TF truth, falsity, and indeterminacy P is truth-functionally true iff it has the value T for any truth-value assignment. P is truth-functionally false iff it has the value F for any truth-value assignment. P is tf-false iff ~P is tf-true P is truth-functionally indeterminate iff it has the value T for some truth-value assignments, and the value F for some other truth-value assignments. P is tf-indeterminate iff it is neither tf-true nor th-false.
TF equivalence and consistency P and Q are truth-functionally equivalent iff P and Q do not have different truth-values for any truth-value assignment. A set of sentences is truth-functionally consistent iff there is a truth- value assignment that on which all the members of the set have the value T. A set of sentences is truth-functionally inconsistent iff it is not tf- consistent.
TF entailment and validity A set of SL sentences truth-functionally entails a sentence P iff there is no truth-value assignment on which every member of is true and P false.
TF entailment and validity A set of SL sentences truth-functionally entails a sentence P iff there is no truth-value assignment on which every member of is true and P false. An argument of SL is truth-functionally valid iff there is no truth- value assignment on which all the premises are true and the conclusion false.
TF entailment and validity A set of SL sentences truth-functionally entails a sentence P iff there is no truth-value assignment on which every member of is true and P false. An argument of SL is truth-functionally valid iff there is no truth- value assignment on which all the premises are true and the conclusion false. An argument of SL is truth-functionally invalid iff it is not tf-valid.
TF entailment and validity A set of SL sentences truth-functionally entails a sentence P iff there is no truth-value assignment on which every member of is true and P false. An argument of SL is truth-functionally valid iff there is no truth- value assignment on which all the premises are true and the conclusion false. An argument of SL is truth-functionally invalid iff it is not tf-valid. An argument is tf-valid iff the premises tf-entail the conclusion.
TF properties P is truth-functionally true iff it has the value T for any truth-value assignment. P is truth-functionally false iff ~P is tf-true. P is truth-functionally indeterminate iff P is neither tf-true nor tf-false. P and Q are truth-functionally equivalent iff P and Q do not have different truth- values for any truth-value assignment. A set of sentences is truth-functionally consistent iff there is a truth-value assignment that on which all the members of the set have the value T. A set of SL sentences truth-functionally entails a sentence P iff there is no truth-value assignment on which every member of is true and P false. An argument of SL is truth-functionally valid iff there is no truth-value assignment on which all the premises are true and the conclusion false.
