# Logic: The Laws of Truth

NICHOLAS J. J. SMITH
Pages: 576
https://www.jstor.org/stable/j.ctt7sr96

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. PREFACE
(pp. xi-xiv)
4. ACKNOWLEDGMENTS
(pp. xv-xvi)
5. PART I Propositional Logic
• 1 Propositions and Arguments
(pp. 3-31)

Somebody who wants to do a good job of measuring up a room for purposes of cutting and laying carpet needs to know some basic mathematics—but mathematics is not the science of room measuring or carpet cutting. In mathematics one talks about angles, lengths, areas, and so on, and one discusses the laws governing them: if this length is smaller than that one, then that angle must be bigger than this one, and so on. Walls and carpets are things that have lengths and areas, so knowing the general laws governing the latter is helpful when it comes to...

• 2 The Language of Propositional Logic
(pp. 32-48)

In this chapter we introduce a symbolic language PL (the language of Propositional Logic). Why do we need it? Why can’t we continue to work in English (or some other natural language), as we did in the previous chapter when we looked at connectives? The main reason is that we are primarily interested in propositions. As noted in §1.2.2, English sentences provide a means of expressing propositions: we can make claims about the world by uttering sentences in contexts. However, from the point of view of someone interested in the propositions themselves—in particular, in their structure and the role...

• 3 Semantics of Propositional Logic
(pp. 49-62)

When we first introduced the notion of a connective in §1.6, we said that connectives are of interest to us because of their relationship to truth and falsity. More specifically, we said that we shall focus on a particular kind of relationship to truth and falsity: the kind where the truth or falsity of the compound proposition is determined by the truth or falsity of its component propositions (in a particular way characteristic of the given connective). In other words, our focus is ontruth-functionalconnectives. In our discussion in the previous chapter of the connectives of PL, we described...

• 4 Uses of Truth Tables
(pp. 63-78)

According to the intuitive conception laid out in §1.4, an argument is valid if it is NTP by virtue of its form: that is, if the structure of the argument guarantees that it is impossible for the premises to be true and the conclusion false. Let us focus just on the NTP part of the intuitive notion of validity for a moment and use it to motivate an analysis of validity in terms of truth tables. (We shall return to the issue of form in Chapter 5, where we shall see that this analysis also captures the “by virtue of...

• 5 Logical Form
(pp. 79-96)

In §1.4 we presented the intuitive idea of validity, which comprises two aspects: (i) NTP (it is not possible for the premises to be true and the conclusion false), (ii) by virtue of form. In §4.1, we gave a precise definition of validity (for arguments in PL) in terms of truth tables: an argument is valid iff, in the joint truth table for the premises and conclusion, there is no row in which the premises are true and the conclusion false. As we discussed in §4.1, the precise definition captures the first part of the intuitive idea of validity (the...

• 6 Connectives: Translation and Adequacy
(pp. 97-133)

In this chapter we examine two topics concerning connectives. First, in §§6.2–6.5, we take a closer look at issues surrounding translation from English into PL. More specifically, for each connective in PL, we have mentioned one or more words of English that typically express that connective—but we have noted that the correlations are not perfect: for example, “and” does not always express conjunction, and conversely, conjunction is often expressed using words other than “and.” In these sections we engage in a more detailed discussion of the relationships between connectives of PL and expressions in English. Before discussing particular...

• 7 Trees for Propositional Logic
(pp. 134-160)

Using truth tables, we can frame a precise definition of validity (an argumentα₁, …,αn/∴βis valid iff there is no row in their joint truth table in whichα₁, …,αnare true andβis false), and we can also test whether a given argument is valid (we write out its truth table and check whether there is any such row). Similar remarks apply to the other central logical notions: using truth tables, we can both give precise definitions of these notions and test for their presence (a formulaαis a tautology or logical truth iff...

6. PART II Predicate Logic
• 8 The Language of Monadic Predicate Logic
(pp. 163-188)

One of our aims is to come up with a method for determining whether any given argument is valid. In Part I we made a good start on this, but there are arguments that the account presented there cannot handle, because the propositions making up those arguments cannot be represented adequately in PL. In Part II of this book we therefore extend our logical language. We do this in three phases:

1. monadic predicate logic (Chapters 8–10),

2. general predicate logic (Chapter 12),

3. general predicate logic with identity (Chapter 13).

