Logical Equivalence, Logical Truths, and Contradictions 3-1. LOGICAL EQUIVALENCE I introduced logic as the science of arguments. But before turning to ar- guments, we need to extend and practice our understanding of logic's. basic tools as I introduced them in chapter 1. For starters, let's look at the truth table for 'A', '-A', and the negation of the negation of 'A', namely, '--A' This uuth table exhibits the special situation which I mentioned at the end of the last chapter: The auth value of '--A' is always the same as that of 'A'. Logicians say that 'A' and '--A' are Logicdl' Equrualtnt. As we will see in a moment, much more complicated sentences can be logically equivalent to each other. To get dear on what this means, let us review some of the things that truth tables do for us. Suppose we are looking at a compound sentence, perhaps a very complicated one which uses many sentence letters. When we write out the truth table for such a 30 Logical Equivdmcc, Logical Truths, and Contradictions sentence, we write out all the possible cases, that is, all the possible assign- ments of truth values to sentence letters in all possible combinations. In 1 each one of these possible cases our original sentence has the truth value i t or the truth value f. Now suppose that we look at a second sentence which uses the same sentence letters, or perhaps only some of the sentence letters that the first sentence uses, and no new sentence letters. We say that the two sentences are logically equivalent if in each possible case, that is, for each line of the these eventualities, we must draw the circles representing X and Y as truth table, they have the same truth value. overlapping, as in Figure 3-2 and 3-3. Two sentences of sentence logic are Logically Equivalent if and only if in each possible case (for each assignment of truth values to sentence letters) the two sentences have the same truth value. What we said about the double negation of 'A' naturally holds quite generally: I X&Y XvY The Law of Double Negation (DN): For any sentence X, X and --X are logically equivalent. Figure 3-2 Figure 3-3 Here are two more laws of logical equivalence: The conjunction X&Y is true in just those cases represented by points that lie inside both the X and Y circles, that is, the shaded area in Figure De Morgan's Laws (DM): For any sentences X and Y, -(X&Y) is logically 3-2. The disjunction XVY is true in just those cases represented by points equivalent to -XV-Y. And -(XvY) is logically equivalent to -X&-Y. that lie inside either the X or the Y circle (or both), that is, the shaded area in Figure 3-3. Now we can use Venn diagrams to check De Morgan's laws. Consider Thus 'Adam is not both ugly and dumb.' is logically equivalent to 'Either first a negated conjunction. Look for the area, shown in Figure 3-4, Adam is not ugly or Adam is not dumb.' And 'Adam is not either ugly or which represents -(X&Y). This is just the area outside the shaded lens in dumb.' is logically equivalent to 'Adam is not ugly and Adam is not Figure 3-2. dumb.' You should check these laws with truth tables. But I also want to show you a second, informal way of checking them which allows you to "see" the laws. This method uses something called Van Diagram. A Venn diagram begins with a box. You are to think of each point inside the box as a possible case in which a sentence might be true or FIgule 3-4 false. That is, think of each point as an assignment of truth values to sentence letters, or as a line of a truth table. Next we draw a circle in the box and label it with the letter 'X', which is supposed to stand for some arbitrary sentence, atomic or compound. The idea is that each point in- side the circle represents a possible case in which X is true, and each point Let us compare this with the area which represents -XV-Y. We draw outside the circle represents a possible case in which X is false. overlapping X and Y circles. Then we take the area outside the first circle Look at Figure 3-1. What area represents the sentence -X? The area (which represents -X; see Figure 3-5), and we take the area outside the outside the circle, because these are the possible cases in which X is false. second (which represents -Y; see Figure 3-6). - Now let's consider how Venn diagrams work for compound sentences Finally, we put these two areas together to get the area representing the built up from two components, X and Y. Depending on what the sen- disjunction -XV-Y, as represented in Figure 3-7. tences X and Y happen to be, both of them might be true, neither might Notice that the shaded area of Figure 3-7, representing -XV-Y, is the be true, or either one but not the other might be true. Not to omit any of same as that of Figure 3-4, representing -(X&Y). The fact that the same 32 Logical EquivaJcnce, Logical Truths, and Contradietions 3-2. Subrtirurion of Logical Equiualrntr and Some Mon Lcnus 33 'Fill in the areas to represent YvZ, and then indicate the area which represents the conjunction of this with X. In a separate diagram, first fill in the areas representing X&Y and X&Z, and then find the area corre- sponding to the disjunction of these. If the areas agree, you will have demonstrated logical equivalence. Do the second of the distributive laws similarly. Also, if you feel you need more practice with truth tables, prove Figure 3-5 Figure 3-6 these laws using truth tables. EXERCISES 3-1. Prove the second of De Morgan's laws and the two distributive laws using Venn diagrams. Do this in the same way that I proved the first of De Morgan's laws in the text, by drawing a Venn diagram Figure 3-7 for each proof, labeling the circles in the diagram, and explaining in a few sentences how the alternate ways of getting the final area give shaded area represents both -XV-Y and -(X&Y) means that the two the same result. Use more than one diagram if you find that helpful sentences are true in exactly the same cases and false in the same cases. in explaining your proof. In other words, they always have the same truth value. And that is just what we mean by two sentences being logically equivalent. Now try to prove the other of De Morgan's laws for yourself using Venn diagrams. 3-2. SUBSTITUTION OF LOGICAL EQUIVALENTS Here are two more laws of logical equivalence: AND SOME MORE LAWS The Dishibutive Laws: For any three sentences, X, Y, and Z, X&(YvZ) is We can't do much with our laws of logical equivalence without using a logically equivalent to (X&Y)v(X&Z).And Xv(Y&Z)is logically equivalent to very simple fact, which our next example illustrates. Consider (XW&(XvZ). For example, 'Adam is both bold and either clever or lucky.' comes to the same thing as 'Adam is either both bold and clever or both bold and '--A' is logically equivalent to 'A'. This makes us think that (1) is logically lucky.' You should prove these two laws for yourself using Venn dia- equivalent to grams. To do so, you will need a diagram with three circles, one each representing X, Y, and Z. Again, to make sure that you omit no possible (2) AVB. combination of truth values, you must draw these so that they all overlap, as in Figure 3-8. This is right. But it is important to understand why this is right. A compound sentence is made up of component sentences, which in turn may be made up of further component sentences. How do subsentences (components, or components of components, or the like) affect the truth value of the original sentence? Only through their truth values. The only way that a subsentence has any effect on the truth values of a larger sen- tence is through the subsentence's truth value. (This, again, is just what we mean by saying that compound sentences are truth -functions.) But if only the truth values matter, then substituting another sentence which always has the same truth value as the first can't make any difference. 34 Logical Equiwlmee, Logical Truth, and ConhndictiOnr I'll say it again in different words: Suppose that X is a subsentence of (--A&--B)v(--A&-B) DM,SLE some larger sentence. Suppose that Y is logically equivalent to X, which means that Y and X always have the same truth value. X affects the truth (As before, 'DM' on the right means that we have used one of De Mor- value of the larger sentence only through its (i.e., X's) truth value. So, if gan's laws. 'SLE' means that we have also used the law of substitution of we substitute Y for X, there will be no change in the larger sentence's logical equivalents in getting the last line from the previous one.) truth value. Now we can apply the law of double negation (abbreviated 'DN') to But this last fact is just what we need to show our general point about '--A' and to '--B' and once more substitute the results into the larger logical equivalence.