logic: conditional & biconditional · josh engwer (ttu) logic: conditional & biconditional...

Logic: Conditional & BiconditionalContemporary Math

Josh Engwer

TTU

21 July 2015

Josh Engwer (TTU) Logic: Conditional & Biconditional 21 July 2015 1 / 33

Truth Tables for the Conditional & Biconditional

Truth Table for Conditional (IF...THEN):

P Q P −→ QT T TT F FF T TF F T

Truth Table for Biconditional (IF AND ONLY IF):

P Q P←→ QT T TT F FF T FF F T


Logic Connectives (Order of Operations)

It’s important to know the ”order of operations” of logic connectives.Otherwise, statements would require too many parentheses & brackets.

DOMINANCE: CONNECTIVES:MOST DOMINANT Biconditional ←→

2nd DOMINANT Conditional −→

3rd DOMINANT Conjunction ∧Disjunction ∨

LEAST DOMINANT Negation ∼

REMARK: Since conjunction & disjunction has equal dominance, statementsinvolving several of them require parentheses & square brackets!

For example:P ∧ Q ∨ R is ambiguous! It needs to changed to one of the following:? (P ∧ Q) ∨ R? P ∧ (Q ∨ R)? WARNING: The above two statements have different truth tables!

(∼ P ∨ ∼ Q) ∧ ∼ R is equivalent to [(∼ P) ∨ (∼ Q)] ∧ (∼ R)(Q ∧ ∼ P) −→ ∼ R is equivalent to [(Q ∧ (∼ P))] −→ (∼ R)


Truth Table involving the Conditional (Example)

WEX 3-3-1: Construct a truth table for the statement: (∼ P −→ Q) −→∼ R




P Q R ∼ P ∼ P −→ Q ∼ R (∼ P −→ Q) −→∼ R




P Q R ∼ P ∼ P −→ Q ∼ R (∼ P −→ Q) −→∼ RTTTTFFFF




P Q R ∼ P ∼ P −→ Q ∼ R (∼ P −→ Q) −→∼ RT TT TT FT FF TF TF FF F




P Q R ∼ P ∼ P −→ Q ∼ R (∼ P −→ Q) −→∼ RT T TT T FT F TT F FF T TF T FF F TF F F




P Q R ∼ P ∼ P −→ Q ∼ R (∼ P −→ Q) −→∼ RT T T FT T F FT F T FT F F FF T T TF T F TF F T TF F F T




P Q R ∼ P ∼ P −→ Q ∼ R (∼ P −→ Q) −→∼ RT T T F TT T F F TT F T F TT F F F TF T T T TF T F T TF F T T FF F F T F




P Q R ∼ P ∼ P −→ Q ∼ R (∼ P −→ Q) −→∼ RT T T F T FT T F F T TT F T F T FT F F F T TF T T T T FF T F T T TF F T T F FF F F T F T




P Q R ∼ P ∼ P −→ Q ∼ R (∼ P −→ Q) −→∼ RT T T F T F FT T F F T T TT F T F T F FT F F F T T TF T T T T F FF T F T T T TF F T T F F TF F F T F T T


Truth Table involving the Biconditional (Example)

WEX 3-3-2: Construct a truth table for: ∼ (P ∧ ∼ Q)←→ P ∨ ∼ Q




P Q ∼ Q P ∧ ∼ Q ∼ (P ∧ ∼ Q) P ∨ ∼ Q ∼ (P ∧ ∼ Q)←→ P ∨ ∼ Q




P Q ∼ Q P ∧ ∼ Q ∼ (P ∧ ∼ Q) P ∨ ∼ Q ∼ (P ∧ ∼ Q)←→ P ∨ ∼ QTTFF




P Q ∼ Q P ∧ ∼ Q ∼ (P ∧ ∼ Q) P ∨ ∼ Q ∼ (P ∧ ∼ Q)←→ P ∨ ∼ QT TT FF TF F




P Q ∼ Q P ∧ ∼ Q ∼ (P ∧ ∼ Q) P ∨ ∼ Q ∼ (P ∧ ∼ Q)←→ P ∨ ∼ QT T FT F TF T FF F T




P Q ∼ Q P ∧ ∼ Q ∼ (P ∧ ∼ Q) P ∨ ∼ Q ∼ (P ∧ ∼ Q)←→ P ∨ ∼ QT T F FT F T TF T F FF F T F




P Q ∼ Q P ∧ ∼ Q ∼ (P ∧ ∼ Q) P ∨ ∼ Q ∼ (P ∧ ∼ Q)←→ P ∨ ∼ QT T F F TT F T T FF T F F TF F T F T




P Q ∼ Q P ∧ ∼ Q ∼ (P ∧ ∼ Q) P ∨ ∼ Q ∼ (P ∧ ∼ Q)←→ P ∨ ∼ QT T F F T TT F T T F TF T F F T FF F T F T T




P Q ∼ Q P ∧ ∼ Q ∼ (P ∧ ∼ Q) P ∨ ∼ Q ∼ (P ∧ ∼ Q)←→ P ∨ ∼ QT T F F T T TT F T T F T FF T F F T F FF F T F T T T


More about Conditionals....

In English, conditionals can be worded various ways:

P −→ Q”If P, then Q”

”Q if P””P only if Q”

”P is sufficient for Q””Q is necessary for P”

Definition(More about the Conditional)

Given the conditional P −→ Q,

P is sometimes known as the hypothesis (or antecedent)Q is sometimes known as the conclusion (or consequent)


Converses, Inverses, Contrapositives of Conditionals

Definition(Converses, Inverses, Contrapositives)

The converse of conditional P −→ Q is Q −→ PThe inverse of conditional P −→ Q is ∼ P −→∼ QThe contrapositive of conditional P −→ Q is ∼ Q −→∼ P

Proposition(Logical Equivalence w.r.t. Conditionals)

(a) ∼ Q −→∼ P ⇐⇒ P −→ Q

(b) ∼ P −→∼ Q ⇐⇒ Q −→ P






(a) ∼ Q −→∼ P ⇐⇒ P −→ Q

(b) ∼ P −→∼ Q ⇐⇒ Q −→ P

PROOF: (a)

P Q ∼ Q ∼ P ∼ Q −→∼ P P −→ QT T F F T TT F T F F FF T F T T TF F T T T T






(a) ∼ Q −→∼ P ⇐⇒ P −→ Q

(b) ∼ P −→∼ Q ⇐⇒ Q −→ P

PROOF: (b)

P Q ∼ P ∼ Q ∼ P −→∼ Q Q −→ PT T F F T TT F F T T TF T T F F FF F T T T T



WEX 3-3-3: Given the statement ”If roses are red, then violets are blue”:

(a) Find the converse.

(b) Find the inverse.

(c) Find the contraposition.




Let P ≡ ”Roses are red”, Q ≡ ”Violets are blue”







Let P ≡ ”Roses are red”, Q ≡ ”Violets are blue”Then, ”If roses are red, then violets are blue” ≡ P −→ Q









Q −→ P


∼ P −→∼ Q


∼ Q −→∼ P






Q −→ P ⇐⇒ ”If violets are blue, then roses are red”


∼ P −→∼ Q ⇐⇒ ”If roses are not red, then violets are not blue”


∼ Q −→∼ P ⇐⇒ ”If violets are not blue, then roses are not red”


Fin

Fin.


logic: conditional & biconditional · josh engwer (ttu) logic: conditional & biconditional...

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