Truth Tables and Validity

Question 1

What are all the operators thus far introduced (&, ∨, ¬, and →)

Answer 1

Truth functional operators

Question 2

Given a sentence A & B, what can we work out?

Answer 2

The truth value just by knowing whether A is true and B is true.

Question 3

The truth value of any sentence built from truth-functional operators is completely determined by what?

Answer 3

The truth value of the basic propositions in it

Question 4

What is the way of doing logic with truth function operators called?

Answer 4

Truth Functional Logic (TFL)

Question 5

What 3 things does the conditional not say concerning the sentence:
‘IF I’m in Paris THEN I have blue hair’?

Answer 5

1. It doesn’t say ‘Whenever to go to Paris, I always have blue hair’
2. It doesn’t say ‘If I were to go to Paris then I would have blue hair.’;
3. It doesn’t say ‘IN any situation in which I’m in Paris, I have blue hair.’

Question 6

What does the sentence ‘IF I’m in Paris THEN I have blue hair’ actually say?

Answer 6

If it’s true that I’m in Paris then it’s true that I have blue hair

Question 7

When we think about whether an argument is valid, we think about what?

Answer 7

All possible situations and whether we can imagine a way to make the premises true and the conclusion false.

Question 8

When we think about whether a Conditional is true in some situation, we think about what?

Answer 8

Whether the antecedent and the conditional are true or false in that particular situation.

Question 9

What are the four types of conditional?

Answer 9

1. Rule;
2. Decision;
3. Cause;
4. General;
5. Inference

Question 10

Concerning the inferential form of the conditional, say you have an argument ‘If the fingerprints match then you’re guilty’ and the fingerprints do not match, is it valid to claim that the person is not guilty?

Answer 10

Invalid

Question 11

Why is that invalid?

Answer 11

Because the person could be proven guilty for other reasons.

Question 12

Formalise the argument:
P1 ‘IF the fingerprints match, then you’re guilty’
P2 ‘The fingerprints do not match’
Conc: ‘You’re not guilty’

Answer 12

P1: F => G
P2: ¬ F
Conc: ¬ G

Question 13

With this argument, what row of the truth table for the conditional proves it incorrect?

Answer 13

Row three

Question 14

How can we express a stronger conditional?

Answer 14

With the biconditional

Question 15

What is the biconditional in natural language?

Answer 15

‘If and only if’

Question 16

What is the logical operator for the biconditional?

Answer 16

<=>

Question 17

If we were to rewrite the argument ‘The fingerprints match if and only if you’re guilty’, what would happen to the aforementioned issue?

Answer 17

Then if the fingerprints did not match, you would not be guilty.

Question 18

What are the four possibilities we have to consider with the biconditional?

Answer 18

T T | T
T F | F
F T | F
F F | T

Question 19

What do we look for to prove an argument’s validity?

Answer 19

A counterexample

Question 20

Define a counterexample

Answer 20

A situation in which all the premises are true and the conclusion is false

Question 21

Define a valid argument

Answer 21

In any situation in which all the premises are true and the conclusion is also true

Question 22

What is meant by ‘situation’ here?

Answer 22

‘A way the world could be’ - it could be a realistic scenario or something completely fantastical

Question 23

For any situation we consider, what is the sole thing we care about?

Answer 23

Whether the premises and conclusion are true

Question 24

What is the truth table for the argument
P1: A => B
P2: A
Conc: B

Answer 24

________P1 P2 Conc
A B | A => B | A | B
———————————-
T T | T | T | T
T F | F | T | F
F T | T | F | T
F F | T | F | F

P1 | P2|Conc

Question 25

How, with a truth table, do we see if it valid?

Answer 25

Check every row in which all the premises are true and if the conclusion is true then the argument is valid

Question 26

What is the truth table for the argument
P1: A => B
P2: ¬B
Conc: ¬A

Answer 26

_______P1 P2 Conc
A B | A => B | ¬B | ¬A
———————————-
T T | T | F | F
T F | F | T | F
F T | T | F | T
F F | T | T | T

Question 27

What is the truth table for the argument
P1: A => B
P2: ¬A
Conc: ¬B

Answer 27

_______P1 P2 Conc
A B | A => B | ¬A | ¬B
———————————-
T T | T | F | F
T F | F | F | F
F T | T | T | F
F F | T | T | T

Question 28

In this example, why is it invalid?
_______P1 P2 Conc
A B | A => B | ¬A | ¬B
———————————-
T T | T | F | F
T F | F | F | F
F T | T | T | F
F F | T | T | T

Answer 28

It is invalid because row 3 and row 4, all premises are yet they don’t both have a true conclusion

Question 29

What is the form of this complex sentence:
‘It will be sunny on Wednesday and
it will rain on Thursday or it will rain on Friday’

Answer 29

A & (B V C)
A stands for ‘It will be sunny on Wednesday’;
B stands for ‘It will rain on Thursday’;
C stands for ‘It will rain on Friday

Question 30

The brackets in A & (B V C) are important. Why?

Answer 30

To remove ambiguity. For A & (B V C) is different from (A & B) V C

Question 31

We can take any two complex sentences and join them together with an operator to make a bigger sentence. Translate:
‘If the train is late or I forget my passport then
I’ll miss my flight and I’ll be sad.’

Answer 31

(A V B) => ( C & D)
A stands for ‘The train is late’;
B stands for ‘I forgot my passport’;
C stands for ‘I’ll miss my flight’; and,
D stands for ‘I’ll be sad’

Question 32

Concerning this argument what do we want to know?
P1: (A => B) & (B => A)
Conc A <=> B

Answer 32

The truth values of (A => B) & (B => A) and A <=> B

Question 33

To work out the truth value of (A => B) & (B => A), what do we first have to know?

Answer 33

The truth values of (A => B) and (B => A)

Question 34

So before we fill in the column for the premise, what do we have to fill in?

Answer 34

The intermediate columns: (A=>B) and (B=>A)

Question 35

What are intermediate columns for?

Answer 35

Our own convenience

Question 36

Why do we only need intermediate columns for?

Answer 36

Because we cannot jump straight to figuring out the values of the premises and conclusion

Question 37

Once the intermediate columns are figured out, what are the values in the intermediate columns?

Answer 37

Not important anymore