UNIT 1 FORMAL PROOF OF VALIDITY: RULES OF INFERENCE Contents 1.0 Objectives 1.1 Introduction 1.2 Formal Proof of Validity – Meaning 1.3 Rules of Inference 1.4 Testing the Validity of Arguments 1.5 Testing the Validity of Arguments (Verbal) 1.6 Let Us Sum Up 1.7 Key Words 1.8 Further Readings and References 1.9 Answers to Check Your Progress 1.0 OBJECTIVES The main objective of this unit is: to make explicit the art of testing arguments. This is being achieved in two ways; the limitations of traditional logic are exposed and at the same time the necessity of traditional logic is being demonstrated. to compare verbal form of arguments and its symbolic form. to assess the relative merits and demerits of two forms if any. to translate symbolic representation to verbal form and verbal form to symbolic form. 1.1 INTRODUCTION The primary function of logic is to classify arguments into good and bad. This can be done by testing the validity of arguments. As we know only limited types of arguments are covered by classical logic. Even those arguments which are within the range of modern logic are not alike in all respects. Some are simple enough so that the truth-table technique is adequate for the purpose of testing. Now, what is this truth-table technique of determining the validity of arguments? Let us take an argument form, for example: If p, the q p Therefore, q Its truth table can be constructed as follows: we need the initial columns of the statement or proposition variable p and q; then we need a column for the first premise here which is an implicative statement (second premise and conclusion are the initial columns themselves); since 1 there are only two variable in this argument we need only four rows, as we have learned earlier. The truth-table of the above argument is as follows: p q p =>q 1 1 1 1 2 1 0 0 3 0 1 1 4 0 0 1 Truth-table technique uses the principle that in a valid argument the conclusion is implied in the premises and so from true premises only true conclusions follow. In the above truth table, only in the first row the premises are true (see under p => q and p) and there under q we see the truth value as true. Hence we can say, in this truth-table no substitution instances (i.e., different rows) with true premises and false conclusion is seen and so it is a valid argument. This is a mechanical method; just construct a truth table for any given argument and see whether there are instances with true premises and false conclusions. Students can attempt the same for the following elementary arguments forms given below as rules of inference. Generally, any argument, which consists of two or three simple but different propositions, is regarded as amenable to the truth-table method. But if the argument consists of more than three different propositions, then the truth-table method is of no avail. It is mainly because of its manoeuvrability, i.e., if there are three propositions, we need 8 rows in the truth-table; if four, then 16; if 5, the 32; if 6, the 64, and so on. In such circumstances we have to look for an alternative. Formal proof helps us here in quickly determining the validity of arguments. See the following example: 1. A => B 2. B => C 3. C => D 4. ⌐D 5. A v E / ∴ E 6. A => C 7. A => D 8. ⌐A 9. E In this argument we have five propositions like A, B, C, D, E; if we construct truth-table for it, we need 32 rows. Now we can prove its validity by applying certain rules of inference in just four lines. This is exactly the advantage of formal proofs. An argument, which is complex in this sense, is nothing but an aggregate of several simple (by simple, in this context, we mean short) arguments. Examples make this point clear. 1) p => q 2) q => r 3) p =>q p q q => r ∴q ∴ r ∴ p => r 2 In classical logic also we have „complex‟ type of argument in the form of sorites. (We should remember that the terms complex, simple, etc. are relative). An example for sorites is given: 1 All Indians are Asians. All Hindus are Indians. All Kannadigas are Hindus. ∴ All Kannadigas are Asians. There are three premises and a conclusion. Hence, it is a polysyllogistic argument. As a matter of fact, a sorites consists of at least two syllogistic arguments and therefore, two conclusions. So it is more complex than an ordinary syllogism. This point becomes clear when we break sorites into constituent syllogisms. 