. But if the tree is open on at least one branch then p does not entail q. (Note how this is the same thing as the test for validity.) Two trees: one to check if p entails q, and one to check if q entails p. If both formulas entail each other then they are logically equivalent. A tree for the set of formulas has at least one open branch. The columns under the main connectives for p and q are identical. Entailment is another word for validity. If the argument that has p as premise and q as conclusion is valid, then p entails q. On a table, p entails q, as long as there is no row on which p is True and q is False, that is, no counterexample from p to q. Contingency Entailment Logical Equivalence Consistency A set of formulas is logically consistent if they don't contradict each other, that is, if it's possible for them all to be true at the same time. On a table you can determine that a set is consistent if there is at least one row on which all formulas are True. A tree for the set of formulas is closed on all branches. (Mutual Entailment) Inconsistency A set of formulas is inconsistent if there is not a row on which all the formulas are True. 4.11 Unusual Valid Argument Forms Over the last weeks we have seen a variety of unusual valid arguments. On this page I have assembled a list of examples in the hope of illuminating the concepts and dispelling (somewhat) the strangeness. ‡&LUFXODU$UJXPHQWV p p Jello is the world's most wiggly dessert. Jello is the world's most wiggly dessert. A circular argument is one in which the conclusion is a mere repetition of one or more of the premises. All circular arguments are valid and they can even be sound, but they aren't very useful. Of coure, we should note that in all valid arguments (except in the case of the trivial validity discussed below) the content of the conclusion is always contained in the premises in some form or another. Thus, one might say that there is an element of circularity in every valid argument. ‡:HGJHLQ$UJXPHQWV p pq Jello is translucent. Jello is translucent or poisonous. 2QUHÁHFWLRQZHGJHLQUHDVRQLQJPDNHVJRRG sense, but because we rarely encounter instances of this pattern, it initially seems odd. The main reason that we don't encounter instances of this pattern is WKDWWKHSUHPLVHLVVLJQLÀFDQWO\VWURQJHU RUPRUH informative) than the conclusion. In practice it's rarely appropriate to argue from a stronger claim to a weaker one. ‡$UJXPHQWVWKDWGHSHQGRQWKHWUXWKFRQGLWLRQVRIWKHDUURZ p qAp I am making jello today. If 2+2=4, then I am making jello today. 2+2 =4. If the moon is made of pudding, then 2+2=4. p ~pAq I am making jello today. If I am not make jello today, then 2+2=4. 2+2 =4. ,IȴWKHQWKHPRRQLVPDGHRISXGGLQJ As we have seen, the truth conditions for the arrow are not very intuitive. The result is peculiar valid arguments OLNHWKHRQHVVKRZQKHUH,QVRPHUHVSHFWVWKHVHFDVHVDUHOLNHZHGJHLQ,QIDFWLQDOORIWKHVHFDVHVRQH might say that the conclusion is weaker (or less informative) than the premise. So we could simply claim that these are examples of good reasoning, they're just unfamiliar. But the situation seems worse than it does with the wedge. The reasons for this are complex and controversial, but there are at least two elements worth thinking about. First, in natural language and reasoning, conditionals with false antecedents are often taken to have indeterminate truth values; they may even be treated like nonsense. So, for instance, if asked whether any of the conclusions above are true or false, many people would be inclined to say that they are neither. But in logic we cannot have truth value gaps, everything has to be true or false. Second, when we hear a conditional, we tend to LPDJLQHWKDWWKHDQWHFHGHQWDQGWKHFRQVHTXHQWKDYHVRPHHVVHQWLDOFRQQHFWLRQFDXVDORUWHPSRUDORURWKHUwise. But in logic the arrow merely combines the truth values of the antecedent and the consequent, causal and WHPSRUDOUHODWLRQVDUHLUUHOHYDQW7KHVLJQLÀFDQFHRIWKHVHGLVFUHSDQFLHVEHWZHHQRXUQDWXUDOXQGHUVWDQGLQJRI conditionals and their logical interpretation is the subject of ongoing philosophical inquiry. ‡7ULYLDO9DOLGLW\ ‡$Q\WKLQJIROORZVIURPDFRQWUDGLFWLRQ ‡$WDXWRORJ\IROORZVIURPDQ\WKLQJ p&~p This jello is red and this jello is not red. q The moon is made of pudding. q The moon is made of pudding. p~p This jello is red or this jello is not red. 7KHVHPD\EHWKHPRVWSHFXOLDUYDOLGSDWWHUQVRIDOO7KH\VDWLVI\WKHGHÀQLWLRQRIYDOLGLW\EXWLQSUDFWLFHWKH\ PDNHORXV\DUJXPHQWV:KDWLVPRVWSHFXOLDULQWKHVHFDVHVLVWKDWWKHUHLVQRFRQQHFWLRQEHWZHHQWKHSUHPLVHV and the conclusion. One way to make sense of these arguments is to note the correlation between their forms and the truth conditions of the conditional. A conditional is true whenever its antecedent is false. Similarly, an argument is valid whenever the premises cannot be true, as is always the case when one or more premises are contradictory. Likewise, a conditional is true whenever its consequent is true. This corresponds to arguments WKDWKDYHWDXWRORJLHVIRUFRQFOXVLRQV)RUWKHUHFRUGQRWHWKDWWKHÀUVWSDWWHUQDERYHFRXOGQHYHUFRUUHVSRQGWR a sound argument, because its premise could never be true. However, there is no reason that the second pattern could not correspond to a sound argument, it merely requires a true premise. 7HVW‡3UDFWLFH3UREOHPV 4.12 ,&RQVWUXFWD7UXWK7DEOHIRUWKHIROORZLQJVHWVRIIRUPXODVDQGWKHQDQVZHUWKHTXHVWLRQV :KLFKIRUPXODVLIDQ\DUHWDXWRORJLHV" D$B :KLFKIRUPXODVLIDQ\DUHFRQWUDGLFWLRQV" Ea a$ a%  :KLFKIRUPXODVLIDQ\DUHFRQWLQJHQFLHV" F%A(AAB) :KLFKIRUPXODVLIDQ\DUHORJLFDOO\HTXLYDOHQWWRIRUPXODD" Ga a a$ % AB) :KLFKIRUPXODVLIDQ\DUHORJLFDOO\HTXLYDOHQWWRIRUPXODE" e. ~~(ACA) :KLFKIRUPXODVLIDQ\DUHORJLFDOO\HTXLYDOHQWWRIRUPXODF" Da a5AS)&~~R Ea a6AR)&~R) F5C(RR) Ga 5S)AR :KLFKIRUPXODVLIDQ\DUHHQWDLOHGE\IRUPXODD" :KLFKIRUPXODVLIDQ\DUHHQWDLOHGE\IRUPXODE" :KLFKIRUPXODVLIDQ\DUHHQWDLOHGE\IRUPXODF" :KLFKIRUPXODVLIDQ\DUHHQWDLOHGE\IRUPXODG" ,VWKLVVHWRIIRUPXODVORJLFDOO\FRQVLVWHQW" ,,8VHWKH7UXWK7DEOHPHWKRGWRVKRZZKHWKHUWKHIROORZLQJDUJXPHQWVDUHYDOLGRULQYDOLG 1. ~(~A&B)A(~BC) ~((~BA)C~(C&B)), AAC "
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