The Foundations: Logic and Proof, Sets, and Foundations PROPOSITIONS A proposition is a declarative sentence that is either True or False, but not the

The Foundations: Logic and Proof, Sets, and FoundationsPROPOSITIONS•A proposition is a declarative sentence that is either True or False, but not the both.•Example:• 1. Washington is the capital of United states of America,• 2. Toronto is the capital of Canada.• 3. 1+1 = 2• 4. 2+2 = 3

•Proposition 1 and 3 are True whereas 2 and 4 are False.

The Foundations: Logic and Proof, Sets, and Foundations• Consider the following sentences• 1. What time is it?• 2. Read this carefully.• 3. x+1 = 2• 4. x+y = 3

• 1 and 2 are not Propositions because they are not declarative sentences. 3 and 4 are not Propositions because they are neither True nor False.• Truth value of proposition is true, denoted by T and truth value of

proposition is false denoted F.• The area of logic that deals with propositions is called the propositional

calculus or propositional logic.• Developed by the Greek philosopher Aristotle 2300 years back.

Logical Connectives

Operator Symbol

Usage

Negation not

Conjunction

and

Disjunction or

Exclusive or

xor

Conditional

if,then

Biconditional

iff

Compound Propositions: Examples

p = “Cruise ships only go on big rivers.”q = “Cruise ships go on the Hudson.”r = “The Hudson is a big river.”

r = “The Hudson is not a big river.”

pq = “Cruise ships only go on big rivers and go on the Hudson.”

pq r = “If cruise ships only go on big rivers and go on the Hudson, then the Hudson is a big river.”

Negation

This just turns a false proposition to true and the opposite for a true proposition.

EG: p = “23 = 15 +7”p happens to be false, so p is true.

Negation – truth table

Logical operators are defined by truth tables –tables which give the output of the operator in the right-most column.

Here is the truth table for negation:

p p

FT

TF

Conjunction

Conjunction is a binary operator in that it operates on two propositions when creating compound proposition. On the other hand, negation is a unary operator (the only non-trivial one possible).

Conjunction

• Conjunction is supposed to encapsulate what happens when we use the word “and” in English. I.e., for “p and q ” to be true, it must be the case that BOTH p is true, as well as q. If one of these is false, than the compound statement is false as well.

Conjunction

EG. p = “Clinton was the president.” q = “Hilary was the president.” r = “The meaning of is is important.”

Assuming p and r are true, while q false.

Out of pq, pr, qr

only pr is true.

Conjunction – truth table

p q pqTTFF

TFTF

TFFF

Disjunction – truth table

Conversely, disjunction is true when at least one of the components is true:

p q pqTTFF

TFTF

TTTF

Disjunction – Example

EG. p = “Clinton was the president.” q = “Hilary was the president.” r = “The meaning of is is not important.”

Assuming p is true, while q and r are false.

Out of p q, p r, q r

only q r is false.

Review NOT, AND, OR

• What is the disjumction --- or• What is the conjunction – and• Negation – not• Construct the TT for all above

Disjunction – caveat

A: The entrée is served with soup or salad.

Most restaurants definitely don’t allow you to get both soup and salad so that the statement is false when both soup and salad is served. To address this situation, exclusive-or is introduced next.

Exclusive-Or – truth table

p q p q

TTFF

TFTF

FTTF

Conditional (Implication)

This one is probably the least intuitive. It’s only

partly akin to the English usage of “if, then” or

“implies”.

DEF: p q is true if q is true, or if p is false. In the

final case (p is true while q is false) p q is false.

Semantics: “p implies q ” is true if one can

mathematically derive q from p.

Conditional -- truth table

p q p q

TTFF

TFTF

TFTT

Biconditional

Let P and Q be propositions. The biconditional P ↔ Q is the proposition that is True when P and Q have the same truth values, and is False otherwise.

P if and only if Q

P→Q and Q→ PP is necessary and sufficient for Q

Biconditional truth table

P Q P↔Q

TTFF

TFTF

TFFT

• Let P and Q be the propositionsP: It is below freezing.Q: It is snowing.Express each of these propositions as an English sentences

PQ, PQ, PQ, PQ, P→Q, P↔Q

Answers:It is below freezing and snowing.It is below freezing but not snowing.It is not below freezing and it is not snowing.It is below freezing or snowing.If it is below freezing then it is snowing.It is below freezing iff it is snowing

Propositional Equivalences

Tautologies, contradictions, contingencies

DEF: A compound proposition is called a tautology if no matter what truth values its atomic propositions have, its own truth value is T.

EG: p ¬p (Law of excluded middle)The opposite to a tautology, is a compound proposition

that’s always false –a contradiction.EG: p ¬p On the other hand, a compound proposition whose truth

value isn’t constant is called a contingency.

EG: p ¬p

Tautologies and contradictions

The easiest way to see if a compound proposition is a tautology/contradiction is to use a truth table.

TF

FT

pp

TT

p p

TF

FT

pp

FF

p p

Tautology example

Demonstrate that[¬p (p q )]q

is a tautology in two ways:1. Using a truth table – show that [¬p (p q )]q is always

true2. Using a proof (will get to this later).

Tautology by truth table

p q ¬p p q ¬p (p q ) [¬p (p q )]q

T T

T F

F T

F F


p q ¬p p q ¬p (p q ) [¬p (p q )]q

T T F

T F F

F T T

F F T


p q ¬p p q ¬p (p q ) [¬p (p q )]q

T T F T

T F F T

F T T T

F F T F


p q ¬p p q ¬p (p q ) [¬p (p q )]q

T T F T F

T F F T F

F T T T T

F F T F F


p q ¬p p q ¬p (p q ) [¬p (p q )]q

T T F T F T

T F F T F T

F T T T T T

F F T F F T

Logical Equivalences

DEF: Two compound propositions p, q are logically equivalent if their biconditional joining p q is a tautology. Logical equivalence is denoted by p q.

