tf truth, falsity, and indeterminacy p is truth-functionally true iff it has the value t for any...

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) D)

T T

T F

F T

F F

3.2E 1j

~ B ((B D) D)

F T T

F T F

T F T

T F F

3.2E 1j

~ B ((B D) D)

F T T T

F T T F

T F T

T F F

3.2E 1j

~ B ((B D) D)

F T T T

F T T F

T F T

T F F T

3.2E 1j

~ B ((B D) D)

F T T T

F T T F

T F T T T

T F F T

3.2E 1j

~ B ((B D) D)

F T T T

F T T F

T F T T T T

T F T F T

3.2E 1j

~ B ((B D) D)

F T T T

F T T F

T F T T T T

T F T F T

3.2E 1l

(M ~ N) & (M N)

T T

T F

F T

F F

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 F F T

T T T F

F T F T

F F T F

3.2E 1l

(M ~ N) & (M N)

T F F T T

T T T F F

F T F T F

F F T F T

3.2E 1l

(M ~ N) & (M N)

T F F T T

T T T F F

F T F T F

F F T F T

3.2E 1l

(M ~ N) & (M N)

T F F T F T

T T T F F F

F T F T F F

F F T F F T

3.3E 1d

(C & (B A)) ((C & B) A)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

3.3E 1d

(C & (B A)) ((C & B) A)

T T T

T T F

T F T

T F F

F F T T

F F T F

F F F T

F F F F

3.3E 1d

(C & (B A)) ((C & B) A)

T T T

T T F

T F T

T F F F

F F T T

F F T F

F F F T

F F F F

3.3E 1d

(C & (B A)) ((C & B) A)

T T T T

T T T F

T F T T

T F F F

F F T T

F F T F

F F F T

F F F F

3.3E 1d

(C & (B A)) ((C & B) A)

T T T T T

T T T T F

T T F T T

T F F F F

F F T T

F F T F

F F F T

F F F F

3.3E 1d

(C & (B A)) ((C & B) A)

T T T T T T

T T T T F

T T F T T T

T F F F F

F F T T T

F F T F

F F F T T

F F F F

3.3E 1d

(C & (B A)) ((C & B) A)

T T T T T T

T T T T F T

T T F T T T

T F F F F F

F F T T T

F F T F F

F F F T T

F F F F F

3.3E 1d

(C & (B A)) ((C & B) A)

T T T T T T

T T T T F T T

T T F T T T

T F F F F F F

F F T T T

F F T F F F

F F F T T

F F F F F F

3.3E 1d

(C & (B A)) ((C & B) A)

T T T T T T

T T T T F T T

T T F T T T

T F F F F F F

F F T T T

F F T F F F

F F F T T

F F F F F F

3.3E 1d

(C & (B A)) ((C & B) A)

T T T T T T

T T T T F T T

T T F T T T

T F F F F F F

F F T T F T

F F T F F F

F F F T T

F F F F F F

3.4E 1f

U (W & H) W (U H) H ~H

1 T T T

2 T T F

3 T F T

4 T F F

5 F T T

6 F T F

7 F F T

8 F F F

3.4E 1f

U (W & H) W (U H) H ~H

1 T T T T T

2 T T T F T

3 T T F T F

4 T T F F F

5 F T T T T

6 F F T F F

7 F F F T F

8 F F F F T

3.4E 1f

U (W & H) W (U H) H ~H

1 T T T T T T2 T T T F T T3 T T F T F T4 T T F F F T5 F T T T T T6 F F T F F T7 F F F T F T8 F F F F T T

3.5E1b

B (A ~ C) (C A) B ~ B A ~ (A C)

1 T T T

2 T T F

3 F T T

4 F T F

5 T F T

6 T F F

7 F F T

8 F F F

3.5E1b

B (A ~ C) (C A) B ~ B A ~ (A C)

