1 of 31

Logical Operators

There are five logical operators:

~, ∨, ·, ⊃, ≡

1

P	~P
T	F
F	T

P	Q	P·Q
T	T	T
T	F	F
F	T	F
F	F	F

P	Q	P ∨ Q
T	T	T
T	F	T
F	T	T
F	F	F

P	Q	P≡Q
T	T	T
T	F	F
F	T	F
F	F	T

P	Q	P⊃Q
T	T	T
T	F	F
F	T	T
F	F	T

2 of 31

Computation Procedures

There are just two steps:

Enter the truth values of the simple components.
Compute the truth values of the compound components.

2

If there is more then one compound component, calculate the truth values of them one by one following the principle of “from the innermost to the outermost brackets” until the one under the main operator.

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Computation Procedures

P≡(Q ∨ R)

Suppose P and Q are true, R is false.

3

P	≡	(Q	∨	R)

T

F

4 of 31

Computation Procedures

A complete truth table for proposition illustrates the truth values of it in all logical possibilities.

4

P	≡	(Q	∨	R)
T	T	T	T	T
T	T	T	T	F
T	T	F	T	T
T	F	F	F	F
F	F	T	T	T
F	F	T	T	F
F	F	F	T	T
F	T	F	F	F

This →

5 of 31

Truth Tables for Propositions

To construct a complete truth table, the first thing we have to do is to determine the number of rows of it.
One row = one logical possibility
The number of rows can be understood by the principle of 2ⁿ, where n is the number of different simple components.

5

6 of 31

Truth Tables for Propositions

6

P	~P
T	F
F	T

P	Q	Q ∨ R
T	T	T
T	F	T
F	T	T
F	F	F

P	Q	R	P≡(Q ∨ R)
T	T	T	T
T	T	F	T
T	F	T	T
T	F	F	F
F	T	T	F
F	T	F	F
F	F	T	F
F	F	F	T

2 = 2¹

2³ = 8

4 = 2²

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Truth Tables for Propositions

Truth table for wff with one simple component.

7

G	(G	⊃	G)	≡	G

T

F

T

F

T

F

T

F

T

F

8 of 31

Truth Tables for Propositions

Truth table for wff with two simple components.

8

A	H	~	A	≡	H

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

9 of 31

Truth Tables for Propositions

9

P	Q	R	P	≡	(Q	∨	R)

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

10 of 31

Exercise (1): Answers

10

W	X	Y	(~	W	·	X)	·	(Y	·	W)
T	T	T
T	T	F
T	F	T
T	F	F
F	T	T
F	T	F
F	F	T
F	F	F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

Self-contradictory

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Exercise (1): Answers

11

P	Q	R	~	(P	·	Q)	∨	(R	⊃	Q)
T	T	T
T	T	F
T	F	T
T	F	F
F	T	T
F	T	F
F	F	T
F	F	F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

Tautology

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Exercise (1): Answers

12

S	T	U	(S	⊃	T)	·	(U	·	~	T)
T	T	T
T	T	F
T	F	T
T	F	F
F	T	T
F	T	F
F	F	T
F	F	F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

Contingent

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Relations of Propositions

We may use truth tables to compare different propositions and determine their relations.
There are four relations between propositions:

Logically Equivalent
Contradictory
Consistent
Inconsistent

13

14 of 31

Relations of Propositions

Logically Equivalent

For any wffs α and β, α and β are logically equivalent just in case they have the same truth value in every logical possibility.

14

P	Q	P	⊃	Q	/	~	P	∨	Q
T	T	T	T	T		F	T	T	T
T	F	T	F	F		F	T	F	F
F	T	F	T	T		T	F	T	T
F	F	F	T	F		T	F	T	F

15 of 31

Relations of Propositions

Contradictory

For any wffs α and β, α and β are contradictory just in case α and β have different truth values in every logical possibility.

15

P	Q	P	⊃	Q	/	P	·	~	Q
T	T	T	T	T		T	F	F	T
T	F	T	F	F		T	T	T	F
F	T	F	T	T		F	F	F	T
F	F	F	T	F		F	F	T	F

16 of 31

Relations of Propositions

Consistent

For any wffs α and β, α and β are consistent just in case there is at least one logical possibility that α and β are both true.

16

P	Q	P	·	Q	/	P	∨	Q
T	T	T	T	T		T	T	T
T	F	T	F	F		T	T	F
F	T	F	F	T		F	T	T
F	F	F	F	F		F	F	F

T

17 of 31

Relations of Propositions

Inconsistent

For any wffs α and β, α and β are inconsistent just in case there is no logical possibility that α and β are both true.

