Proposition

A proposition or statement is a sentence that is either true or false, but not both or neither, or true sometimes and false sometimes.
Example propositions:
- \(\sqrt2\) is an irrational number.
- \(1 < 1\).
- Some integers \(x, y, z\) satisfy \(x^3 + y^3 = z^3\).
- There are no positive integers \(x, y, z\) such that \(x^4 + y^4 = z^4\).
Example non-propositions:
- What do you mean by that?
- How pretty!
- Please give me a call.
- Abraham Lincoln was an honest president.
- 23 is an interesting number.
- \(x^2 + 3x - 4 = 0\).
- This statement is false.

Propositional variables and Truth values

We denote propositions using short names like \(P\), \(Q\), \(R\), \(\dots\), \(P_1\), \(P_2\), \(P_3\), \(\dots\), \(P'\), \(Q'\), \(R'\), \(\dots\), \(p\), \(p\), \(p\), \(\dots\), \(p_1\), \(p_2\), \(p_3\), \(\dots\), \(p'\), \(p'\), \(p'\), etc.
Atomic propositions are called propositional variables or boolean variables and should be denoted by lowercase letters.
When a proposition is written symbolically, it’s usually called a formula.
The truth value of a proposition is usually denoted by T (for true), and F (for false).
Computer scientists also use \(1\) for true, and 0 for false.
When T appears in a formula, it stands for any arbitrary true proposition. Similarly, an F stands for any arbitrary false proposition.

Compound vs Atomic Propositions

A compound proposition is obtained by connecting other propositions by the logical connectives (or boolean operators) \(\neg\), \(\land\), \(\lor\), \(\Rightarrow\), \(\Leftrightarrow\), \(\oplus\), etc.

Name	Formula	English
negation	\(\neg P\),~\(P\), \(\overline{P}\)	not P
conjunction	P\(\land\)Q	P and Q
disjunction	P\(\lor\)Q	P or Q
implication	P\(\Rightarrow\)Q	P implies Q
biconditional	P\(\Leftrightarrow\)Q	P if and only if Q

In the implication P\(\Rightarrow\)Q, the proposition P is called premise or hypothesis or antecedent, and the proposition Q is called conclusion or consequence or consequent.

Truth Tables

The truth values of compound propositions are defined using truth tables.
P \(\neg\)P

F T

T F
P Q P\(\land\)Q

F F F

F T F

T F F

T T T
P Q P\(\lor\)Q

F F F

F T T

T F T

T T T

P	\(\neg\)P
F	T
T	F

P	Q	P\(\land\)Q
F	F	F
F	T	F
T	F	F
T	T	T

P	Q	P\(\lor\)Q
F	F	F
F	T	T
T	F	T
T	T	T

Truth Tables (continued)

P Q P\(\Rightarrow\)Q

F F T

F T T

T F F

T T T
P Q P\(\Leftrightarrow\)Q

F F T

F T F

T F F

T T T
P Q P\(\oplus\)Q

F F F

F T T

T F T

T T F

P	Q	P\(\Rightarrow\)Q
F	F	T
F	T	T
T	F	F
T	T	T

P	Q	P\(\Leftrightarrow\)Q
F	F	T
F	T	F
T	F	F
T	T	T

P	Q	P\(\oplus\)Q
F	F	F
F	T	T
T	F	T
T	T	F

Associativity, Order of Precedence

When a formula contains the same connectives several times, e.g., \(P \land Q \land R\), we assiciate the operator to the left. So this expression means \((P\land Q) \land R\).
When a formula contains several different connectives, it becomes ambiguous which connectives should be applied first, e.g., \(P \lor Q\land R \Rightarrow S\). Order of precedence then determines which connectives to apply first.
Connective Precedence

\(\neg\) \(1\)

\(\land\) 2

\(\lor\) 3

\(\Rightarrow\) 4

\(\Leftrightarrow\) 5
A smaller numbered connective precedes a larger numbered connective.
So the above formula means \((P \lor (Q\land R)) \Rightarrow S\).
When in doubt, parenthesize!

Connective	Precedence
\(\neg\)	\(1\)
\(\land\)	2
\(\lor\)	3
\(\Rightarrow\)	4
\(\Leftrightarrow\)	5

Truth table calculation

Calculate in class to show that the formula \(A\lor \overline{A} \land B\) means the same as \(A\lor B\).

