Logic: The Basics

Jc Beall, Shay Allen Logan

• Publisher: Routledge
• Year: 2017
• ISBN:
  9781317528609
• Edition: 2
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

Logic: The Basics is an accessible introduction to several core areas of logic. The first part of the book features a self-contained introduction to the standard topics in classical logic, such as:

· mathematical preliminaries

· propositional logic

· quantified logic (first monadic, then polyadic)

· English and standard 'symbolic translations'

· tableau procedures.

Alongside comprehensive coverage of the standard topics, this thoroughly revised second edition also introduces several philosophically important nonclassical logics, free logics, and modal logics, and gives the reader an idea of how they can take their knowledge further. With its wealth of exercises (solutions available in the encyclopedic online supplement), Logic: The Basics is a useful textbook for courses ranging from the introductory level to the early graduate level, and also as a reference for students and researchers in philosophical logic.

I

BACKGROUND IDEAS

1

CONSEQUENCES

Everybody, sooner or later, sits down to a banquet of consequences.

– Robert Louis Stevenson

‘Watch what you say,’ a parent often advises, ‘because what you say has consequences.’ In saying as much, parents are right, and doubly so. There are two senses in which what one says has consequences. One sense, not terribly relevant for present purposes (not terribly relevant for logic), is captured in the familiar dictum that actions have consequences. To say something is to do something, and doing something is an action. Actions, in turn, are events, and events, as experience tells, have consequences, namely, their causal effects. (Example: a consequence – a causal effect – of your drinking petrol is your being ill, at least other things being equal.) So, in the causal effects sense of ‘consequences’, the parents’ dictum is perfectly right, but that sense of ‘consequence’ has little to do with logic.

For present purposes, there is a more relevant sense in which what one says has consequences. What one says, at least in the declarative mode,¹ has logical consequences, namely, whatever logically follows from what one said, or whatever is logically implied by what one said. Suppose, for example, that you’re given the following information.

1. Agnes is a cat.

2. All cats are smart.

A consequence of (1) and (2), taken together, is that Agnes is smart. In other words, that Agnes is smart logically follows from (1) and (2); it is implied by (1) and (2), taken together.

1.1 RELATIONS OF SUPPORT

Logical consequence is a relation on sentences of a language, where ‘sentence’, unless otherwise indicated, is short for ‘meaningful, declarative sentence’.²

Logical consequence is one among many relations over the sentences of a language. Some of those relations might be called relations of support. For example, let A₁,…,A_n and B be arbitrary sentences of some given language – say, English. Here is one such way that sentences in a given language can support other sentences in the same language:

R1. If all of A₁,…,A_n are true, then B is probably true.

Consider, for example, the following sentences.

S1. Max took a nap on Day 1.

S2. Max took a nap on Day 2.

S3. Max took a nap on Day 3.

⋮

Sn. Max took a nap on Day n (viz., today).

Sm. Max will take a nap on Day n + 1 (viz., tomorrow).

On the surface, sentences (S1)–(Sn) support sentence (Sm) in the sense of (R1): taken together, (S1)–(Sn) make (Sm) more likely. Similarly, (3) supports (4) in the same way.

3. The sun came up every day in the past.

4. The sun will come up tomorrow.

If (3) is true, then (4) is probably true too.

The relation of support given in (R1) is important for empirical science and, in general, for rationally navigating about our world. Clarifying the (R1) notion of ‘support’ is the job of probability theory (and, relatedly, decision theory), an area beyond the range of this book.

1.2 LOGICAL CONSEQUENCE: THE BASIC RECIPE

Logical consequence, the chief topic of logic, is a stricter relation of support than that in (R1). Notice, for example, that while (4) may be very likely true if (3) is true, it is still possible, in some sense, for (3) to be true without (4) being true. After all, the sun might well explode later today.

While (R1) might indicate a strong relation of support between some sentences and another, it doesn’t capture the tightest relation of support. Logical consequence, on many standard views, is often thought to be the tightest relation of support over sentences of a language. In order for some sentence B to be a logical consequence of sentences A₁,…,A_n, the truth of the latter needs to ‘guarantee’ the truth of the former, in some suitably strong sense of ‘guarantee’.

Throughout this book, we will rely on the following (so-called semantic) account of logical consequence, where A₁,…,A_n and B are arbitrary sentences of some given language (or fragment of a language).

Definition 1 (Logical Consequence) B is a logical consequence of A₁,…,A_n if and only if there is no case in which A₁,…,A_n are all true but B is not true.

Notice that the given ‘definition’ has two parts corresponding to the ‘if and only if’ construction, namely,

• If B is a logical consequence of A₁,…,A_n, then there is no case in which A₁,…,A_n are all true but B is not true.

• If there is no case in which A₁,…,A_n are all true but B is not true, then B is a logical consequence of A₁,…,A_n.

Also notable is that the given ‘definition’ is really just a recipe. In order to get a proper definition, one needs to specify two key ingredients:

• what ‘cases’ are;

• what it is to be true in a case.

Once these ingredients are specified, one gets an account of logical consequence. For example, let A₁,…,A_n and B be declarative sentences of English. If we have a sufficiently precise notion of possibility and, in turn, think of ‘cases’ as such possibilities, we can treat ‘true in a case’ as ‘possibly true’ and get the following account of logical consequence – call it ‘necessary consequence’.

• B is a (necessary) consequence of A₁,…,A_n if and only if there is no possibility in which A₁,…,A_n are all true but B is not true. (In other words, B is a consequence of A₁,…,A_n if and only if it is impossible for each given A_i to be true without B being true.)

Presumably, this account has it that, as above, ‘Agnes is smart’ is a consequence of (1) and (2). After all, presumably, it’s not possible for (1) and (2) to be true without ‘Agnes is smart’ also being true. On the other hand, (4) is not a necessary consequence of (3), since, presumably, it is possible for (3) to be true without (4) being true.

Of course, taking ‘cases’ to be ‘possibilities’ requires some spec- ification of what is possible, or at least some class of ‘relevant possibilities’. The answer is not always straightforward. Is it possible to travel faster than the speed of light? Well, it’s not physically possible (i.e., the physical laws prohibit it), but one might acknowledge a broader sense of ‘possibility’ in which such travel is possible – for example, coherent or imaginable or the like. If one restricts one’s ‘cases’ to only physical possibilities, one gets a different account of logical consequence from an account that admits of possibilities that go beyond the physical laws.

In subsequent chapters, we will be exploring different logical theories of our language (or fragments of our language). A logical theory of our language (or a fragment thereof) is a theory that specifies a relation that models (in a sense to be made more precise) the logical consequence relation over that language (or fragment). Some fragments of our language seem to call for some types of ‘cases’, while other fragments call for other (or additional) types. Subsequent chapters will clarify this point.

1.3 VALID ARGUMENTS AND TRUTH

In general, theses require arguments. Consider the thesis that there are feline gods. Is the thesis true? An argument is required. Why think that there are feline gods? We need to examine the argument – the reasons that purport to ‘support’ the given thesis.

Arguments, for our purposes, comprise premises and a con- clusion. The latter item is the thesis in question; the former purport to ‘support’ the conclusion. Arguments may be evaluated according to any relation of support (over sentences). An argument might be ‘good’ relative to some relation of support, but not good by another. For example, the argument from (3) to (4) is a good argument when assessed along the lines of (R1); however, it is not good when assessed in terms of (say) necessary consequence, since, as noted above, (4) is not a necessary consequence of (3).

In some areas of rational inquiry, empirical observation is often suffi...