Philosophy 160: Formal Logic                                                                                                             Winter 2002

LESSON 03: TRUTH TABLES AND THE SEMANTICS OF SENTENTIAL LOGIC

PURPOSE

In Lesson 1, we began our study of the semantics of SL; in this lesson we shall complete it. OBJECTIVES After completing this lesson, you should:

1. Be able to write out the truth-tables for sentences of SL, and to determine the truth value of a sentence on a given truth-value assignment.

2. Be able to determine if a sentence is truth-functionally true, truth-functionally false, or truth-functionally indeterminate, and whether or not two sentences are truth-functionally equivalent.

3. Be able to determine if a set of sentences is truth-functionally consistent, whether or not a given set of sentences truth-functionally entails a sentence, and whether or not an SL argument is truth-functionally valid.
 
 

 READING ASSIGNMENT The Logic Book, pages 67-101 COMMENTARY

Commentary on The Logic Book, pages 67-75:

The semantics of SL is concerned with the relations between the symbols of the language and the world. The key notion here is that of truth. A sentence of English, for example, is true if and only if it bears a certain relation to the world. In the case of natural languages, the study of semantics crucially involves the study of meaning. In the case of SL, we can get by with focussing on truth, because, in essence, we are concerned only with the meaning of the truth-functional connectives, and their meanings are given by their characteristic truth-tables (see Lesson 1). Consider the case of 'and', which we symbolize with '&'. The idea is that you have grasped the meaning of 'and' once you realize that the conjunction of two sentences (i.e., the sentence formed by placing 'and' between them) is true if and only if they themselves are both true. And this is summed up in the characteristic truth table for '&' (see page 57 of the text): note that ‘P & Q’ is false unless P is true and Q is true.

Recall that we use only truth-functional connectives in SL (see, e.g., pages 43-44 of the text). This means that the truth-value of a sentence (i.e., whether it is true or false) of SL is determined by the truth-values of the atomic sentences (the sentence letters) it contains. A truth-value assignment assigns truth-values (i.e., T's or F's) to all the sentence letters. Thus a truth-value assignment determines the truth-values of all the sentences of SL.

A truth-table for a sentence gives its truth-value on each truth-value assignment. Consider the sentence:

(A & B) Ú C What is its truth-value on a truth-value assignment that assigns T to 'A', F to 'B' and F to 'C'?

First we must isolate the main connective, in this case 'Ú '. So the sentence is of the form:

P Ú Q Thus (by consulting the characteristic truth-table for 'Ú ') we know that our sentence is true (i.e. assigned T) under our truth-value assignment if and only if: '(A & B)' is assigned T by the truth-value assignment, or 'C' is assigned T by the truth-value assignment, or both. 'C' is assigned F by the truth-value assignment, thus the truth-value of '(A & B) Ú C ' hangs upon the truth-value of '(A & B)': if the latter is assigned F, then '(A & B) ÚC ' is assigned F; if the latter is assigned T, then '(A & B) Ú C ' is assigned T. The main connective of '(A & B)' is '&', so '(A & B)' is of the form: P & Q Thus (by consulting the characteristic truth-table for '&') we know that '(A & B)' is assigned T under our truth-value assignment if and only if: 'A' is assigned T and 'B' is assigned T. But 'B' is assigned F, thus '(A & B)' is assigned F, and hence '(A & B) Ú C' is assigned F.

Spelling matters out makes the process of determining truth-values under assigments appear very laborious. However, it becomes quite quick and easy with practice. In our example, write things out as follows.

We are given the truth-values of the sentence letters, so write those in first:
 
A B C (A & B) Ú C
T F F T ? F ? F

 
 
 

We cannot determine what to write under the 'Ú ' until we have decided what to write under the '&', so we do that next:
 
A B C (A & B) Ú C
T F F T T F ? F

Finally, we determine the truth-value of the 'V' (the main connective):
 
A B C (A & B) Ú C
T F F T F F F F

In determining, in this fashion, the truth-value of '(A & B) ÚC ' under that truth-value assignment which assigns T to 'A', F to 'B' and F to 'C', we have produced one row of the truth-table for '(A & B) ÚC ' (what the text refers to as a 'shortened truth-table'). [Note: strictly, one row of a truth table corresponds to an infinite number if truth-value assignments, which are alike with respect to the atomic sentences in question.]

The complete truth-table shows the truth-value of '(A & B) ÚC ' under all possible truth-value assignments. Here it is:
 
  A B C (A & B) Ú C
  T T T T T T T T
  T T F T T T T F
  T F T T F F T T
  T F F T F F F F
  F T T F F T T T
  F T F F F T F F
  F F T F F F T T
  F F F F F F F F

Each row of the truth-table represents a truth-value assignment – a possible combination of truth values. For example, the top row represents the truth-value assignment that assign T to 'A', T to 'B' and T to 'C'). Every truth-value assignment is represented: given any truth-value assignment, it either assigns T to 'A', T to 'B' and T to 'C' (Row 1) or T to 'A', T to 'B' and F to 'C' (Row 2) or . . . or F to 'A', F to 'B' and F to 'C' (Row 8).
 