Within each phase we follow the same three-step process:

1. We introduce...

• 9 Semantics of Monadic Predicate Logic
(pp. 189-210)

In this chapter we turn to the task of developing a semantics for our new language MPL. As mentioned at the beginning of Chapter 8, what our semantics needs to do is (i) give a precise account of what a “possible way of making propositions true or false” is, and (ii) tell us how the truth value of each proposition of MPL is determined in each such scenario. Once these tasks are done, we can then formulate precise accounts of our key logical notions, such as validity (in every possible scenario in which the premises are true, the conclusion is...

• 10 Trees for Monadic Predicate Logic
(pp. 211-241)

As discussed in §9.5, we now have precise analyses of the notions of satisfiability, logical truth, validity, and so on (for propositions of MPL, arguments composed of such propositions, and the like)—but we do not yet have a systematic method of finding out the answers to questions framed in terms of these concepts, such as “Is this argument valid?” or “Is this proposition a logical truth?” In this chapter we extend the system of tree proofs (Chapter 7) to give us such a method.

The basic idea behind the tree method is just the same as it was for...

• 11 Models, Propositions, and Ways the World Could Be
(pp. 242-263)

In this chapter we pause the development of the machinery of predicate logic to reflect on the significance of the machinery now in hand. In particular, we wish to investigate how the logical apparatus relates to the guiding ideas set out in Chapter 1:

Logic is the science of truth.

Our primary objects of study in logic are those things that can be true or false: propositions.

A proposition is a claim about how things are—it represents the world as being some way; it is true if the world is that way, and otherwise it is false.

Among our...

• 12 General Predicate Logic
(pp. 264-297)

We said in §8.1.1 that our development of predicate logic would proceed as follows: we start with the simplest kind of basic proposition and distinguish its parts—a name and a predicate; we then see how far we can get representing further propositions using the connectives from propositional logic plus names and predicates; we find that we need more resources—quantifiers and variables; we then see how far we can get representing further propositions using connectives, names, predicates, quantifiers, and variables; we eventually find that we need even more resources, and so on. We come now to the point where,...

• 13 Identity
(pp. 298-354)

We said in §8.1.1 that our development of predicate logic would proceed as follows. We start with the simplest kind of basic proposition and distinguish its parts—a name and a predicate; we then see how far we can go representing further propositions using the connectives from propositional logic plus names and predicates. We find that we need more resources—quantifiers and variables; we then see how far we can go representing further propositions using connectives, names, predicates, quantifiers, and variables. We eventually find that we need even more resources—many-place predicates; we then see how far we can go...

7. PART III Foundations and Variations
• 14 Metatheory
(pp. 357-384)

At the end of §1.4, we said that one of our goals was to find a method of assessing arguments for validity that is both:

1. foolproof: it can be followed in a straightforward, routine way, without recourse to intuition or imagination—and it always gives the right answer; and

2. general: it can be applied to any argument.

In this chapter we consider the extent to which this goal has been achieved. It turns out that there is an essential tension between the desiderata: generally speaking, the more general a system of logic is—the greater its expressive power—the less...

• 15 Other Methods of Proof
(pp. 385-437)

In §9.5 we contrasted precise analyses of logical properties (e.g., validity) with methods for showing that a given object possesses or does not possess one of these properties (e.g., that a given argument is valid or is not valid). In the case of propositional logic, we looked at two methods of proof: truth tables and trees. In the case of predicate logic, we have so far examined only one method: trees. But there are many other proof methods that have been developed, and in this chapter we look at the three most commonly used among them: axiomatic proof, natural deduction,...

• 16 Set Theory
(pp. 438-466)

This chapter—more in the nature of an appendix—explains basic concepts from set theory, some of which have been employed earlier in this book; it is not a full introduction to the field of set theory.

Asetis a collection of objects. These objects are said to bemembersorelementsof the set, and the set is said tocontainthese objects.

If we are in a position to name all elements of a set, we can name the set itself by putting braces (“{” and “}”) around them. For example, we denote the set containing the...

8. NOTES
(pp. 467-508)
9. REFERENCES
(pp. 509-514)
10. INDEX
(pp. 515-528)