A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that sentences are either true or false. That is, we assume that sentences have one or other of two truth values, T or F. We will also need some sentence connectives or logical operators in order to build more complex sentences out of elementary or basic sentences. The sentence connectives correspond roughly to logical expressions found in English (and other languages, of course). Here are a few sentential connectives: Symbol Name Example Rough English Equivalent ~, Negation A It is not the case that A. A: not. & Conjunction A & B A and B. Disjunction A B A or B (or both)., Conditional A B If A, then B., Biconditional A B A if and only if B. A iff B. In the conjunction A & B, we call the two component sentences A, B conjuncts. In the disjunction A B, we call the two component sentences disjuncts. In the (material) conditional A B, A is the antecedent (of the conditional) and B is the consequent (not, please note, the consequence). As logicians, we are not concerned with what sentences are actually true or actually false. It is the job of empirical scientists (among others) to try to determine these facts. As logicians what we do is consider all possibilities of truth or falsity for some given sentence or set of sentences. Take, for example, just one sentence, A. A may be true, or A may be false. (Remember that we are consciously idealizing here. Sentences in ordinary daily usage may be vague, may have more than two truth values or perhaps lack truth value altogether. We are considering only the rather broad class of sentences that have one of two truth values, that are bivalent. There are different logical systems that try to accommodate these
other sorts of sentences.) But we can then define negation by means of the following quite natural table: A T F A F T Note that since negation changes the truth value of a sentence from T to F or from F to T, then two negation signs in a row have the effect of leaving the truth value of a sentence unchanged. In classical deductive logic two negation signs in a row can be dropped or omitted, often simplifying complex expressions. (There are non-classical logics in which this simple rule does not hold.) If we combine two statements, then we have four possible ways that the two component statements may be true or false. We can then define the sentence connectives &,, and by means of the following table of possible truth values of the component sentences (or truth table, for short) and the resulting truth value for the whole sentence: A B A & B A B A B A B T T T T T T T F F T F F F T F T T F F F F F T T The truth table for & is quite natural. The truth table for is also what one would expect. However, there may be a use of or in English in which A or B excludes one of A or B. That is, in this meaning of or, A or B is false if both A and B are true. This kind of or is called the exclusive or. What we have represented in the truth table above is the inclusive or, which is true even if both A and B are true. It turns out that it is convenient to take the inclusive or as basic and define the exclusive or later, should we need it. The truth table for may seem a bit unintuitive, but one might note that it is the same as the truth table for (A & B), which, if you think about it a while (and you should), says: If A, then B. The biconditional has the same truth table as: (A B) & (B A). We can now see that in sentential or propositional logic, if we have a complex sentence made up of sentence letters and sentence connectives, 2
then given any assignment of truth values to the sentence letters, we can find the truth value of the whole complex sentence. (Some sentences in natural language do not have this simple truth functional form. Consider, for example, the sentence: Henry believes that A. ) It is worth noting briefly that, if we were developing fully a system of sentential logic, we would introduce parentheses to distinguish sentences that would be ambiguous without them. Consider, as a simple example, the string of symbols ~A B. Is this sentence a conditional with a negated antecedent or the negation of a conditional? (There is a difference! In a world in which both A and B are true, the former is true while the latter is false.) With rules concerning parentheses in place, we might write (~A) B for the former and ~(A B) for the latter. In the first sentence, the negation sign governs only the antecedent (and has, as we say, narrow scope). In the second sentence, the negation sign governs the whole sentence and so has broad scope. Keeping track of scope can be vitally important in evaluating the correctness or cogency of arguments, as we shall see. Below is an example of a truth table for a sentence with three sentence letters. The table provides two extra columns indicating intermediate steps between the assignment of truth-values to the component letters and the resulting truth-value of the statement as a whole (for each assignment). A truth table need only contain the assignments of truth-values and the final result. A B C A B C (A B) ( C) T T T T F F T T F T T T T F T T F F T F F T T T F T T T F F F T F T T T F F T F F T F F F F T T For practice, it will be useful for you to work out truth tables for a few examples, like: (A B) (A & B), (A & B) (A B), ((A B) & A) B, ((A B) & B) A, (A B) ( B A). If you do these examples, you will notice that in some of them (the second, fourth, and fifth) the final row of the truth table will contain all Ts. Sentences that have this property are called tautologies or logical truths, 3
