Chapter 8: Symbolic Logic• Modern Logic and Its Symbolic Language• The Symbols for Conjunction, Negation, and Disjunction• Conditional Statements and Material Implication

Modern Logic and Its Symbolic Language

Theory of Deduction

Classical/Aristotelian Logic Modern/Symbolic Logic

Its modern development began with George Boole in the 19th century.

Why Symbolic Logic?1. Ordinary everyday language is flowery

and ambiguous (because of equivocation, amphiboly, accent, vagueness, etc.)

2. Symbolic logic introduces high degree of clarity and simplicity.

3. There is also economy of space and time.

Symbolic logic begins by first identifying the fundamental logical “connectives” on which deductive argument depends.

The Symbols for Conjunction, Negation, and Disjunction

• Consider the following simple arguments:The blind prisoner has a red hat or the blind prisoner has a white hat.The blind prisoner does not have a red hat.Therefore the blind prisoner has a white hat.

AndIf Mr. Robinson is the brakeman’s next-door neighbor, then Mr. Robinson lives halfway between Detroit and Chicago.Mr. Robinson does not live halfway between Detroit and Chicago.Therefore, Mr. Robinson is not the brakeman’s next-door neighbor.

• In studying such arguments we divide all propositions into two general categories:1. Simple Propositions2. Compound propositions

Simple vs Compound• Simple Propositions– Statements which cannot be broken down

without a loss in meaning.• E.g. “Ali and Sara is a couple” (if break it down to “Ali

is a couple” and “Sara is a couple”)• But “Ali and Shah are diligent students” is not a

simple sentence because it can be broken down without a change in meaning.

• “Ali is a diligent student.” “Shah is a diligent student.”

– This is an example of a “Compound Proposition.”

Propositions and Operators• Representation in symbolic logic:• “Ali and Sara is a couple.” A• “Ali and Shah are diligent students.” A • S

A • S

1- Conjunction• There are several types of compound statements,

each requiring its own logical notation.• The first type of compound statement is the

“conjunction”• We can form the conjunction of two statements

by placing the word “and” between them.• Thus, the compound statement, “Ali and Shah are

diligent students” is a conjunction• Symbol “•” is used to represent “and” thus:• “Ali and Shah are diligent students.” A • S

Truth Value• We know that every statement has a truth

value, where the truth value of a true statement is true and the truth value of a false statement is false

• Hence, the truth value of the conjunction of two statement is determined wholly and entirely by the truth values of its two conjuncts.

• If both conjuncts are true, the conjunction is true; otherwise it is false.

Truth Table• Given any two statement, p and q, there are only

four possible sets of truth values they can have i.e.

• Other words to conjoin: “but”, “yet”, “also”, “still”, “although”, “however”, “moreover”, nevertheless”, comma, and the semi colon, etc.

p q p • qT T TT F FF T FF F F

2- Negation• The negation (contradictory or denial) of a statement is

formed by the insertion of a “no” in the original statement.• Alternatively, it can also be written as “it is false that” or

“it is not the case that”• Thy symbol “ ~ ” called a curl or a tilde, to form the

negation of a statement.• Examples: Where M symbolizes the statement “All humans

are mortal,” the various statements:– “Not all humans are mortal,” – “Some humans are not mortal,” – “It is false that all humans are mortal,” and – “ It is not the case that all humans are mortal”

• are symbolized as ~M.• The negation of any true statement is false, and the

negation of any false statement is true.

• These can be presented very simple and clearly by means of a truth table:

Disjunction• The disjunction (or alternation) of two statement is

formed by inserting the word “or” between them.• The word “or” is ambiguous, having two related but

distinguishable meanings.• E.g. “Premiums will be waived in the event of

sickness or unemployment.”• The intent here is that premiums are waived not only

for sick persons and for unemployed person, but also for person who are both.

• In this sense, “or” is called weak or inclusive.• An inclusive disjunction is true if one or the other or

both disjuncts are true; only if both disjuncts are false is their inclusive disjunction false.

• The word “or” is also used in a strong or exclusive sense, in which the meaning is not “at least one” but “at least one and at most one”.

• Where a restaurant lists “salad or dessert” on its dinner menu, “but not both” is often added.

• We interpret the inclusive disjunction of two statement as an assertion that at least one of the statements is true, and we interpret their exclusive disjunction as an assertion that at least one of the statements is true but not both are true.

• Wage “V” is used for inclusive disjunction

Punctuation• Punctuation is required if complicated statements

are to be clear.• E.g. quite different meanings attach to “The

teacher says John is a fool” when it is given different punctuations.

• Punctuation is equally necessary in mathematics.• E.g. 2 x 3 + 5• In the language of symbolic logic punctuation

marks such as (), [], {} are essential for compound statements.

• E.g. p . q v r is ambiguous.

• E.g. Jamal and Derek will not both be elected. and Jamal and Derek will both not be elected.

• The first denies the conjunction of J.D and may be symbolized as ~(J.D)

• The second says that each one of the two will not be elected, and is symbolized as ~(J).~(D)

Exercises (page#315 )• A. Using the truth-table definitions of the dot, the wedge, and the

curl, determine which of the following statements are true:

B. If A, B, and C are true statements and X, Y, and Z are false statements, which of the following are true?

B. If A, B, and C are true statements and X, Y, and Z are false statements, which of the following are true?

Assignment• Solve the whole exercises A (25

questions), and B (25 questions): Page 315-318 (14th Edition)