17

P	Q	P	·	Q	/	~	Q	≡	P
T	T	T	T	T		F	T	F	T
T	F	T	F	F		T	F	T	T
F	T	F	F	T		F	T	T	F
F	F	F	F	F		T	F	F	F

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Exercise (2): Answers

18

M	N	~	M	⊃	N	/	M	⊃	~	N
T	T
T	F
F	T
F	F

S	T	(S	⊃	T)	∨	S	/	~	(T	∨	~	S)	·	T
T	T
T	F
F	T
F	F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

Consistent

Contradictory

Inconsistent

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Exercise (2): Answers

19

X	Y	Z	X	·	(Y	≡	Z)	/	Y	≡	(X	·	~	Z)
T	T	T
T	T	F
T	F	T
T	F	F
F	T	T
F	T	F
F	F	T
F	F	F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

Inconsistent

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Truth Tables for Arguments

The procedures are more or less the same as constructing a truth table for proposition.
In a truth table for argument, we need to list out all premises and conclusion in the truth table.
To clearly indicate premises and conclusion, we use “ / ” to separate the premises and “// ” to separate the last premise and the conclusion.

20

21 of 31

Truth Tables for Arguments

Consider the following argument:

21

1. P⊃Q

2. P

3. Q

P	Q	P	⊃	Q	/	P	//	Q
T	T	T	T	T		T		T
T	F	T	F	F		T		F
F	T	F	T	T		F		T
F	F	F	T	F		F		F

P1. If the MTR is delayed, I will be late for class.

P2. The MTR is delayed.

C. I will be late for class.

Among all logical possibilities, there is no combination with all true premises and a false conclusion.

→ It is not possible for this argument to have all true premises and a false conclusion.

→ Valid

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Truth Tables for Arguments

Consider the following argument:

22

P1. Either Natalie Portman or Scarlett Johansson is an actress.

P2. Scarlett Johansson is an actress.

C. Natalie Portman is not an actress.

1. P ∨ Q

2. Q

3. ~P

P	Q	P	∨	Q	/	Q	//	~	P
T	T	T	T	T		T		F	T
T	F	T	T	F		F		F	T
F	T	F	T	T		T		T	F
F	F	F	F	F		F		T	F

→ Invalid

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Exercise (3): Answers

23

M	N	(M	⊃	N)	⊃	~	(M	⊃	N)	//	~	N
T	T
T	F
F	T
F	F

T

F

T

F

T

F

T

F

T

Valid

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

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Exercise (3): Answers

24

R	S	T	~	T	⊃	S	/	~	S	∨	(~	T	⊃	R)	/	~	T	//	R
T	T	T
T	T	F
T	F	T
T	F	F
F	T	T
F	T	F
F	F	T
F	F	F

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

Valid

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Indirect Truth Tables

Indirect truth tables provide faster way for checking validity of arguments.
The idea of it is to focus on checking whether the argument is possible to be invalid. If it is possible, then the argument is invalid. Otherwise, valid.

25

26 of 31

Indirect Truth Tables

Consider the following argument:

A contradiction has been found, which means it is not possible for this argument to be invalid.
That means this argument is valid.

26

P	⊃	Q	/	~	Q	//	~	P

T

F

T

F

27 of 31

Indirect Truth Tables

Where the contradiction lies may depend on your filling order.

As long as the argument is valid, a contradiction must be able to find.

27

Q	≡	R	/	P	⊃	Q	//	~	R	⊃	~	P

T

F

T

F

T

F

T

F

T

F

T

28 of 31

Indirect Truth Tables

Sometimes we need to make some assumptions as we do not know what value to put in.

In this case, if a contradiction has been found, you CANNOT stop the computation.

You need to exhaust other possibilities.

28

P	⊃	Q	/	Q	⊃	P	//	P	≡	Q

P	⊃	Q	/	Q	⊃	P	//	P	≡	Q

T

F

T

F

T

F

T

F

T

F

F_A

T_A

29 of 31

Exercise (4): Answers

29

F

M·N / (M ∨ N)⊃P // ~P

M	·	N	/	(M	∨	N)	⊃	P	//	~	P

A⊃(E ∨ B) / C⊃(B ∨ D) / (A ∨ B)⊃C / D⊃(E·~A) // B≡C

A	⊃	(E	∨	B)	/	C	⊃	(B	∨	D)	/	(A	∨	B)	⊃	C	/	D	⊃	(E	·	~	A)	//	B	≡	C

Invalid

T

F

T

F

T

F

T

F

T

F

T

F

T

F

F_A

T_A

T

F

30 of 31

Indirect Truth Tables

We can also use indirect truth table to check whether a set of wffs are consistent.

The procedures are more or less the same.

This time, we focus on checking whether the argument is possible to be consistent. If it is possible, then the argument is consistent. Otherwise, inconsistent.

30

J	·	(L	⊃	K)	/	(J	⊃	K)	·	~	K

T

F

T

F

T

F

31 of 31

Aside: Strategies

There are five logical operators:

~, ∨, ·, ⊃, ≡

31

P	~P
T	F
F	T

P	Q	P·Q
T	T	T
T	F	F
F	T	F
F	F	F

P	Q	P ∨ Q
T	T	T
T	F	T
F	T	T
F	F	F

P	Q	P≡Q
T	T	T
T	F	F
F	T	F
F	F	T

P	Q	P⊃Q
T	T	T
T	F	F
F	T	T
F	F	T

T

F

T

F

T