T2.4 Printable Version:Chapter Two, Tutorial Four SL
Symbolization We have already seen that symbolization of English statements into SL can be less than straightforward. It is time to look at a few more of the complications. Conjunction In Tutorial 2.1, we saw that there were a number of different ways in English to express a conjunction. Just to remind yourself, look at the following list, and pick out the one connective which does NOT express conjunction: (Click on the connective which canNOT be symbolized by the ampersand, '&'.)
Good. The words 'and' and 'both' are often used to express conjunction. When you see them, you should expect to use the ampersand, '&', to symbolize. Just as reminder, our first example, was: 1. Agnes is to attend law school and, as well, her husband Bob will attend law school. which we symbolized with: A&B Now, we didn't have to pick 'A' for "Agnes is to attend law school". We could have picked any capital letter. As long as you (or I) decide in advance which "atomic" sentences of SL are used to represent which simple sentences of English, there will be no confusion. But, of course, it is often easiest to be mnemonically correct and choose a capital letter corresponding to the subject or a verb of the English sentence to be represented. Think about a different example. Suppose we need to symbolize 2. Agnes will attend law school but her husband Bob will not. Here the connective is 'but', the first component will be symbolized by 'A' and the second by '~B'. So a first approximation is: A but ~B. We just need to express the 'but' with a connective. Which one? None seem to fit at first. But think about truth conditions. 2 is true if Agnes does what? and Bob does what? You should have answered: Agnes attends law school AND Bob doesn't. Make sure you see why! So, 2 needs to be symbolized as follows. A&~B and "but" is just another word for "and". Does this seem right? It may not at first. But as far as the truth conditions go, "and" and "but" are synonymous. Thus, both can be symbolized with the '&'. Still, there is a difference. "But", unlike "and", indicates a contrast. We say that one thing is true, but (in contrast) another is not. Now, from the following list, click on all other words that are much like "but" and can be symbolized by the '&'.
T2.4: 2 of 8 Disjunction We use the wedge, 'v', to express a disjunction. For example, consider any of the following: Either Agnes will attend law school or Bob will. These sentences can all be symbolized as AvB These are fairly straightforward. But consider this more difficult one: 3. Agnes will attend law school unless Bob attends. How should we symbolize this statement which has "unless" as its English connective? What SL symbolization will have the same truth conditions as 3? In fact we will see there are a number of equally good symbolizations. But perhaps the easiest to remember involves the wedge. Think about 3 for just a moment. Again: 3. Agnes will attend law school unless Bob attends. This clearly means that if one of the two will not attend law school, then the other will. Thus, it means that at least one of the two will attend, so it's just another way to express disjunction and can be symbolized as AvB It is time to consider an objection. We said that 'v' represents inclusive "or", meaning "one or the other or both". Can 3 be symbolized with inclusive "or"? It may seem that "unless" in 3 indicates that one of the two will attend law school, but not both. We need not settle this matter just yet. But it may be worth considering a second example to see why one might claim that wedge (inclusive "or") is right for symbolizing "unless". Suppose our team is ahead near the end of some game. Then we might truly say: Our team wins unless the other team scores. Here we clearly do NOT want to say "our team wins or the other team scores but not both." Instead, it remains open that they score and then we score again to win. Thus, 'v' appropriately renders "unless" in at least some cases. (And we'll argue later that it's generally appropriate.) In any case, for simplicity, let's stipulate that 'v' is correct for "unless". Bottom line: every time you see unless, you may think about symbolizing it with the wedge. Now, which of the following should be symbolized with the wedge? Click on all of the following which should be seen as disjunctions of the form 'AvB': Either
Agnes or Bob will attend law school. T2.4: 3 of 8 Negation The English phrase "it's not the case that" is much like SL's tilde, '~'. Both are unary connectives which take a single sentence and produce a new one. So, our easy example, It's not the case the Agnes will attend law school, can be symbolized as ~A But we've seen a number of English alternatives expressing negation. All of the following will express the same thing as 3 and can be symbolized as '~A'. Agnes will not attend law school. And so forth. The different ways to express negation are limited only by one's imagination. Which of the following are negations? Click on all correct answers. Bob
isn't to attend law school.
