Formal Logic Or, If and only if

Question 1

What symbol do we use for ‘or’?

Answer 1

v

Question 2

What type of logical operator is ‘or’?

Answer 2

Disjunction

Question 3

What are some synonyms of ‘or’?

Answer 3

‘either…or’, ‘alternatively’, ‘unless…’

Question 4

What is the truth table for ‘v’ (inclusive)?

Answer 4

A¦ B¦AvB¦
T¦ T ¦ T ¦
T¦ F ¦ T ¦
F¦ T ¦ T ¦
F¦ F ¦ F ¦

Question 5

What is meant by inclusive or?

Answer 5

AvB is true if either A is true, or B is true or both A and B are true

Question 6

What is the truth table for ‘v’ (exclusive)?

Answer 6

A¦ B¦A_v_B¦ (underlined v)
T¦ T ¦ F ¦
T¦ F ¦ T ¦
F¦ T ¦ T ¦
F¦ F ¦ F ¦

Question 7

What is meant by exclusive or?

Answer 7

A_v_B is true if A is true or B is true but not both

Question 8

How do we express exclusive or in our language (as we only use inclusive or)?

Answer 8

(AvB) & ~ (A&B)

Question 9

How can a trilemma or a dilemma be a good form of argument using or?

Answer 9

Trilemma (argument with three options)
Dilemma (argument with two options)

-Enumerate (list off one by one) options as a disjunction (this OR this)
-Show why none of the options work
-Conclude the opponent is wrong

Question 10

What is meant by the fallacy called ‘false dilemma’/’false dichotomy’?

Answer 10

-Someone enumerates two options
-They claim those are the only two options
-It is false when there are more options

Question 11

What symbol do we use for ‘if and only if’?

Answer 11

<->

Question 12

What type of logical operator is ‘if and only if’?

Answer 12

the biconditional

Question 13

What are other ways of saying ‘if and only if’?

Answer 13

‘iff’, ‘just in case’, ‘when and only when’

Question 14

What are the truth conditions for ‘If and only if’?

Answer 14

For A<->B to be true, we need both A if B and A only if B to be true so we must combine both of their truth tables.

A if B says if B is true then A is also true

A only if B says that A will not be true if B is false

Combine:
A if and only if B says that if B is true then A is also true, and if B is false then A is also false

SO A if and only if B is true if both A and B are true or both false

Question 15

What is the truth table for ‘<->’?

Answer 15

A¦ B¦A<->B¦
T¦ T ¦ T ¦
T¦ F ¦ F ¦
F¦ T ¦ F ¦
F¦ F ¦ T ¦

Question 16

What are necessary conditions?

Answer 16

Things that are necessary for things to exist/to be as they are,
‘x’ cannot be ‘x’ without the necessary condition

B->A implies that B is necessary for A

e.g. Being a mammal is a necessary condition for being a horse
Note: but it is not a sufficient condition as cats and dogs are also mammals

Question 17

What are sufficient conditions?

Answer 17

Things that are enough (suffice) for things to exist/to be as they are
‘x’ can suffice to be ‘x’ with a sufficient condition

A->B implies that A is sufficient for B

e.g. being born in the UK is a sufficient condition for being a UK citizen
(being born there is enough for citizenship)
Note: but it’s not necessary, you may be born elsewhere to British parents or you may be naturalised

Question 18

’<->’ is useful for giving necessary and sufficient conditions, what does that mean?

Answer 18

A necessary and sufficient condition is when both it is necessary and enough for something to exist/to be as it is

A<->B implies that A is necessary and sufficient for B and B is necessary and sufficient for A

e.g. being a valid argument with true premises is a necessary and sufficient condition for being a sound argument
we say:
an argument is sound iff it is a valid argument and its premises are true

Question 19

What are other ways of signalling the conclusion?

Answer 19

‘so’, ‘hence’, ‘thus’, ‘therefore’, ‘it follows that…’