SYMBOLIZATION

Statement: A sentence that makes a claim about the world.

Simple Statement: A statement that makes a single claim about the world. It is a single piece of content.

The sky is blue. Butane is a gas.

Parakeets are colorful. Logic is fun.

Simple statements are symbolized with single capital letters, i. e. each piece of content is represented by a single capital letter.

The sky is blue. – S

Butane is a gas. – B

Parakeets are colorful. – P

Logic is fun. – FOne may use any capital letter so long as the

same letter is not used to symbolize two different simple statements.

Complex (Compound) Statement: A statement that contains a logical operator or connective.

Negation Operator (~): Logical operator that takes you from a statement to its negation (its direct opposite).

Not

The sky is not blue. ~S

It’s not the case that butane is a gas. ~B

The tilde always goes in front.

Conjunction (·): Logical connective used to join two statements when asserting that both statements are true. The statements that comprise a conjunction are called conjuncts.

and but moreover still

although nevertheless however

Bush is President, and Cheney is Vice President. B · C

Although Bush is President, Gore is not Vice President. B · ~G

Disjunction (v): Logical connective used to join two statements when asserting that at least one of them is true. The statements that comprise a disjunction are called disjuncts.

(Either) . . . or unless

Paul will not write the play, unless Betty directs it.

~P v B

Although Richard or Marie is the winner, Steve and Carol will get the prize money.

(R v M) · (S · C)

Mike and Paul will return, unless Jolene doesn’t; nevertheless, Karl or Teresa will remain.

[(M · P) v ~J] · (K v T)

Material Equivalence (): Logical connective used to join two statements when asserting they always have the same truth value.

if and only if

sufficient and necessary conditionRichard’s singing or Paula’s dancing is a

sufficient and necessary condition for Mike’s acting and Laura’s not dancing.

(R v P) (M · ~L)

Henry falls in love, if and only if, Betty does too but Steve does not; moreover, Chris loses that loving feeling unless Marie finds it, although Karl will never have it.

[H (B · ~S)] · [(C v M) · ~K]The dosey dotes or the mersey dotes gimble

in the wabe, if and only if, the vorpels ning nang and ping pang; nevertheless, the wam bams ling lang and ting tang, unless tam tams sam sing or rau rau.

[(D v M) (N · P)] · [(L · T) v (S v R)]

Material Implication (): Logical connective used to join two statements when asserting the truth of one (the antecedent) is sufficient by itself for the truth of the other (the consequent).

When you symbolize, the antecedent always goes to the left of the horseshoe. The consequent always goes to the right of the horseshoe.

Antecedent Consequent

Antecedent

if . . .

given that . . .

just in case . . .

provided that . . .

. . . implies

. . . entails

. . . is a sufficient condition

Consequent

then . . .

only if . . .

…is a necessary condition

Taylor will enter the contest but Lyle won’t, if Mary’s not entering is a necessary condition for Paula’s entering.

(P ~M) (T · ~L)

Hank and Tom’s finding true love is a sufficient condition for Mary or Jolene’s losing true love; nevertheless, Barbara will marry only if Richard and Patrice get a divorce.

[(H · T) (M v J)] · [B (R · P)]

Not William but Carol will direct the play.

~W · C

Not both William and Carol will direct the play.

~(W · C)

William and Carol will not direct the play.

~W · ~C

Not either (Neither) William or (nor) Carol will direct the play.

~(W v C)

William or Carol will not direct the play.

~W v ~C