University of the Cordilleras

Introduction to LogicAtty. Nestor Mondoc

Feb 21, 2015Symbolic Logic

Nagulman, Harmony T.

Symbolic Logic

Two bodies of logical theory:

1. Classical or Traditional logic (chapter 5 through 7)Fundamental elements were terms and the relation of classes of things is central.Traditional logic is the system of logic originally formulated by Aristotle, the Greek philosopher, in the fourth century BC. Traditional logic involves mostly the study of the classical syllogism.Traditional logic has also been called term logic, since it deals primarily (but not exclusively) with the relation of terms in an argument (in this case, the terms man, mortal, and Socrates). Whether the reasoning is valid depends on the proper arrangement of these terms in an argument.[footnoteRef:1] [1: http://vereloqui.blogspot.com/2012/12/the-difference-between-traditional-and.html]

Here is a classic example of a simple syllogism, which we will use shortly as a way to see how the two systems of traditional and modern logic are different:

All men are mortalSocrates is a manTherefore, Socrates is mortal[footnoteRef:2] [2: 14th Edition ofIntroduction to Logic by Copi, Cohen & McMahon]

2. Modern logicAlthough traditional categorical logic can be used to represent and assess many of our most common patterns of reasoning, modern logicians have developed much more comprehensive and powerful systems for expressing rational thought. These newer logical languages are often called "symbolic logic," since they employ special symbols to represent clearly even highly complex logical relationships.[footnoteRef:3] [3: http://www.philosophypages.com/lg/e10a.htm]

Modern 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.There are five connectives: negation, conjunction, disjunction, conditional, and biconditional. In the notation of symbolic logic, these connectives are represented byoperators.With symbols we can perform some logical operations almost mechanically, with eye, which might otherwise demand great effort. It may seem paradoxical, but a symbolic language therefore helps us to accomplish some intellectual tasks without having to think too much.The system of modern logic is in some ways less elegant than analytical syllogistic, but it is more powerful. Using the approach taken by modern logic with its more versatile symbolic language, we can pursue the aims of deductive analysis directly and permit more efficient achievement of the central aim of deductive logic: discriminating between valid and invalid argument.[footnoteRef:4] [4: Copi, Cohen chap 8]

StatementTo understand the symbolic representation used in propositional logic, we must distinguish Simple statements from Compound statements.

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.[footnoteRef:5] [5: Patrick Hurley]

S P Compoundstatement It 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 the capital of Italy.12

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:a. Negation (~)Truth function:p ~p

T F

F T

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 below.[footnoteRef:6] [6: http://www.philosophypages.com/lg/e10a.htm]

The English expression "It is not the case that . . ." serves the same function, though of course we have many other methods of negating an assertion in ordinary languagesometimes the single word "not" embedded in a sentence is enough to do the job.

The tilde ~ symbol is used to translate any negated simple or compound statements.

Rolex does not make computers~R

It is not the case that Rolex make computers~R

It is false that Rolex makes computers~R

Jamal and Derek will not both be elected~ (J D)

It is not the case that Rolex makes computer nor 1 Honda makes computer.~ (R H)2Morgans lawThe rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation.

Examples:

~ (R H) ~(R) v ~ (H)

~ [~(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.[footnoteRef:7] [7: Patrick Hurley and Copi, Cohen chap 8]

Argument Forms and refutation by Logical Analogy

This 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 and; (2) Has true premises and false conclusions.

This method is based on the fact that validity and invalidity are purely formal characteristics of arguments, which is to say that any two arguments that have exactly the same form are either both valid or both invalid, regardless of any differences in the subject matter with which they are concerned[footnoteRef:8] [8: Copi, Cohen chap 8]

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 argument

If it rained last nightthen the ground is wetIf P then QIt rained last night____________________ P_________Therefore, the ground is wet therefore Q

Fallacy of affirms the consequent Not valid

If it rained last night, then the ground is wetIf P then QThe ground is wet_______________________________Q_________Therefore, It rained last nighttherefore P