1 T T F T F

2 T T T F F

3 F T F T T

4 F T T F T

5 T F F T F

6 T F T F F

7 F F F T T

8 F F T F T

3.5E1b

B (A ~ C) (C A) B ~ B A ~ (A C)

1 T T T F T F

2 T T T T F F

3 F T F T T

4 F T T F T

5 T T F F T F

6 T T F T F F

7 F F F T T

8 F F T F T

3.5E1b

B (A ~ C) (C A) B ~ B A ~ (A C)

1 T T T F T F

2 T T T T F F

3 F T F F T T

4 F T T T F T

5 T T F F T F

6 T T F T F F

7 F F F F T T

8 F F F T F T

3.5E1b

B (A ~ C) (C A) B ~ B A ~ (A C)

1 T T T F T F

2 T T T T F F

3 F F T F F T T

4 F T T T T F T

5 T T F F T F

6 T T F T F F

7 F F F F F T T

8 F F F F T F T

3.5E1b

B (A ~ C) (C A) B ~ B A ~ (A C)

1 T T T F T F

2 T T T T F T F

3 F F T F F T T

4 F T T T T F T T

5 T T F F T F

6 T T F T F T F

7 F F F F F T T

8 F F F F T F T T

3.5E1b

B (A ~ C) (C A) B ~ B A ~ (A C)

1 T T T F T T F

2 T T T T F T F

3 F F T F F T T T

4 F T T T T F T T

5 T T F F T F

6 T T F T F T F

7 F F F F F T T

8 F F F F T F T T

3.5E1b

B (A ~ C) (C A) B ~ B A ~ (A C)

1 T T T F T T F

2 T T T T F T F

3 F F T F F T T T

4 F T T T T F T T

5 T T F F T F F

6 T T F T F T F

7 F F F F F T F T

8 F F F F T F T T

3.5E1b

B (A ~ C) (C A) B ~ B A ~ (A C)

1 T T T F T T T F

2 T T T T F T T F

3 F F T F F T T F T

4 F T T T T F T F T

5 T T F F T F F F

6 T T F T F T T F

7 F F F F F T F T T

8 F F F F T F T F T

3.5E1b

B (A ~ C) (C A) B ~ B A ~ (A C)

1 T T T F T T T F T

2 T T T T F T T F T

3 F F T F F T T F T T

4 F T T T T F T F T T

5 T T F F T F F F

6 T T F T F T T F

7 F F F F F T F T T T

8 F F F F T F T F T T

3.5E1b

B (A ~ C) (C A) B ~ B A ~ (A C)

1 T T T F T T T F T

2 T T T T F T T F T

3 F F T F F T T F T T

4 F T T T T F T F T T

5 T T F F T F F F F

6 T T F T F T T F F

7 F F F F F T F T T T

8 F F F F T F T F T T

3.5E1b

B (A ~ C) (C A) B ~ B A ~ (A C)

1 T T T F T T T F T T

2 T T T T F T T F T T

3 F F T F F T T F T T T

4 F T T T T F T F T T T

5 T T F F T F F F F T

6 T T F T F T T F F

7 F F F F F T F T T T T

8 F F F F T F T F T T

3.5E1b

B (A ~ C) (C A) B ~ B A ~ (A C)

1 T T T F T T T F T T

2 T T T T F T T F T T

3 F F T F F T T F T T T

4 F T T T T F T F T T T

5 T T F F T F F F F T

6 T T F T F T T F F F

7 F F F F F T F T T T T

8 F F F F T F T F T T F

3.5E1b

B (A ~ C) (C A) B ~ B A ~ (A C)

1 T T T F T T T F T F T

2 T T T T F T T F T F T

3 F F T F F T T F T T F T

4 F T T T T F T F T T F T

5 T T F F T F F F F F T

6 T T F T F T T F F T F

7 F F F F F T F T T T F T

8 F F F F T F T F T T T F

3.5E1b

B (A ~ C) (C A) B ~ B A ~ (A C)