EG: The contrapositive of a logical implication is the reversal of the implication, while negating both components. I.e. the contrapositive of p q is ¬q ¬p . As we’ll see next: p q ¬q ¬p

Logical Equivalence of Conditional and Contrapositive

The easiest way to check for logical equivalence is to see if the truth tables of both variants have identical last columns:

p qqp

Q: why does this work given definition of ?

¬q¬pp ¬pq ¬q



TFTT

TFTF

TTFF

p qqp


¬q¬pp ¬pq ¬q



TFTT

TFTF

TTFF

p qqp


TTFF

¬q¬pp ¬p

TFTF

q ¬q



TFTT

TFTF

TTFF

p qqp


TTFF

¬q¬pp ¬p

TFTF

q

FTFT

¬q



TFTT

TFTF

TTFF

p qqp


TTFF

¬q¬pp

FFTT

¬p

TFTF

q

FTFT

¬q



TFTT

TFTF

TTFF

p qqp


TFTT

TTFF

¬q¬pp

FFTT

¬p

TFTF

q

FTFT

¬q

Logical Equivalences

A: p q by definition means that p q is a tautology. Furthermore, the biconditional is true exactly when the truth values of p and of q are identical. So if the last column of truth tables of p and of q is identical, the biconditional join of both is a tautology.

Logical Non-Equivalence of Conditional and ConverseThe converse of a logical implication is the reversal of the implication. I.e. the converse of p q

is q p.EG: The converse of “If Donald is a duck then Donald is a bird.” is “If Donald is a bird then Donald

is a duck.” As we’ll see next: p q and q p are not logically equivalent.

Logical Non-Equivalence of Conditional and Converse

p q p q q p (p q) (q p)



TTFF

TFTF



TTFF

TFTF

TFTT



TTFF

TFTF

TFTT

TTFT



TTFF

TFTF

TFTT

TTFT

TFFT

Derivational Proof Techniques

When compound propositions involve more and more atomic components, the size of the truth table for the compound propositions increases

Q1: How many rows are required to construct the truth-table of:( (q(pr )) ((sr)t) ) (qr )

Q2: How many rows are required to construct the truth-table of a proposition involving n atomic components?


A1: 32 rows, each additional variable doubles the number of rowsA2: In general, 2n rowsTherefore, as compound propositions grow in complexity, truth tables

become more and more unwieldy. Checking for tautologies/logical equivalences of complex propositions can become a chore, especially if the problem is obvious.


EG: consider the compound proposition (p p ) ((sr)t) ) (qr )

Q: Why is this a tautology?


A: Part of it is a tautology (p p ) and the disjunction of True with any other compound proposition is still True:

(p p ) ((sr)t )) (qr ) T ((sr)t )) (qr ) TDerivational techniques formalize the intuition of this example.

Tables of Logical Equivalences

• Identity lawsLike adding 0

• Domination lawsLike multiplying by 0

• Idempotent lawsDelete redundancies

• Double negation“I don’t like you, not”

• Commutativity Like “x+y = y+x”

• AssociativityLike “(x+y)+z = y+(x+z)”

• DistributivityLike “(x+y)z = xz+yz”

• De Morgan

Tables of Logical Equivalences

• Excluded middle• Negating creates opposite• Definition of implication in

terms of Not and Or

DeMorgan Identities

DeMorgan’s identities allow for simplification of negations of complex expressions• Conjunctional negation:

(p1p2…pn) (p1p2…pn)

• Disjunctional negation:(p1p2…pn) (p1p2…pn)

Tautology example (1.2.8.a) Part 2

Demonstrate that[¬p (p q )]q

is a tautology in two ways:1. Using a truth table (did above)2. Using a proof.

Tautology by proof[¬p (p q )]q


[(¬p p)(¬p q)]q Distributive



[ F (¬p q)]q ULE



[ F (¬p q)]q ULE [¬p q ]q Identity



[ F (¬p q)]q ULE [¬p q ]q Identity ¬ [¬p q ] q ULE




[¬(¬p) ¬q ] q DeMorgan





[p ¬q ] q Double Negation






p [¬q q ] Associative







p [q ¬q ] Commutative







p [q ¬q ] Commutative p T ULE







p [q ¬q ] Commutative p T ULE T Domination

Exercises• Construct the truth table for the following Propositions:• 1, (PQ) →Q• 2, (PQ) →(PQ)• 3, (P→Q) (P→Q)• 4, (PQ) (PQ)• 5, (P↔Q)(P↔Q)

• Show that the following by constructing the truth table:• 1, (P↔Q) (P→Q) (Q→P)• 2, (P→Q) PQ• 3, P(Q R) (PQ) (PR)

Exercises

• Show that (PQ)→(PQ) is a tautology.• Show that [P(PQ)]→Q is a tautology.• Determine whether the proposition [P(P → Q)]→Q is a tautology or not.• Determine whether the proposition [P(P → Q)]→Q is a tautology or not.• Determine whether the proposition [Q(P → Q)]→ Q is a tautology or

not.• Show that (P→Q)→Q is a tautology.• Show that (P→Q)→P is a tautology.• Show that (P(PQ)) and PQ are logically equivalent.

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The Foundations: Logic and Proof, Sets, and Foundations PROPOSITIONS A proposition is a declarative sentence that is either True or False, but not the