Formal logic 'not'

Question 1

What is ‘Not’?

What does it express?

What symbol do we use to represent it and how?

Answer 1

Not is a logical operator (connects two or more expressions)

Not expresses ‘negation’

We use the symbol ‘~’ to express Negation

You can put ~ before a word or a sentence (meaning ‘it is not the case that’)

Eg. ~ London is north of Manchester

Question 2

Truth Table for ‘Not’

Answer 2

Not goes in front of a single letter like ~A so it is a singulary/unary operator. Unlike & which is binary, conjoining two letters. Therefore the truth table is short

Question 3

When is ‘~~A’ true?

Answer 3

~ negates whatever it is applied to, it can be applied to a sentence already including a negation sign.

If ~ A is true then ~~ A is false and vice versa

~~ A is equivolent to A so we can use a rule called ‘double negation elimination’

Question 4

Good Arguments using Not

Answer 4

‘Sharapova hasn’t won Wimbledon since 2004. But it’s not the case that she’s not suited to the tournament; it’s just that the players she’s faced in the past have been too good.’ (Radio Times.)

• ‘It’s not the case that she is not suited to the tournament’ becomes, with disambiguation and logical operators:
• ~~ Sharapova is suited to the Wimbledon tournament
• So we argue as follows, using double negation elimination:
• P1 ~~ Sharapova is suited to the Wimbledon tournament
• ____________________________________
• Sharapova is suited to the Wimbledon tournament

Question 5

Bad arguments using not

Answer 5

• there is a difference between ‘we don’t know whether A is true’ and ’it is not the case that A is true’.
• One bad style of argument you sometimes hear is an appeal to ignorance: ‘We don’t know whether A is true, so A is false (or neither true nor false’.