while their negations, which have all Fs in the final row of their truth tables, are called contradictions or logical falsehoods. Such sentences are of particular interest to logicians, who, while they are not interested in which sentences are true or false in any given particular world, certainly are interested in which sentences are true (or false) in all (possible) worlds, like the tautologies and contradictions. In addition to helping us find logical truths and falsehoods, truth tables can be of use in evaluating argument forms (and logic surely should have some connection to evaluating arguments). For our purposes, an argument is just a set of statements, some of which we designate as premises and one of which we call the conclusion. An argument is a valid deductive argument if, should its premises all be true, its conclusion is true as well. Consider, for example, the argument form (and I say form, since I will use letters instead of particular sentences): If H, then E. It is not the case that E. Therefore, it is not the case that H. (An argument is sound if, in addition to being valid, its premises are true. Suppose that we can accurately represent this argument in our logical notation in the following way: H E, E, H (where the symbol indicates the conclusion of an argument, just as the abbreviation QED often does.) Then consider the following table: H E H E E H T T T F F T F F T F F T T F T F F T T T The bottom row of this table is the only row (that is, the only assignment of truth values to the constituent sentence letters) in which both premises are true. In this row, the conclusion is true as well. The argument (form) we are considering is therefore deductively valid. This argument form is often called modus tollens or denying the consequent. You should construct tables like the one above to convince yourself of the following claims: 1. The argument form H E, H, E is another valid argument form. It is often called modus ponens or affirming the antecedent. 2. The argument forms H E, E, H (affirming the consequent) and H E, H, E (denying the antecedent) are not valid argument forms. (The third row of the table just above shows that denying the 4
antecedent is not a valid argument form. If you have any trouble seeing this, just switch the last two columns.) It is also worthwhile to convince yourself that the following are also a valid argument forms: (A & B) A B, (A & B & C) A B C, etc. When philosophers say that some condition A is a sufficient condition for some other condition B, they usually just mean that A B is true. Similarly, to say that A is a necessary condition for B is to say A B, which can also be written as B A, since this sentence has exactly the same truth table as A B and so, to the eyes of a logician, is indistinguishable from it. If A is both a necessary and sufficient condition for B, we often say A if and only if B, and we sometimes abbreviate this as A iff B. Truth functional logic can also be extended to inferences containing all and some. We will omit details, but it is essential to become familiar with the basic notation. Variables, or placeholders for names, are usually written as x, y, z, Predicates are often indicated by F, G, H, So Fx says that x has property F, where neither the particular thing x nor the particular property F is specified. Then to Fx we can attach an operator that we call the universal quantifier and write (x)fx, [or, in older literature ( x)fx ] which says that everything is F. One might read this sentence to oneself as: For every x, x is F. Similarly (x)(fx Gx) says that for every x if that thing is F, then it s G. That is, all Fs are Gs. Also, we can say that something is an F by attaching an existential quantifier to Fx. That is usually written nowadays as (Ex)Fx. In older literature, you ll often find it written as ( x)fx. One might read this sentence to oneself as: There is an x such that x is F. Alternatively: there is at least one F. It is useful to think through the intended meanings of these symbols and see that the following equivalences must be true: ~(Ex)~Fx iff (x)fx and ~(x)~fx iff (Ex)Fx. We can also represent relations amongst two or more things (as opposed to properties that apply to one thing) by using other letters, like R or L. So, for example, we could represent x loves y as Lxy or x gave y to z as Gxyz. This logical notation is quite subtle and flexible, since one can attach (or bind) different variables to different quantifiers (and in different orders). For instance, we could write the (ambiguous) English sentence Everyone loves someone as either (x)(ey)lxy or (Ey)(x)Lxy. Can you see how the logical notation captures the ambiguity (by shifting the order of the quantifiers)? These two sentences are quite distinct. The latter entails the former, but not vice versa. That is, informally, if the latter sentence is true, so must the former sentence be true. If the former is true, however, the latter need not 5
be true. If each person, for instance, loves their own mother, then there need not be one person who is loved by all. As a useful exercise, try to disambiguate these two sentences: (1) Everybody is somebody s fool, and (2) You can fool some of the people all of the time. Logic can be extended beyond truth-functional logic. For instance, logicians write P for it is necessary that P or that P is necessary, and they write P for P is possible. The precise formal definitions of these terms requires more logical machinery than we will employ here, but the following two assertions should be intuitively plausible: P iff P and P iff P. (Can you convince yourself that the logic of possible and necessary will not be truth-functional?) In the most popular or widely-adopted system of modal sentential logic, S5, the following are all theorems: P P, P P, P P, and P P. The effect of these theorems is that, if one adopts this system of modal logic, whenever a string of modal operators precedes a given formula, the string can always be replaced by just the last operator to the right (preserving logical equivalence). There are many subtly differing shades of meaning of necessary (must), possible (can), if then or and even and. We shall see that one must exercise great care in coordinating sentences in natural language with sentences in logical formalism, which can be used to disambiguate them. 6