12 MS 2015-16
Review of Logic
Proposition: a proposition is a statement, which may be true, false or
indeterminate.
e.g.𝑝: it is raining
𝑞: it is windy
Truth Value: the truth value of a proposition is whether it is true or false.
Negation: the opposite truth value, “not p”, written ¬𝑝
Truth table: columns showing the truth values of propositions
e.g.
𝑝 ¬𝑝
T F
F T
Negation works like the complement of a set does, so if a proposition 𝑝 is true,
then the negation ¬𝑝 is false, and vice versa.
Compound propositions: compound propositions are formed using a variety of
connectives defined below.
Conjunction: “and”, 𝑝⋀𝑞, only true when both propositions 𝑝 and 𝑞 are true.
e.g. It is raining and windy.
𝑝 𝑞 𝑝⋀𝑞
T T T
T F
F
F T
F
F F
F
Disjunction: “or”, 𝑝⋁𝑞, true when either proposition 𝑝 or 𝑞, or both are true.
e.g. It is raining or windy.
𝑝 𝑞 𝑝⋁𝑞
T T T
T F
T
F T T
F F
F
Exclusive Disjunction: “exclusive or”, 𝑝 V 𝑞 is true when either 𝑝 or 𝑞 is true, but
NOT if both are true.
e.g. Either it is raining, or it is windy, but not both.
𝑝 𝑞 𝑝V𝑞
T T
F
T F
T
F T
T
F F
F
Tautology: a compound proposition is a tautology if all the values in the truth table
are true.
Logical contradiction: a compound proposition is a logical contradiction if all the
values in the truth table are false.
Logical equivalence: two compound propositions are equivalent if they have the
same truth table.
Implication: “if… then…”, 𝑝 ⟹ 𝑞, 𝑝 implies 𝑞.
e.g. If it is raining, then it is windy. The broken promise.
𝑝 𝑞 𝑝⟹𝑞
T T
T
T F
F
F T
T
F F
T
12 MS 2015-16
e.g. If you pass your driving test then your parents will give you a car.
If you pass the test and you get the car, then you are happy. T implies T gives T
If you fail the test and do not get the car then you cannot complain. F implies F is T
If you fail the test but are still given the car, then you are definitely happy with
that. F implies T is T
However, if you pass the test and are NOT given the car, then you are justly very
unhappy! This is the broken promise. T implies F is F.
Equivalence: “if… then… and vice versa”, 𝑝 ⟺ 𝑞, “if and only if”.
e.g. If it is raining, then it is windy, and if it is windy, then it is raining.
It is raining, if and only if it is windy.
𝑝 𝑞 𝑝⟺𝑞
T T
T
T F
F
F T
F
F F
T
Converse: the converse of 𝑝 ⟹ 𝑞 is 𝑞 ⟹ 𝑝
Given 𝑝 ⟹ 𝑞 means if it is raining then it is windy,
the converse 𝑞 ⟹ 𝑝 means if it is windy then it is raining.
e.g. Pythagoras’ Theorem: if the triangle is right angled, then the square on the
hypotenuse is equal to the sum of the squares on the other two sides.
The converse is also true: if the square on the hypotenuse is equal to the sum of
the squares on the other two sides, then the triangle is right angled.
We use the converse of Pythagoras’ Theorem to test whether a triangle has a right
angle.
Inverse: the inverse of 𝑝 ⟹ 𝑞 is ¬𝑝 ⟹ ¬𝑞
Given 𝑝 ⟹ 𝑞 means if it is raining then it is windy,
the inverse ¬𝑝 ⟹ ¬𝑞 means if it is not raining then it is not windy.
Contrapositive: the contrapositive of 𝑝 ⟹ 𝑞 is ¬𝑞 ⟹ ¬𝑝
Given 𝑝 ⟹ 𝑞 means if it is raining then it is windy,
the contrapositive ¬𝑞 ⟹ ¬𝑝 means if it is not windy then it is not raining.
Valid argument: this is a set of propositions, the premise, which leads to a
conclusion.
Using the previous propositions,
𝑝: You pass your driving test
𝑞: Your parents will give you a car
Premise:
If you pass your driving test, then your parents will give you a car.
and
You pass your driving test.
Conclusion: Therefore your parents will give you a car.
Showing this in a truth table:
𝑝 𝑞 𝑝 ⟹ 𝑞 (𝑝 ⟹ 𝑞)⋀ 𝑝 (𝑝 ⟹ 𝑞)⋀𝑝 ⟹ 𝑞
T T
T
T
T
T F
F
F
T
F T
T
F
T
F F
T
F
T
This is a tautology, which shows that the argument is valid. This means that the
conclusion follows logically from the premise. For the purposes of the argument,
the actual truth of any statement or conclusion is irrelevant; it is only concerned
with whether the conclusion itself follows logically from the premise.