I
BACKGROUND IDEAS
1
CONSEQUENCES
Everybody, sooner or later, sits down to a banquet of consequences.
â Robert Louis Stevenson
âWatch what you say,â a parent often advises, âbecause what you say has consequences.â 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 â a causal effect â of your drinking petrol is your being ill, at least other things being equal.) So, in the causal effects sense of âconsequencesâ, the parentsâ dictum is perfectly right, but that sense of âconsequenceâ 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 youâre 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 âsentenceâ, unless otherwise indicated, is short for âmeaningful, declarative sentenceâ.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,âŠ,An and B be arbitrary sentences of some given language â 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,âŠ,An 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.
âź
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)â(Sn) support sentence (Sm) in the sense of (R1): taken together, (S1)â(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 âsupportâ 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 doesnât 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,âŠ,An, the truth of the latter needs to âguaranteeâ the truth of the former, in some suitably strong sense of âguaranteeâ.
Throughout this book, we will rely on the following (so-called semantic) account of logical consequence, where A1,âŠ,An 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,âŠ,An if and only if there is no case in which A1,âŠ,An are all true but B is not true.
Notice that the given âdefinitionâ has two parts corresponding to the âif and only ifâ construction, namely,
âą If B is a logical consequence of A1,âŠ,An, then there is no case in which A1,âŠ,An are all true but B is not true.
âą If there is no case in which A1,âŠ,An are all true but B is not true, then B is a logical consequence of A1,âŠ,An.
Also notable is that the given âdefinitionâ is really just a recipe. In order to get a proper definition, one needs to specify two key ingredients:
âą what âcasesâ are;
âą 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,âŠ,An and B be declarative sentences of English. If we have a sufficiently precise notion of possibility and, in turn, think of âcasesâ as such possibilities, we can treat âtrue in a caseâ as âpossibly trueâ and get the following account of logical consequence â call it ânecessary consequenceâ.
âą B is a (necessary) consequence of A1,âŠ,An if and only if there is no possibility in which A1,âŠ,An are all true but B is not true. (In other words, B is a consequence of A1,âŠ,An if and only if it is impossible for each given Ai to be true without B being true.)
Presumably, this account has it that, as above, âAgnes is smartâ is a consequence of (1) and (2). After all, presumably, itâs not possible for (1) and (2) to be true without âAgnes is smartâ 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 âcasesâ to be âpossibilitiesâ requires some spec- ification of what is possible, or at least some class of ârelevant possibilitiesâ. The answer is not always straightforward. Is it possible to travel faster than the speed of light? Well, itâs not physically possible (i.e., the physical laws prohibit it), but one might acknowledge a broader sense of âpossibilityâ in which such travel is possible â for example, coherent or imaginable or the like. If one restricts oneâs âcasesâ 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 âcasesâ, 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 â the reasons that purport to âsupportâ 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 âsupportâ the conclusion. Arguments may be evaluated according to any relation of support (over sentences). An argument might be âgoodâ 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...