Philosophy 160 (002): Formal Logic Winter 2002

SECTION I - SENTENTIAL LOGIC

LESSON 01

INTRODUCTION AND THE TRUTH-FUNCTIONAL CONNECTIVES

PURPOSE

This lesson will acquaint you with the basic notions of logic, and introduce you to the focus of Section I: Sentential Logic. OBJECTIVES After completing this lesson, you should: 1. Be able to define the following terms: argument; logical truth; logical falsity; logical indeterminacy; logical consistency; logical equivalence; deductive validity; truth functional connective; conjunction; disjunction; negation; material conditional; material biconditional.

2. Understand the truth-functional connectives.

3. Be able to do simple Sentential Logic symbolizations -- i.e., represent the logical form of simple sentences of English in the language of Sentential Logic.

4. Know the characteristic truth-tables for the five truth-functional connectives.

READING ASSIGNMENT The Logic Book, pages 1-45 COMMENTARY Commentary on The Logic Book, pages 1-24:

Logic, as a subject, is concerned with a family of related notions. The main business of Chapter One is to introduce these notions. You’ll find them defined in the glossary on page 24. Of the nine concepts defined there, we will regard six as being fundamental to the subject of logic:

Logical truth, logical falsity and logical indeterminacy; logical consistency and logical equivalence; deductive validity.

The goal of good reasoning is true belief. Logic is about the relations among the truth values of sentences that can express beliefs. Such sentences, sometimes called "declarative sentences" are of the kind that must be either true or false:

Examples

‘It is raining’ is a sentence that is either true or false. It expresses a possible belief.

‘What time is it?’ is a sentence that does not have a truth value. It does not express any possible belief.

Logical truth

Definition: a sentence is logically true if and only if it is not possible for it to be false.

The truth of most sentences is contingent on circumstance. What makes ‘it is raining’ true (when it is true) is a contingent set of meteorological circumstances. But some sentences cannot be false in any circumstance.

E.g. ‘all mammals are animals’

Such a sentence is, by the above definition, logically true. A sentence that is explicitly logically true is sometimes called a tautology.

E.g. ‘all animals are animals’

Logical falsity

Definition: a sentence is logically false if and only if it is not possible for it to be true.

Although the sentence ‘Al Gore is the President’ is false, it could have been true had circumstances been different. However some sentences had to be false.

E.g. ‘two plus two equals five’

Such a sentence is logically false.

Logical indeterminacy

Definition: a sentence is logically indeterminate if and only if it is neither logically true nor logically false.

As we have seen, many sentences - although true – could have been false and vice versa. These sentences, even if true, are not logically true by definition, and even if false are not logically false. Such sentences are logically indeterminate.

Logical consistency

Definition: a set of sentences is logically consistent if and only if it is possible for all the members of that set to be true.

A consistent set of sentences is a set all of which can be true together -- or, as logicians are fond of saying: there is some possible world in which they are all true. This is not the same as saying that they are all true, merely that it is possible that this be so. If a set of sentences is inconsistent, then, it is not possible for them all to be true -- at least one of them must be false.

E.g. these three sentences, expressing possible beliefs, cannot all be true at once:

‘The Bible says I may not eat shellfish’, ‘Everything the Bible says is true’, ‘I may eat shellfish’

Thus these sentences form a logically inconsistent set. At least one of them must be false. Note that pointing out the inconsistency of a set of sentences does not tell us which of them is to be rejected as false.

Logical equivalence

Definition: a pair of sentences are logically equivalent if and only if it is not possible for one of the sentences to be true while the other sentence is false.

A pair of sentences may turn out true under exactly the same circumstances.

E.g. ‘this creature has a heart’ and ‘this creature has kidneys’ are true in all the same cases and false in all the same cases.

Thus they are logically equivalent. Note that this doesn’t mean that they "say the same thing". The two sample sentences above express quite different beliefs.

Deductive validity

Definition: an argument is deductively valid if and only if it is not possible for the premises to be true and the conclusion (at the same time) to be false.

To understand this, we need to grasp the logician’s concept of an argument.

Definition: an argument is a set of sentences one of which (the conclusion) is taken to be supported by the others (the premises).

If there is no possible circumstance in which the premises of an argument can be true while the conclusion is false, then the argument is deductively valid. This is because anyone who accepts (believes) the premises is logically committed to accepting the conclusion.

E.g. All women are intelligent
Robin is a woman
Robin is intelligent

The premises are listed above the line; the conclusion is below it. In this case, although one or other of the premises might be held to be false, anyone who accepts them must accept the conclusion. There is no way for the premises to be true while at the same time the conclusion is false.