1 T T T F T T T F T F T

2 T T T T F T T F T F T

3 F F T F F T T F T T F T

4 F T T T T F T F T T F T

5 T T F F T F F F F F T

6 T T F T F T T F F T F

7 F F F F F T F T T T F T

8 F F F F T F T F T T T F

3.5E 1D

~ (Y A) ~ Y ~ A W & ~ W

1 F T T T F F T F F

2 F T T T F F F F T

3 T T F F F T T F F

4 T T F F F T F F T

5 T F F T T F T F F

6 T F F T T F F F T

7 F F T F T T T F F

8 F F T F T T F F T

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)

F F F

shortened truth tables

Show that ~B(B&~B) is not tf-true

B ~ B (B & ~ B)

F T F F

shortened truth tables

Show that ~B(B&~B) is not tf-true

B ~ B (B & ~ B)

T F T F T F T

shortened truth tables

Show that ~B(B&~B) is not tf-true

B ~ B (B & ~ B)

T F T F T F F T

shortened truth tables

Show that (~B~A)&C is not tf-false

A B (~ B ~ A) & B

shortened truth tables

Show that (~B~A)&C is not tf-false

A B (~ B ~ A) & B

T

shortened truth tables

Show that (~B~A)&C is not tf-false

A B (~ B ~ A) & B

T T T

shortened truth tables

Show that (~B~A)&C is not tf-false

A B (~ B ~ A) & B

T T T T

shortened truth tables

Show that (~B~A)&C is not tf-false

A B (~ B ~ A) & B

T T T T T

shortened truth tables

Show that (~B~A)&C is not tf-false

A B (~ B ~ A) & B

T F T T T T

shortened truth tables

Show that (~B~A)&C is not tf-false

A B (~ B ~ A) & B

T F T T T T T

shortened truth tables

Show that (~B~A)&C is not tf-false

A B (~ B ~ A) & B

F T F T T T F T T

shortened truth tables

Show that (AB) (BA) is not tf-false

A B (A B) (B A)

T

shortened truth tables

Show that (AB) (BA) is not tf-false

A B (A B) (B A)

T T T

F T F

shortened truth tables

Show that (AB) (BA) is not tf-false

A B (A B) (B A)

T T T

T F F T T F F

shortened truth tables

Show that (AB) (BA) is not tf-false

A B (A B) (B A)

T T T

? ? T F F T T F F

COTRADICTION!

shortened truth tables

Show that (AB) (BA) is not tf-false

A B (A B) (B A)

T T T

shortened truth tables

Show that (AB) (BA) is not tf-false

A B (A B) (B A)

T T T T T

F T T T T

F T F T T

shortened truth tables

Show that (AB) (BA) is not tf-false

A B (A B) (B A)

T T T T T

F T F T T T T T F

F T F T T

shortened truth tables

Show that (AB) (BA) is not tf-false

A B (A B) (B A)

T T T T T

F T F T T T T T F

F T F T T

CONTRADICTION!

shortened truth tables

Show that (AB) (BA) is not tf-false

A B (A B) (B A)

T T T T T

F T F T T

shortened truth tables

Show that (AB) (BA) is not tf-false

A B (A B) (B A)

T T T T T T T T T

F T F T T

shortened truth tables

Show that (AB) (BA) is not tf-false

A B (A B) (B A)

T T T T T T T T T

shortened truth tables

Show that (AB) (BA) is not tf-true

A B (A B) (B A)

F

shortened truth tables

Show that (AB) (BA) is not tf-true

A B (A B) (B A)

F F F

shortened truth tables

Show that (AB) (BA) is not tf-true

A B (A B) (B A)

T F F F T F F

shortened truth tables

Show that (AB) (BA) is not tf-true

A B (A B) (B A)

? ? T F F F T F F

CONTRADICTION!

