Embed Size (px)
Citation preview
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