An argument that is deductively valid and has true premises is said to be deductively sound. Nevertheless, deductive soundness is not a purely logical property, since the truth of the premises is (for the most part) not a matter of logic.

Note that deductive validity is a property of arguments; logical truth, falsity, and indeterminacy are properties of sentences; and logical consistency and equivalence are properties of pairs or sets of sentences. To ask, for example, whether a particular sentence is valid makes no sense -- it is a category mistake akin to asking whether a particular carrot is happy.

Note also the discussion of special cases of validity on pages 21-23.

Commentary on The Logic Book, pages 25-45:

Note that the definition of each of the basic notions depends on the concepts of truth (and falsity) and possibility. This notion of possibility is very vague. Successful attempts at logical evaluation depend upon giving a precise content to the notion of logical possibility. That will be our business.

Logicians evaluate sentences and sets of sentences for the main logical properties and relations by developing formal systems, which act as models of reasoning. A model abstracts from every feature of what is modeled that is irrelevant to what is being examined. So a formal system is intended to represent only the logical features of communication.

We will study two formal systems in this course. Each of these formal systems consists in three components.

(1) A formal language consisting of
(a) vocabulary (a set of symbols)
(b) grammar (a set of rules for well-formed expressions.)
(2) A semantic scheme (a way of interpreting or assigning meanings to the expressions of the language.)
(3) Derivation rules (a set of rules for manipulating well-formed expressions of the language.)

We will use these systems to evaluate sentences, sets of sentences and arguments for the main logical properties. As an example, think of deductive validity.

There are infinitely many deductively valid arguments, so we cannot hope to write them all down. And even if we could, this would be uninformative. So how are we to proceed in our study of deduction?

One can think of a deductively valid argument as a sequence of numbered lines, with exactly one sentence appearing on each line. The first few lines will typically be the premises, and any line that is not a premise follows from the premises in the sense that it cannot possibly be false if the premises are all true. We shall proceed by looking first at those features of sentences that are relevant to the logical properties and relations -- their logical forms. The logical form of a sentence is written out with the aid of symbols. By replacing each sentence in an argument with its symbolic logical form, we shall have written out the argument's logical form.

Sentential logic

The first logical system we will develop is the sentential logic. Sentential logicians recognize the importance of sentences in the evaluation of the logical properties and relations. They also notice some significant differences and relationships between natural language sentences.

Take these sentences:

(1) Rover is a dog.

(2) Rover is a dog and he likes cookies.

(3) Either Rover is a dog or he is a wombat.

(1) differs from (2) and (3) in that it is simple whereas they are compound. (2) and (3) are made up from other sentences by the help of connective expressions (we’ll call them ‘sentential connectives’.) Also, the conditions under which a sentence like (2) will be true are different from the conditions under which (3) is true.

SENTENTIAL CONNECTIVE: An expression used to make compound sentences from other sentences.

Truth-functional connectives

Definition: A sentential connective is used TRUTH-FUNCTIONALLY if and only if the compound sentence it generates has a truth-value that is dependent on the truth-values of its components.

Sentential logic recognizes two truth-values – TRUE and FALSE. It is called a bivalent (or classical) logic.

PRACTICE QUESTIONS

The most important questions are the sets from pages 43-45. The rest are to ensure that you grasp the basic concepts.

Pages 14-15, nos. 1a,c,e,g,i,j; 2a,c,e

Pages 20-21, nos. 1a,c; 2a,c,e; 3a,c; 4a,c,e,g,i

Pages 23-24, nos. 1a,c,e,g

Pages 43-45, nos. 1a,c,e,g,i,k,m; 3a,c,e,g,i,k; 4a,c,e; 5a,c,e,g,i,k,m,o,q,s

Hints:

Note that 'unless' is to be symbolized by 'Ú ' (see page 41 of the textbook).

EXERCISE SET FOR CREDIT

[No need to provide explanations even when they are asked for (e.g. p.14 ex.1, p.20 ex. 2, p.23 ex. 1)]

Pages 14-15, nos. 1b,d,f,h; 2b,d

Pages 20-21, nos. 1b; 2b,f

Pages 23-24, nos. 1b,d,f

Pages 43-45, nos. 1b,d,f,h,j,l,n; 5b,d,f,h,j,l,n,p,r,t

Please submit your answers by Friday January 25th. You may turn them in by hand to Lucas 599, by fax to (314) 516 4610, or by email to ablack@umsl.edu.