PRACTICE AND STRATEGY FOR QUIZ 4 (Identifying logically possible and necessary sentences, including tautologies)On one portion of the quiz, I will provide the following diagram of logical ‘regions’ or ‘spaces’ and ask you to identify for several sentences into which logical region each sentence falls (i.e., whether the sentence is merely a logical possibility, a logical truth/necessity, or more specifically a tautology). A: logical possibilityThere is a general two-step procedure to follow for this sort of task:STEP #1: First, determine whether the sentence is merely logically possible (i.e., falls into region A in the diagram) or is necessarily true in all worlds (i.e., regions B or C). The easiest way to determine this is to ask yourself whether the sentence could ever be false in some world. If you can think of a world in which the sentence would be false, then obviously the sentence is not always true in every world, which meansthe sentence is no more than a logical possibility (i.e., falls into region A in the above diagram). If, on the other hand, you convince yourself that this sentence could never be false, then you know that it will fall into logical region B or C of the diagram (i.e., the two basic varieties of logical truths we’ve covered so far—keep in mind that for this quiz I’m leaving out consideration of what the book calls TW-necessities). In the latter case, you’ll need to move on to STEP #2 below in order to narrow down which type of logical necessity it is.Let’s consider an example before going further. Ask yourself whether the following sentence of FOL couldever be false in a world:(1) Larger(a,b) ᴧ ¬Larger(b,a)Sentence (1) is a conjunction, so it says that a is larger than b AND that b is not larger than a. Of course, the second conjunct is redundant because if a is the bigger object, then b obviously can’t also be the bigger object. So, the second half of the sentence is, in a way, simply restating the first half of the sentence, only using different ‘words’. But remember that the question at hand is whether this sentence could be ever be false in some worlds. And the answer is, yes, of course it could be. Imagine a world where b IS larger than a. In that sort of world, sentence (1) is simply false. Don’t fall into the trap of B: logical truthC: tautologythinking that just because a sentence claims something to be true, that this means it must actually be true in every world. The claim here that a is larger (and that b is not larger) is just that—merely a claim, and even though it will turn out true in some worlds, it could clearly be false in others. Therefore, the sentence is just a logical possibility and the correct answer for sentence (1) is that it falls into logical region A of the diagram.But what if you can’t think of any way that a sentence could be false? That would be the case with the following sentence:(2) Larger(a,b) ᴠ ¬Smaller(b,a)Sentence (2) is a disjunction; it says that a is larger than b (that’s the easy part) OR that b isn’t smaller than a. Keep in mind that there are three possible logical scenarios to consider in any world where ‘larger’ and ‘smaller’ relations between a and b are concerned: a is bigger, b is bigger, or neither object is bigger than the other (they’re the same size). The first disjunct in the above sentence covers the first of these three cases. The key now is to realize that the second disjunct of sentence (2) actually covers both of the other two possible cases that the first disjunct doesn’t. That is, if b is not smaller than a, thenb is either bigger than a or the same size as a. So then, your overall disjunction (i.e., the two disjuncts making up the disjunction in sentence (2)) covers all three of the available possibilities, meaning that at least one of the two disjuncts in sentence (2) will always be true in any world, and there would be no way to ever falsify this sentence (i.e., sentence (2) is a logical necessity, a logical truth).If you come across a sentence like this one where there’s no way it could ever be false, then you know you’re dealing with either a logical truth (logical region B in the above diagram) or, more strictly, a tautology (logical region C). To decide which of these regions the sentence occupies, you’ll need to move to STEP #2 below.STEP #2: Look at the sentence again and ask yourself whether its logical necessity (i.e., the fact that it must always be true in every world) derives from the meanings of the atomic sentences and predicates involved (in which case the sentence is a standard logical truth and occupies region B in the diagram) or derives strictly from the way the Boolean connectives structure the parts of the sentence (in which casethe sentence is a tautology and occupies region C). Sometimes the answer to this question will be obvious, but sometimes you’ll need to use a special technique to ‘blind’ yourself to the meanings of the atomic sentences and predicates in the sentence in order to decide whether you can still see the logical necessity of the sentence even when you don’t know what the atomic sentences and predicates mean anymore. If you can still see that the sentence must always be true even when you can’t see what the predicates and atomic parts of the sentence mean, then you know that you’re dealing with a tautology.Let’s reconsider sentence (2) above. You might already see that the reason this sentence is always true is due to the interaction of the meanings of the predicates Larger and Smaller; that is, the negation of Smaller covers all the logical ground that the predicate Larger doesn’t cover. This fact suggests that thenecessity of sentence (2) is driven in part by the meanings of the predicates involved, so that sentence (2) would fall into logical region B of the diagram. But if you’re still not sure, then ‘blind’ yourself to the meanings of the predicates Smaller and Larger by replacing all the atomic sentences (but not the Boolean connectives) with capital letters, a distinct capitalletter for each atomic sentence. When you do this, you get the following version of sentence (2):(2b) P ᴠ ¬QNotice that when you blind yourself to the meanings of the two atomic sentences in this way, you can nolonger see any logical necessity to the sentence. That is, without knowing what P and Q actually mean, there’s no way to know whether this sentence is
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