3.2E 1j ~B ((B D) TT TF FT FF
3.2E 1j ~B ((B D) FTT FTF TFT TFF
3.2E 1j ~B ((B D) FTTT FTTF TFT TFF
3.2E 1j ~B ((B D) FTTT FTTF TFT TFFT
3.2E 1j ~B ((B D) FTTT FTTF TFTTT TFFT
3.2E 1j ~B ((B D) FTTT FTTF TFTTTT TFTFT
3.2E 1j ~B ((B D) FT T T FT T F TF T TTT TF T FT
3.2E 1l (M ~N)&(M N) TT TF FT FF
3.2E 1l (M ~N)&(M N) T F T T T F F F T F T F
3.2E 1l (M ~N)&(M N) T FF T T TT F F TF T F FT F
3.2E 1l (M ~N)&(M N) T FF T T T TT F F F TF T F F FT F T
3.2E 1l (M ~N)&(M N) T FF T T T TT F F F TF T F F FT F T
3.2E 1l (M ~N)&(M N) T FF T F T T TT F F F F TF T F F F FT F F T
3.3E 1d (C&(B A)) ((C&B) A) TTT TTF TFT TFF FTT FTF FFT FFF
3.3E 1d (C&(B A)) ((C&B) A) TTT TTF TFT TFF FFTT FFTF FFFT FFFF
3.3E 1d (C&(B A)) ((C&B) A) TTT TTF TFT TFFF FFTT FFTF FFFT FFFF
3.3E 1d (C&(B A)) ((C&B) A) TTTT TTTF TFTT TFFF FFTT FFTF FFFT FFFF
3.3E 1d (C&(B A)) ((C&B) A) TTTTT TTTTF TTFTT TFFFF FFTT FFTF FFFT FFFF
3.3E 1d (C&(B A)) ((C&B) A) TTTTTT TTTTF TTFTTT TFFFF FFTTT FFTF FFFTT FFFF
3.3E 1d (C&(B A)) ((C&B) A) TTTTTT TTTTFT TTFTTT TFFFFF FFTTT FFTFF FFFTT FFFFF
3.3E 1d (C&(B A)) ((C&B) A) TTTTTT TTTTFTT TTFTTT TFFFFFF FFTTT FFTFFF FFFTT FFFFFF
3.3E 1d (C&(B A)) ((C&B) A) TTTTTT TTTTFTT TTFTTT TFFFFFF FFTTT FFTFFF FFFTT FFFFFF
3.3E 1d (C&(B A)) ((C&B) A) TTTTTT TTTTFTT TTFTTT TFFFFFF FFTT F T FFTFFF FFFTT FFFFFF
3.4E 1f U (W&H)W (U H)H ~H 1TTT 2TTF 3TFT 4TFF 5FTT 6FTF 7FFT 8FFF
3.4E 1f U (W&H)W (U H)H ~H 1T T TT T 2T T TF T 3T T FT F 4T T FF F 5F T TT T 6F F TF F 7F F FT F 8F F FF T
3.4E 1f U (W&H)W (U H)H ~H 1T T TT T T 2T T TF T T 3T T FT F T 4T T FF F T 5F T TT T T 6F F TF F T 7F F FT F T 8F F FF T T
3.5E1b B (A ~C)(C A) B~B A~(A C) 1 TTT 2 TTF 3 FTT 4 FTF 5 TFT 6 TFF 7 FFT 8 FFF
3.5E1b B (A ~C)(C A) B~B A~(A C) 1TTFTF 2TTTFF 3FTFTT 4FTTFT 5TFFTF 6TFTFF 7FFFTT 8FFTFT
3.5E1b B (A ~C)(C A) B~B A~(A C) 1TTTFTF 2TTTTFF 3FTFTT 4FTTFT 5TTFFTF 6TTFTFF 7FFFTT 8FFTFT
3.5E1b B (A ~C)(C A) B~B A~(A C) 1TTTFTF 2TTTTFF 3FTFFTT 4FTTTFT 5TTFFTF 6TTFTFF 7FFFFTT 8FFFTFT
3.5E1b B (A ~C)(C A) B~B A~(A C) 1TTTFTF 2TTTTFF 3FFTFFTT 4FTTTTFT 5TTFFTF 6TTFTFF 7FFFFFTT 8FFFFTFT
3.5E1b B (A ~C)(C A) B~B A~(A C) 1TTTFTF 2TTTTFTF 3FFTFFTT 4FTTTTFTT 5TTFFTF 6TTFTFTF 7FFFFFTT 8FFFFTFTT
3.5E1b B (A ~C)(C A) B~B A~(A C) 1TTTFTTF 2TTTTFTF 3FFTFFTTT 4FTTTTFTT 5TTFFTF 6TTFTFTF 7FFFFFTT 8FFFFTFTT
3.5E1b B (A ~C)(C A) B~B A~(A C) 1TTTFTTF 2TTTTFTF 3FFTFFTTT 4FTTTTFTT 5TTFFTFF 6TTFTFTF 7FFFFFTFT 8FFFFTFTT
3.5E1b B (A ~C)(C A) B~B A~(A C) 1TTTFTTTF 2TTTTFTTF 3FFTFFTTFT 4FTTTTFTFT 5TTFFTFFF 6TTFTFTTF 7FFFFFTFTT 8FFFFTFTFT