Expressing Implications in English

These are some of the ways to express the implication P\(\Rightarrow\)Q in English.
- if P then Q
- P implies Q
- Q is implied by P
- P only if Q
- Q if P
- P is sufficient for Q
- Q is necessary for P
- Q whenever P

Expressing a Biconditional in English

These are some of the ways to express the biconditional \(p\Leftrightarrow q\) in English.
- P if and only if Q
- P is necessary and sufficient for Q
- Q is necessary and sufficient for P
- P implies Q and vice versa
- if P then Q and vice versa

Terminology on implications

An implication is said to be trivially true when its conclusion is true.
An implication is said to be vacuously true when its hypothesis is false.
The converse of \(P\Rightarrow Q\) is \(Q\Rightarrow P\).
The contrapositive of \(P\Rightarrow Q\) is \(\neg Q\Rightarrow \neg P\).
The inverse of \(P\Rightarrow Q\) is \(\neg P\Rightarrow \neg Q\).

Terminology

A tautology is a proposition that’s always true, e.g., when they have the formula \(P\lor \neg P\).
A contradiction is a proposition that’s always false, e.g., when they have the formula \(P\land \neg P\).
A contingency is proposition that’s sometimes true and sometimes false, e.g., when they have the formula \(P \lor P\).

Practice

Question: which of these are tautology? contradiction? contingency?
- P \(\Rightarrow\) P
- \((\neg P) \Rightarrow P\)
- \([(P \Rightarrow Q) \land P] \land (\neg Q)\)
- \([P \lor (P \land Q)] \Rightarrow P\)
- \([P \land (P \lor Q)] \Leftrightarrow \neg P\)

Terminology (continued)

Two propositions are logically equivalent when they have the same truth value for all possible combinations of the truth values of their constituent propositions.
A formula is valid if the proposition it represents is a tautology.
A formula is satisfiable if the proposition it represents is sometimes true.

Logical Equivalences

Here are some important logical equivalences. Verify!
- \((P \land Q) \land R \equiv P \land (Q \land R)\) (Associativity)
- \((P \lor Q) \lor R \equiv P \lor (Q \lor R)\) (Associativity)
- \(P \Leftrightarrow Q \equiv Q \Leftrightarrow P\) (Commutativity)
- \(P \land Q \equiv Q \land P\) (Commutativity)
- \(P \lor Q \equiv Q \lor P\) (Commutativity)
- \(P \land (Q \lor R) \equiv (P \land Q) \lor (P \land R)\) (Distributivity)
- \(P \lor (Q \land R) \equiv (P \lor Q) \land (P \lor R)\) (Distributivity)
- \(P \land \mathbf{T} \equiv P\) (T is the identity for conjunction)
- \(P \lor \mathbf{F} \equiv P\) (F is the identity for disjunction)
- \(P \land \mathbf{F} \equiv \mathbf{F}\)
- \(P \lor \mathbf{T} \equiv \mathbf{T}\)
- \(\neg (P \land Q) \equiv (\neg P) \lor (\neg Q)\) (De Morgan’s)
- \(\neg (P \lor Q) \equiv (\neg P) \land (\neg Q)\) (De Morgan’s)
- \(\neg (\neg P) \equiv P\) (Double negation)
- \(P \lor (P\land Q) \equiv P\) (Absorption)
- \(P \land (P\lor Q) \equiv P\) (Absorption)
- \(P \Rightarrow Q \equiv (\neg Q) \Rightarrow (\neg P)\)
- \(P \Rightarrow Q \equiv (\neg P) \lor Q\)
- \(\neg (P\Rightarrow Q) \equiv P \land (\neg Q)\)
- \((\neg P) \Rightarrow \mathbf{F}\equiv P\)
- \(P \Leftrightarrow Q \equiv (P \Rightarrow Q) \land (Q \Rightarrow P)\)
- \(P \Leftrightarrow Q \equiv (P \land Q) \lor (\neg P \land \neg Q)\)
- \((P \lor Q) \Rightarrow R \equiv (P \Rightarrow R) \land (Q \Rightarrow R)\)
- \((P \land Q) \Rightarrow R \equiv P \Rightarrow (Q \Rightarrow R)\)

Propositional Logic

San Skulrattanakulchai

September 9, 2021

Proposition

Propositional variables and Truth values

Compound vs Atomic Propositions

Truth Tables

Truth Tables (continued)

Associativity, Order of Precedence

Truth table calculation

Expressing Implications in English

Expressing a Biconditional in English

Terminology on implications

Terminology

Practice

Terminology (continued)

Logical Equivalences

Algebra of Propositions