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Understanding Biconditional and Exclusive Or Statements in Logic, Schemes and Mind Maps of Logic

University of Alaska SystemLogic

The concepts of biconditional and exclusive or statements in logic, including their definitions and truth tables. It also provides exercises to prove the equivalence of certain logical statements. Misusing conditional statements and disjunctions are common errors in logic, and this document clarifies the differences between these logical operators.

Typology: Schemes and Mind Maps

2021/2022

Uploaded on 09/27/2022

mortimer
mortimer 🇺🇸

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Part%2%Module%2%Extension:%The%Biconditional%%and%the%Exclusive%Or%
!
Two!common!sources!of!error!in!logic!involve!misusing!conditional!statements!and!
misuing!disjunctions.!
!
A!typical!misuse!of!conditional!statements!is!confusing!a!conditional!with!its!
converse!or!its!inverse.!
!
A!typical!misuse!of!disjunctions!is!failure!to!realize!that!“or”!in!logic!is!inclusive.!
!
Anothrer!way!!of!stating!it!is!to!say!that!a!typical!error!in!logic!is!confusing!a!
conditional!statement!with!a!biconditional*statement,!and!a!second!typical!error!is!
confusing!a!disjunction!with!an!exclusive*disjunction.!
!
Biconditional*statements*
*
A!biconditional!statement!is!a!statement!of!the!form!“p,#if#and#only#if#q.”!
!
This!is!denoted!p!↔!q,!and!is!sometimes!abbreviated!“p#iff#q.”!
!
Definition:#A#biconditional#statement#is#true,#only#when#the#two#terms#have#the#same#
value.#
!
*
*
Exclusive*disjunctions%
%
An!exclusive*disjunction,!more!simply!called!an!exlcusive*or,!!is!a!statement!of!the!
form!“p#or#q#(but#not#both).”!
!
This!is!denoted!p!⊕!q,!and!is!sometimes!abbreviated!“p#xor#q.”!
!
Definition:!An#exclusive#or#statement#is#true,#only#when#exactly#one#of#the#two#terms#is#
true.#
!
This!truth!table!illustrates!the!definitions!of!the!biconditional!and!exclusive#or!
propositions.!
!
p#
q#
p!↔!q!
p!⊕!q!
T!
T!
T!
T!
F!
F!
F!
T!
F!
F!
F!
T!
!
!
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Part 2 Module 2 Extension: The Biconditional and the Exclusive Or Two common sources of error in logic involve misusing conditional statements and misuing disjunctions. A typical misuse of conditional statements is confusing a conditional with its converse or its inverse. A typical misuse of disjunctions is failure to realize that “or” in logic is inclusive. Anothrer way of stating it is to say that a typical error in logic is confusing a conditional statement with a biconditional statement, and a second typical error is confusing a disjunction with an exclusive disjunction. Biconditional statements A biconditional statement is a statement of the form “ p, if and only if q .” This is denoted p ↔ q , and is sometimes abbreviated “ p iff q .” Definition : A biconditional statement is true, only when the two terms have the same value. Exclusive disjunctions An exclusive disjunction , more simply called an exlcusive or , is a statement of the form “ p or q (but not both) .” This is denoted p ⊕ q , and is sometimes abbreviated “ p xor q .” Definition: An exclusive or statement is true, only when exactly one of the two terms is true. This truth table illustrates the definitions of the biconditional and exclusive or propositions. p q (^) p ↔ q p ⊕ q T T T F T F F T F T F T F F T F

Exercises 1 -­‐3: Use a truth table to prove each of the following:

!

p ↔ q ≡ ( p → q )∧ ( q → p )

p ↔ q ≡ ( p → q )∧ (~ p →~ q )

~ ( p ↔ q ) ≡ p ⊕ q

  1. Let p be the statement “Thomasville is the capitol of Georgia.” Let q be the statement “Sopchoppy is the capitol of Florida.” Determine the truth values of: a. (^) !p → q b. (^)! ¬ p → q c. (^) !p ↔ q d. (^)!! ~ p ↔ q e. (^) !p ∨ q f. (^)!! ~ p ∨ ~ q g. (^) !p ⊕ q h. !! ~ p ⊕ ~ q