Symbolic Logic

Symbolic LogicChapter 8

Classical or Traditional logic deals primarily (but not exclusively) with the relation of terms in an argument.

Whether the reasoning is valid depends on the proper arrangement of these terms in an argument.

The relation of classes of things are central concern.

Classic example of a simple syllogism

All men are mortalSocrates is a manTherefore, Socrates is mortal

Modern logicModern Logic begins by first identifying the fundamental logical connectives on which deductive arguments depends. Using these connectives, a general account of such arguments is given, and methods for testing the validity of arguments are developed.

These newer logical languages are often called "symbolic logic," since they employ special symbols to represent clearly even highly complex logical relationships.

There are five connectives: negation, conjunction, disjunction, conditional, and biconditional. In the notation of symbolic logic, these connectives are represented byoperators.

StatementWe must distinguish Simple statements from Compound statements to understand the symbolic representation used in propositional logic.

Simplestatement Does not contain another statement as a component.

Contains a subject and a predicate.

e.g.Charlie is neat S P James Joyce wrote Ulysses. S P

CompoundstatementIt contains at least one simple statement as a component along with aconnective.e.g.Either Charlie is neat or Charlie is sweet.1 2Paris is the capital of France and Rome is 1 2the capital of Italy.

Compound statements can be formed by inserting the word NOT, or joining two or more statements with connective words such as AND, OR, IFTHEN, ONLY IF, IF AND ONLY IF. (Will be discussed on the later part)

Logical operators:Negation (~)The ~ signifies logical negation; it simply reverses the truth value of any statement (simple or compound) in front of which it appears. If the original is true, the ~ statement is false, and if the original is false, the ~ statement is true. Thus, its meaning can be represented by the truth-table:

Negation (~) truth tableP~PTFFT

The tilde ~ symbol is used to translate any negated simple or compound statements. Simple :Rolex does not make computers.It is false that Rolex makes computers It is not the case that Rolex make computers.Can be presented as : ~ ( R)

Compound:

It is not the case that Rolex makes computer nor 1Honda makes computer.2

Presented as: ~ (R H)

De Morgans lawThe rules allow the expression of conjunctions and disjunctions purely interms of each other via negation.

~ (R H)

~ (R) v ~ (H)

More examples: ~ (P v Q)

~ P ~ Q

~ [ (R H) v ~ A] ~[ ( R H ) A] ~ (R v ~ A) A

~ [~(R v H) v A]

(R v H) ~ A

As these example shows, the tilde is always placed in front or before the proposition it negate.

All of the other operators are placed between two propositions. Also unlike other propositions, the tilde cannot be used to connect two propositions.

Argument Forms and refutation by Logical AnalogyThis method of refutation by logical analogy points the way to an excellent general technique for testing arguments. To prove the invalidity of an argument, it suffices to formulate another argument that:

(1) Has exactly the same form as the first. (2) Has true premises and false conclusions.

This informal account of validity must now be made more precise. To do this, we introduce the concept of an argument form. Consider the following two arguments:

Modus ponens: affirms an antecedent.

Valid argumentIf it rained last night, then the ground is wet If P then Q It rained last night__________ P_______Therefore, the ground is wet Q

VALID: affirms an antecedent.If P is true, then Q is trueP is trueTherefore, Q is true

Fallacy of affirming the consequent

Not validIf it rained last night, then the ground is wet If P then Q The ground is wet_______ Q_________Therefore, It rained last night P

Not valid: affirms the consequent

If P then QQTherefore, P is true

Nothing follows.

-End-Thanks.