Therefore, the sentence is tf-true

shortened truth tables Practice

Show that:{A(B&C), B (AC), C~C} is tf-consistent

{B(A&~C), (CA) B, ~BA} ~(AC)

shortened truth tables Practice

Show that:{A(B&C), B (AC), C~C} is tf-consistent

{B(A&~C), (CA) B, ~BA} ~(AC)

A B C A (B & C) B (A C) C ~ C

T T T

shortened truth tables Practice

Show that:{A(B&C), B (AC), C~C} is tf-consistent

{B(A&~C), (CA) B, ~BA} ~(AC)

A B C A (B & C) B (A C) C ~ C

T T T T T

shortened truth tables Practice

Show that:{A(B&C), B (AC), C~C} is tf-consistent

{B(A&~C), (CA) B, ~BA} ~(AC)

A B C A (B & C) B (A C) C ~ C

T T T T T T T T T T

shortened truth tables Practice

Show that:{A(B&C), B (AC), C~C} is tf-consistent

{B(A&~C), (CA) B, ~BA} ~(AC)

A B C A (B & C) B (A C) C ~ C

T T T T T T T T T T T T T T T

shortened truth tables Practice

Show that:{A(B&C), B (AC), C~C} is tf-consistent

{B(A&~C), (CA) B, ~BA} ~(AC)

A B C A (B & C) B (A C) C ~ C

T T T T T T T T T T T T T T T F T

shortened truth tables Practice

Show that:{A(B&C), B (AC), C~C} is tf-consistent

{B(A&~C), (CA) B, ~BA} ~(AC)

A B C A (B & C) B (A C) C ~ C

T T T T T T T T T T T T T T T F T

B (A & ~ C) (C A) B ~ B A ~ (A C)

T T T F

shortened truth tables Practice

Show that:{A(B&C), B (AC), C~C} is tf-consistent

{B(A&~C), (CA) B, ~BA} ~(AC)

A B C A (B & C) B (A C) C ~ C

T T T T T T T T T T T T T T T F T

B (A & ~ C) (C A) B ~ B A ~ (A C)

T T T F T

shortened truth tables Practice

Show that:{A(B&C), B (AC), C~C} is tf-consistent

{B(A&~C), (CA) B, ~BA} ~(AC)

A B C A (B & C) B (A C) C ~ C

T T T T T T T T T T T T T T T F T

B (A & ~ C) (C A) B ~ B A ~ (A C)

T T T T T F T

shortened truth tables Practice

Show that:{A(B&C), B (AC), C~C} is tf-consistent

{B(A&~C), (CA) B, ~BA} ~(AC)

A B C A (B & C) B (A C) C ~ C

T T T T T T T T T T T T T T T F T

B (A & ~ C) (C A) B ~ B A ~ (A C)

T T T T T T T F T

shortened truth tables Practice

Show that:{A(B&C), B (AC), C~C} is tf-consistent

{B(A&~C), (CA) B, ~BA} ~(AC)

A B C A (B & C) B (A C) C ~ C

T T T T T T T T T T T T T T T F T

B (A & ~ C) (C A) B ~ B A ~ (A C)

T T T T T F T T F T

shortened truth tables Practice

Show that:{A(B&C), B (AC), C~C} is tf-consistent

{B(A&~C), (CA) B, ~BA} ~(AC)

A B C A (B & C) B (A C) C ~ C

T T T T T T T T T T T T T T T F T

B (A & ~ C) (C A) B ~ B A ~ (A C)

T T T T T F F T T T T T T F T T F

shortened truth tables Practice

Show that:{A(B&C), B (AC), C~C} is tf-consistent

{B(A&~C), (CA) B, ~BA} ~(AC)

A B C A (B & C) B (A C) C ~ C

T T T T T T T T T T T T T T T F T

B (A & ~ C) (C A) B ~ B A ~ (A C)

T T T T T F F T T T T F T T T F T T F