Formal Logic: 'If then'

Question 1

If we have two sentences connected by ‘If then’ what is the logical operator called?

What symbol is used to represent it?

Example of a sentence using it

Answer 1

The logical operator if then is called the ‘conditional’

It is represented by ->

If Serena is muscular, then Serena is Strong

Serena is muscular -> Serena is strong

Question 2

If ‘A -> B then what is A called and what is B called?

Answer 2

A is tje antecedent

and

B is the consequent of the conditional

Question 3

What do students often get wrong about the If then conditional?

What is actually correct? Explain using examples

Answer 3

• Often students’ first answer is that there is a causal relation: ‘If Serena is muscular, then Serena is strong’ means that Serena’s being muscular causes Serena to be strong. But that’s wrong.
• Some philosophers of logic argue that ‘if A then B’ should mean a relevant connection between A and B. But we’re going to use the truth-functional conditional, according to which the truth of ‘if A then B’ is a function of the truth values of A and B.

Eg. “If Miss Marple saw the crime, the criminal is toast” is true either is Miss marple saw the crime or not, but not if she saw it and the criminal is not toas.

Question 4

Truth table for If then ->

Answer 4

Question 5

Good arguments using If then

Modus Ponens argument

Answer 5

P1 If Serena is muscular then Serena is strong.

P2 Serena is muscular

________

Serena is strong

• P1 A -> B
• P2 A

________________

•B

Question 6

Good arguments using if then

Modus Tollens

Answer 6

P1 If Daphne is a cat, then Daphne meows

P2 Daphne does not meow

_____________________

Daphne is not a cat

• P1 A -> B
• P2 ~B

________

•~A

Question 7

Bad arguments using if then

2 examples

Answer 7

Denying the antecedent.

P1 If you smoke, you’re likely to get heart disease •P1 A -> B

P2 I don’t smoke •P2 ~A

____________ ________

I’m not likely to get heart disease •~B

(could be false; my risk might be genetic)

Affirming the consequent

P1 If vaccination is widespread, infections are down •P1 A -> B

P2 Infections are down •P2 B

________ ____________________

Vaccination is widespread •A

• (could be false; it could be lockdown)

Question 8

if v Only If

Answer 8

•A if B’ = ‘if B then A’.

‘A only if B’ = ‘if A then B’. (if A is true then B must be true)

• To say ‘A only if B’ is to say that A will not be true if B is false. This is just what the truth-table for A -> B represents: the condition whereby A is true and B false is the one condition that excludes the possibility of A -> B being true (second row). So this is the right translation of ‘A only if B’.
• The sentence ‘A if B’ says that if B is true then A is also true. But this condition is not met by the truth-table for A -> B: the third row allows for A -> B to be true when B is true but A is false. So A -> B cannot be the right way to translate ‘A if B’.

Question 9

Antecedent v Consequent

Answer 9

Antecedent - if part

Consequent - then