2a). All Indians are Asians. ] → All Hindus are Asians. All Hindus are Indians. All Kannadigas are Hindus. → All Kannadigas are Hindus. ∴All Kannadigas are Asians Fortunately or unfortunately, we hardly encounter such stereotype arguments. So there is need to sharpen and augment the tools of testing. At this critical juncture, it is very important to remember that no rule stipulated by classical logic can be ignored or violated. It is the foundation on which the superstructure, i.e., modern logic is built. For the sake of convenience, let us restrict ourselves only to symbols and go to verbal form when we take up exercise. 1.2 FORMAL PROOF OF VALIDITY: IT’S MEANING In modern logic an argument is regarded as a sequence of statements. When proof is constructed to test the argument, the proof also takes the same form, which the argument takes. In this type of proof there is correspondence between the scheme of the given argument and the scheme of the proof. Every step, which is adduced while constructing proof, is the conclusion of the preceding statements, and in turn, becomes the premise for statements, which follow it (if not all, at least to some). Rules, which govern the process of deducing hidden conclusion, constitute what are known as „Rules of Inference‟ in modern logic. Many of these rules have their origin in traditional logic. There is a certain way of constructing proof in modern logic. More descriptive method, which consumes both space and time, has given way to much shorter and simpler method. Whatever conclusion can be drawn from any two given premises is written on left hand side (LHS) while the rule and the premises to which this particular rule applies to derive the conclusion used in further proof, are written on the right hand side (RHS). A rule of inference is applied to the 3 whole line. This is an important point to note. As an economy measure, instead of premises, corresponding serial numbers are written. Thereby we save time. We must ensure that drawn conclusion, the respective premises and the rule applied are always juxtaposed. This procedure is the simplest and most economical in terms of time and effort to grasp the argument. 1.3 RULES OF INFERENCE Our task, from now onwards, is very simple. Modern logic considers nine rules of inference. They are listed below. 1) Modus Ponens (M.P.) 4) Disjunctive Syllogism 7) Simplification (Simp.) (D.S.) p => q p v q p v q p Λ q p ⌐ p ⌐q ∴ p /q ∴ ∴ ∴ q q or p 2) Modus Tollens (M.T.) 5) Constructive 8) Conjunction (Conj.) Dilemma (C.D.) p => q (p => q) Λ (r => s) p ⌐ q p v r q ∴ ⌐ p ∴q v s ∴ p Λ q 3) Hypothetical 6) Destructive Dilemma 9) Addition (Add.) Syllogism (H.S.) (D.D.) p => q (p => q ) Λ (r => s) p q => r ⌐ q v ⌐ s ∴ p v q ∴ p => r ∴⌐ p v ⌐ r First six rules are standard rules of traditional logic. Last three rules need a little clarification. Consider, for example, simplification. Since p Λ q is given to us, we accept that p is true, and q is true as well. So there is no harm in dropping any of them. The case of conjunction is slightly different. p is given to us, so we take it as true; q is given to us. So we take q also as true. Since both are taken as true we can conveniently conjoin them. The case of addition, again, is different. Suppose that we have only p in the premises. Since it is a premise, we take it as true. Suppose that we require q to be added to p. We do not know whether q is true or not. There is no harm in adding q to p because even if q is false p v q still remains true because p is true. After all, one true component can make disjunction true. But what is important is that conjunction does not mean addition. In logical language, addition means disjunction but not conjunction. 4 The rest of our job is very easy; just apply relevant rules for relevant pairs of lines. It needs only practice and detective‟s eyes to identify relevant lines and the rule applicable to those lines. All arguments, which are required to be tested, are valid only because these are proofs only for validity but not for invalidity. 1.4 TESTING THE VALIDITY OF ARGUMENTS Let us begin with the argument we have seen above: 1) A => B 2) B => C 3) C => D 4) ⌐D 5) A v E / ∴ E 6) A => C 1, 2, H.S. 7) A => D 6, 3, H.S. 8) ⌐A 7, 4, M.T. 9) E 5, 8, D.S.