A Class-Room Introduction to Logic

May 4, 2009

Unit-XVI: Truth-functional Compound Statements

Filed under: Truth functional compound statements — Dr. Desh Raj Sirswal @ 6:21 am

A compound statement whose truth function is wholly determined by the truth values of its components. They also known as truth functional connectives. There are five types of truth-functional compound statements, given below:

1. Conjunction: A conjunction is true just in case both its part (its “conjuncts”) is true. The symbol for conjunction the dot—“·”.  .  The truth table for p AND q (also written p q) is as follows:

Conjunction

p

q

p · q

T

T

T

T

F

F

F

T

F

F

F

F

2. Disjunction: A disjunction is false just in case both its parts (its “disjuncts”) are false. The symbol for  disjunction the wedge—“v”.  The truth table for p OR q (also written p Ú q) is as follows:

 Disjunction

p

q

p Ú q

T

T

T

T

F

T

F

T

T

F

F

F

 

3. Material Implication: A material implication is false only when antecedent is true and consequent  is false. The symbol for conditionals is the horseshoe—“É”.The truth table associated (symbolized as É q) with the logical implication p implies q is as follows:

Material Implication

p

q

p É q

T

T

T

T

F

F

F

T

T

F

F

T

4. Material Equivalence: The symbol for  biconditional is the triple bar—“º”. Biconditionals are true just in case both their components have the same truth value, i.e., either are both true or are both false.The truth table for p EQ q (also written as p = q, p ↔ q, or p ≡ q) is as follows:

Material Equivalence

p

q

p ≡ q

T

T

T

T

F

F

F

T

F

F

F

T

5. Negation:

Negations are easy.  The truth value of the negation of a statement is simply the opposite of the truth value of the original statement.  The sign we will use to represent negation is “tilde”—“~”.  The truth table for NOT p (also written as ~p ) is as follows:

Negation

p

~p

F

T

T

F

Each row in the table represents a distinct distribution of truth values to the components of the compound in question—in the case, a negation.  Since there is only one (proper) component here, there are two possibilities.  (In general, where there are n distinct components, there are 2n distinct distributions of truth values to those components.)  So the table here says that for any statement “p,” if “p” is true, “~p” is false, and if “p” is false, then “~p” is true.  Let us construct truth tables for the other compounds in like fashion.

Alternative Notations

There are many variants available as symbols for the same connectives, these are used in mathematics, computer science, and programming language. Here is a list of equivalent symbols used for these connectives.

Name of Connective

Symbol used in Logic

Some alternative notations

And

(dot)

&,*, ^, &&

Or

Ú  ( vee or vel)

+ ,  |, ||

If- then or material conditional

É  (horseshoe or the hook)

Equivalence of Bioconditional

(triple bar)

Not or It is not the case

~ (tilde or curl)

—  , ¬ , !

NOTE : É  represents implication and /\ represents disjunction and Ú represents conjunction.

 

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