3.5E1b B (A ~C)(C A) B~B A~(A C) 1TTTFTTTFT 2TTTTFTTFT 3FFTFFTTFTT 4FTTTTFTFTT 5TTFFTFFF 6TTFTFTTF 7FFFFFTFTTT 8FFFFTFTFTT
3.5E1b B (A ~C)(C A) B~B A~(A C) 1TTTFTTTFT 2TTTTFTTFT 3FFTFFTTFTT 4FTTTTFTFTT 5TTFFTFFFF 6TTFTFTTFF 7FFFFFTFTTT 8FFFFTFTFTT
3.5E1b B (A ~C)(C A) B~B A~(A C) 1TTTFTTTFTT 2TTTTFTTFTT 3FFTFFTTFTTT 4FTTTTFTFTTT 5TTFFTFFFFT 6TTFTFTTFF 7FFFFFTFTTTT 8FFFFTFTFTT
3.5E1b B (A ~C)(C A) B~B A~(A C) 1TTTFTTTFTT 2TTTTFTTFTT 3FFTFFTTFTTT 4FTTTTFTFTTT 5TTFFTFFFFT 6TTFTFTTFFF 7FFFFFTFTTTT 8FFFFTFTFTTF
3.5E1b B (A ~C)(C A) B~B A~(A C) 1TTTFTTTFTFT 2TTTTFTTFTFT 3FFTFFTTFTTFT 4FTTTTFTFTTFT 5TTFFTFFFFFT 6TTFTFTTFFTF 7FFFFFTFTTTFT 8FFFFTFTFTTTF
3.5E1b B (A ~C)(C A) B~B A~(A C) 1TTTFTTTFTFT 2TTTTFTTFTFT 3FFTFFTTFTTFT 4FTTTTFTFTTFT 5TTFFTFFFFFT 6TTFTFTTFFTF 7FFFFFTFTTTFT 8FFFFTFTFTTTF
3.5E 1D ~(Y A)~Y~AW&~W 1FTTTFFTFF 2FTTTFFFFT 3TTFFFTTFF 4TTFFFTFFT 5TFFTTFTFF 6TFFTTFFFT 7FFTFTTTFF 8FFTFTTFFT
shortened truth tables Show that ~B (B&~B) is not tf-true
shortened truth tables Show that ~B (B&~B) is not tf-true B~B (B&~B) F
shortened truth tables Show that ~B (B&~B) is not tf-true B~B (B&~B) FFF
shortened truth tables Show that ~B (B&~B) is not tf-true B~B (B&~B) FTFF
shortened truth tables Show that ~B (B&~B) is not tf-true B~B (B&~B) TFTFTFT
shortened truth tables Show that ~B (B&~B) is not tf-true B~B (B&~B) TFTFTFFT
shortened truth tables Show that (~B ~A)&C is not tf-false AB(~B ~A)&B
shortened truth tables Show that (~B ~A)&C is not tf-false AB(~B ~A)&B T
shortened truth tables Show that (~B ~A)&C is not tf-false AB(~B ~A)&B TTT
shortened truth tables Show that (~B ~A)&C is not tf-false AB(~B ~A)&B TTTT
shortened truth tables Show that (~B ~A)&C is not tf-false AB(~B ~A)&B TTTTT
shortened truth tables Show that (~B ~A)&C is not tf-false AB(~B ~A)&B TFTTTT
shortened truth tables Show that (~B ~A)&C is not tf-false AB(~B ~A)&B TFTTTTT
shortened truth tables Show that (~B ~A)&C is not tf-false AB(~B ~A)&B FTFTTTFTT
shortened truth tables Show that (A B) (B A) is not tf-false AB(A B) (B A) T
shortened truth tables Show that (A B) (B A) is not tf-false AB(A B) (B A) TTT FTF
shortened truth tables Show that (A B) (B A) is not tf-false AB(A B) (B A) TTT TFFTTFF
shortened truth tables Show that (A B) (B A) is not tf-false AB(A B) (B A) TTT ?? T F F T T F F COTRADICTION!
shortened truth tables Show that (A B) (B A) is not tf-false AB(A B) (B A) TTT
shortened truth tables Show that (A B) (B A) is not tf-false AB(A B) (B A) TTTTT FTTTT FTFTT
shortened truth tables Show that (A B) (B A) is not tf-false AB(A B) (B A) TTTTT FTFTTTTTF FTFTT
shortened truth tables Show that (A B) (B A) is not tf-false AB(A B) (B A) TTTTT FTFTTT TTF FTFTT CONTRADICTION!