 

PRACTICE QUESTIONS

Pages 74-75, nos. 1a,c; 2a,c,e,g,m; 3a,c,e,g,i

Hints:

You should construct as many truth-tables as it takes to ensure that you find the exercise almost trivially easy. If you require more practice, do nos. 4 and 5 on page 75 of the text.

 Commentary on The Logic Book, pages 75-101:

Now that you understand truth-value assignments and truth-tables, the rest of this lesson employs them in defining the various other logical concepts. The major problem with our original definitions of the logical properties and relations is that they all contain the concept of ‘possibility’. This is an exceptionally vague concept. In sentential logic, we use the concept of a truth-value assignment to play the role of a logical possibility, so that each row of a truth table is a logical possibility, and a truth table represents all logical possibilities for a given sentence or set of sentences.

Now we give more precise definitions, relative to sentential logic, of the basic logical properties and relations:

A sentence P of SL is truth-functionally true if and only if (henceforth, 'if and only if' will be abbreviated 'iff') P is true on every truth-value assignment (that is, iff the relevant truth-table has a column of T's under P).

A sentence P of SL is truth-functionally false iff P is false on every truth-value assignment (that is, iff the relevant truth-table has a column of F's under P).

A sentence P of SL is truth-functionally indeterminate iff P is neither truth-functionally true nor truth-functionally false (that is, iff the relevant truth-table has a column containing both T's and F's under P).

Sentences P and Q of SL are truth-functionally equivalent iff there is no truth-value assignment on which P and Q have different truth-values (that is, iff, in the relevant truth-table, the columns under P and under Q are identical).

 The next three properties involve sets of sentences. Roughly speaking, you can think of a set as a collection. ‘{’ and ‘}’ are used to denote sets; so, for example, the set containing 'A', 'B ÚD' and 'A & C' is written: {A, B ÚD, A & C}. Just as we use script letters as metalinguistic variables ranging over all the sentences of SL, so we use capital Greek letters as variables ranging over the sets of sentences of SL. Typically we shall only need to use one such letter, the capital gamma:

                    G

One's beliefs are open to criticism if they are inconsistent: that is, if they cannot possibly all be true. In SL, we come closest to capturing this idea with the notion of truth-functional consistency:
 

A set of sentences G of SL is truth-functionally consistent iff there is at least one truth-value assignment on which all the members of G are true (i.e. in a truth-table representing all of the members of G, there is at least one row in which every member of G has a T entered under it).

A set of sentences G of SL is truth-functionally inconsistent iff there is no truth-value assignment on which all the members of G are true. G is truth-functionally inconsistent iff, in the relevant truth-table, there is no row in which every member of G has a T entered under it.

 The next concept, that of entailment, is of crucial importance because of its relation to deductive validity. Recall that in a deductively valid argument it is impossible for all the premises to be true and the conclusion false. We say that a set of sentences deductively entails a single sentence iff it is impossible for every member of the set to be true, and the single entailed sentence false. Thus an argument is deductively valid iff the set consisting of its premises deductively entails its conclusion. Entailment is the purely logical concept behind validity. In sentential logic, there is a precise truth-functional relation that corresponds to this notion. A set of sentences G of SL truth-functionally entails a sentence P of SL iff there is no truth value assignment on which every member of G is true and P is false (that is, iff, in the relevant truth-table, there is no row in which every member of G has a T entered under it, and P has an F entered under it). Truth-functional entailment is represented by the double turnstile: |=. Thus G truth-functionally entails P is written: G |= P Finally, there is the validity of an argument in SL: An argument of SL is truth-functionally valid iff there is no truth value assignment on which all of the premisses are true and the conclusion false (that is, iff, in the relevant truth-table, there is no row in which every premiss has a T entered under it, and the conclusion has an F entered under it).
 
  Although there is a fair amount of text in this chapter, it is primarily devoted to one thing: using truth tables to show the truth-functional properties and relations. In "Notes and Examples" for this lesson, we will look at cases of each truth-functional concept.
 
 

PRACTICE EXERCISES

Pages 79-80, nos. 1a,c,e,g,i,k; 2a,c,e; 3a,c,e

Pages 83-84, nos. 1a,c,g,e,i; 2a,c,e

Pages 86-87, nos. 1a,c,e,g,i; 2a,c,e

Pages 93-96, nos. 1a,c,e,g,i; 2a,c,e
 
 

 CREDIT EXERCISES Pages 79-80, nos. 1b,j; 2b; 3f

Pages 83-84, nos. 1f; 2f

Pages 86-87, nos. 1b; 2f

Pages 93-96, nos. 1d; 2b

 Please submit your answers by Friday February 22nd. You may turn them in by hand to Lucas 599, by fax to (314) 516 4610, or by email to ablack@umsl.edu.