T2.4: 4 of 8 Often negations are more complicated than the ones we've seen. For instance, we will often see the negation of a conjunction: 4. It's not the case that both Agnes and Bob will attend law school. Sometimes this is put more compactly as 5. Agnes and Bob won't both attend law school. 4. and 5. amount to the same thing. But what is this "same thing" and how is it to be symbolized in SL? What we have done up until now does not provide the answer. It should seem clear that 4 and 5 involve a negation and conjunction (because they involve "not" and "and"). This suggests a couple of ways to symbolize the two. Can you guess the best one? (Click on the symbolization which is correct for both 4 and 5.) Right. 4 and 5 say "not both" and (*) ~(A&B) does too. Why? First, ~A&B is not correct because it means that Agnes won't attend law school while Bob will. But 4 and 5 only say that one of the two (not specifically Agnes) won't attend. So why is ~(A&B) correct? Because this symbolization denies (that's the tilde) the whole 'A&B' just like 4 and 5 in the English. It's the work of the parentheses to group A with B. The tilde then negates the whole group. This gets the right truth conditions. But there is another logically equivalent way to restate both 4 and 5: 6. Either Agnes is not a law student or Bob is not a law student. However this is a disjunction! Shouldn't it be symbolized as (**) ~Av~B The answer is YES. At least, this is one way to do it. But notice that * is just as good as **: they both are true exactly when at least one of A and B is false. The bottom line is that 4, 5, 6, and * and ** all have the same truth conditions. Thus, either * or ** can be used to symbolize any of 4, 5, and 6. Now, a harder one! How would you symbolize 7. Neither Agnes nor Bob will be a law student. Click on all good symbolizations of 7... a. ~Av~B
T2.4: 5 of 8 So,7. Neither Agnes nor Bob will be a law student. can be symbolized in at least two ways. The first is to say that it's not the case that either will be a law student: ~(AvB) The tilde is the "not", what's inside parentheses is the either. Equivalently, 7 can be rendered both will not be law students: ~A&~B I.e., Agnes will not AND Bob will not.
Conditionals There are many ways to express conditional or hypothetical claims in English. We will take the "if...then..." form as our paradigm and insert a horseshoe in place of "then". Our example 8. If Agnes will go to law school, then Bob will not. is symbolized as (*) A>~B But there are lots of other ways to express the same thing: Provided Agnes will go to law school, then Bob will not. Be careful with the next one: 9. Agnes will go to law school if Bob will not. This is symbolized the other way around: (**) ~B>A because the "if" precedes the antecedent that Bob will not attend law school. Now, how about 10. Agnes will go to law school only if Bob will not. (Note: sentence 10 has "only if" as its English connective; 9 has plain "if".) Now, 10 is correctly symbolized as one of * or **, but which? You see, 9 and 10 are different. 9 sets out a condition under which Agnes will go to law school, while 10 places one requirement on her attendance. To answer our question about 10, compare 9 and 10 to: 11. There is fire if oxygen is present. and 12. There is fire only if oxygen is present. 11 is O>F because of the "if" as connective rather than "then". But 12 takes a little thought. It gives a requirement or "necessary condition" for fire: there must be oxygen. So, 12 says, if there is fire, then there must be oxygen. Thus, 12 is symbolized as F>O and "only if" is simply replaced by the horseshoe. Finally, our question was about 10. Agnes will go to law school only if Bob will not. This is to be symbolized as what? Click on the correct symbolization: T2.4: 7 of 8 Okay...symbolizing with the horseshoe is rather difficult because English has so many ways to express conditionals. It will take some practice to get used to these and to see their form. But once we can symbolize them, we will be in good shape to understand exactly what these English expressions mean. To help, you can always go through this tutorial again but the information is also summarized in the reference manual for this section. Biconditionals One way to express the biconditional in English is with the expression "if and only if". So, for example, we gave the definition of "sound" this way: An argument is sound if and only if it is both valid and has only true premises. We might symbolize this definition in SL as S=(V&T) But it's worth seeing that there is another good symbolization suggested by our study of conditionals. Remember that "P only if Q" was to be symbolized as P>Q and that "P if Q" is just the other way around: Q>P. Figuratively, the "only if" expression is P leads to Q, the "if" expression is the other "direction": Q leads to P. So, "P if and only if Q" puts both together: (P>Q)&(Q>P) is as good a symbolization for a biconditional as P=Q Either way will do. The first shows why we call a sentence formed from "if and only if" a "biconditional"; the second is shorter. Now, which of the following can symbolize Agnes will attend law school if and only if Bob does not. Click on all correct symbolizations... A>B
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