shortened truth tables Show that (A B) (B A) is not tf-false AB(A B) (B A) TTTTT FTFTT
shortened truth tables Show that (A B) (B A) is not tf-false AB(A B) (B A) TTTTTTTTT FTFTT
shortened truth tables Show that (A B) (B A) is not tf-false AB(A B) (B A) TTTTTTTTT
shortened truth tables Show that (A B) (B A) is not tf-true AB(A B) (B A) F
shortened truth tables Show that (A B) (B A) is not tf-true AB(A B) (B A) FFF
shortened truth tables Show that (A B) (B A) is not tf-true AB(A B) (B A) TFFFTFF
shortened truth tables Show that (A B) (B A) is not tf-true AB(A B) (B A) ??TFFFTFF CONTRADICTION! Therefore, the sentence is tf-true
shortened truth tables Practice Show that: {A (B&C), B (A C), C ~C} is tf-consistent {B (A&~C), (C A) B, ~B A} ~(A C)
shortened truth tables Practice Show that: {A (B&C), B (A C), C ~C} is tf-consistent {B (A&~C), (C A) B, ~B A} ~(A C) ABC A (B&C)B (A C)C ~C TTT
shortened truth tables Practice Show that: {A (B&C), B (A C), C ~C} is tf-consistent {B (A&~C), (C A) B, ~B A} ~(A C) ABC A (B&C)B (A C)C ~C T T T TT
shortened truth tables Practice Show that: {A (B&C), B (A C), C ~C} is tf-consistent {B (A&~C), (C A) B, ~B A} ~(A C) ABC A (B&C)B (A C)C ~C TTT T T TTT TT
shortened truth tables Practice Show that: {A (B&C), B (A C), C ~C} is tf-consistent {B (A&~C), (C A) B, ~B A} ~(A C) ABC A (B&C)B (A C)C ~C TTT T T TTTT T TTT T T
shortened truth tables Practice Show that: {A (B&C), B (A C), C ~C} is tf-consistent {B (A&~C), (C A) B, ~B A} ~(A C) ABC A (B&C)B (A C)C ~C TTT T T TTTT T TTTT T FT
shortened truth tables Practice Show that: {A (B&C), B (A C), C ~C} is tf-consistent {B (A&~C), (C A) B, ~B A} ~(A C) ABC A (B&C)B (A C)C ~C TTT T T TTTT T TTTT T FT B (A&~C)(C A) B~B A~(A C) TTTF
shortened truth tables Practice Show that: {A (B&C), B (A C), C ~C} is tf-consistent {B (A&~C), (C A) B, ~B A} ~(A C) ABC A (B&C)B (A C)C ~C TTT T T TTTT T TTTT T FT B (A&~C)(C A) B~B A~(A C) TTTFT
shortened truth tables Practice Show that: {A (B&C), B (A C), C ~C} is tf-consistent {B (A&~C), (C A) B, ~B A} ~(A C) ABC A (B&C)B (A C)C ~C TTT T T TTTT T TTTT T FT B (A&~C)(C A) B~B A~(A C) TTTTTFT
shortened truth tables Practice Show that: {A (B&C), B (A C), C ~C} is tf-consistent {B (A&~C), (C A) B, ~B A} ~(A C) ABC A (B&C)B (A C)C ~C TTT T T TTTT T TTTT T FT B (A&~C)(C A) B~B A~(A C) TTTTTTTFT
shortened truth tables Practice Show that: {A (B&C), B (A C), C ~C} is tf-consistent {B (A&~C), (C A) B, ~B A} ~(A C) ABC A (B&C)B (A C)C ~C TTT T T TTTT T TTTT T FT B (A&~C)(C A) B~B A~(A C) TTTTTFTTFT
shortened truth tables Practice Show that: {A (B&C), B (A C), C ~C} is tf-consistent {B (A&~C), (C A) B, ~B A} ~(A C) ABC A (B&C)B (A C)C ~C TTT T T TTTT T TTTT T FT B (A&~C)(C A) B~B A~(A C) TTTTTFFTTTTTTFTTF
shortened truth tables Practice Show that: {A (B&C), B (A C), C ~C} is tf-consistent {B (A&~C), (C A) B, ~B A} ~(A C) ABC A (B&C)B (A C)C ~C TTT T T TTTT T TTTT T FT B (A&~C)(C A) B~B A~(A C) TTTTTFFTTTTFTTTFTTF