PRACTICE EXERCISE ANSWER KEY FOR LESSON #10(Discussion of answers to exercise 4.8 -- pp. 104-105 of textbook)I apologize in advance for the length of the discussions of some of the harder sentences below. There’s simply no ‘easy’ way to do this. I’ve tried, however, to make the discussions as smooth and readable as possible, so sit down in your most comfortable chair, allow yourself an undisturbed half hour or so to carefully read and think through this, and with luck you’ll come out understanding the basic concepts of possibility, necessity, and tautological necessity. As you read through the answer explanations below, you should consult as well slide #9 of the PowerPoint presentation for lesson 08 posted on Blackboard, where there is a graphical answer-key for these 10 sentences showing where each of them falls in the various circles of the Euler diagram representing possibility and necessity on p. 102 of the textbook.1. a = blogically possibleThis one is pretty straightforward: It states that there is one object named by both ‘a’ and ‘b’ (i.e., ‘a’ and ‘b’ are identical; one and the same object). Though this arrangement certainly may be the case in some worlds, we can also imagine many worlds in which ‘a’ and ‘b’ would refer to two distinct objects instead of one and the same object. Therefore, this sentence is merely logically-possible (or TW-possible, since we’re talking about a sentence that can describe worlds in the Tarski’s World program) and not logically necessary (i.e., the sentence doesn’t have to be true in all worlds).2. a = b ᴠ b = blogically necessaryThis sentence is a disjunction of two disjuncts: one being an atomic sentence (a = b) that by itself would be merely logically possible (i.e., the same atomic sentence just discussed under #1 above) and the other being a different atomic sentence (b = b) that by itself is necessarily true in all worlds (i.e., because it expresses the principle of self-identity that any object is necessarily identical to itself). Because only one disjunct of a larger disjunction needs to be true in order for the entire disjunction to be true (recall the truth-table for disjunction) and because the second disjunct (b = b) in this sentence is a logical truth and will always be true, therefore the entire disjunction will always be true in every world, making it in turn a logical truth (logically necessary).3. a = b ᴧ b = blogically possibleThe only difference between the sentence here in #3 and the one in #2 is, of course, that #3 is a conjunction instead of a disjunction. But that makes all the difference, because for the entire conjunction to be true in some world, each and every one of its conjuncts (i.e., the atomic sentences that make it up) must be individually true in that world. So, the sentence here in #3 will be true only in worlds where the first conjunct a = b is true, which, as we saw in the discussion of #1 above, will be the case in some worlds but not all worlds. (The fact that the second conjunct, b = b, will be true in every world is irrelevant here because the entire conjunction won’t be true unless the first conjunct, a = b, is also true, but as I just explained, thatwon’t be the case in every world.) Given, then, that the entire conjunction will be true in some worlds (where a = b is true) but not in other worlds (where a = b is false), the entire conjunction is merely logically possible, not necessary.4. ¬(Large(a) ᴧ Large(b) ᴧ Adjoins(a,b))TW-necessaryThis sentence is necessarily true in every world you can build with Tarski’s World only because ofa peculiarity of Tarski’s World itself, namely, the restriction built into the software that large objects may not adjoin each other. Of course, this requirement isn’t otherwise any sort of logicalnecessity (i.e., we can easily imagine an alternative version of Tarski’s World or some other software that would contain no restriction against large objects adjoining), so it is a TW-necessity, not a normal logical necessity.5. Larger(a,b) ᴠ ¬Larger(a,b)tautologically necessary (i.e., a tautology)This sentence says that ‘a’ is larger than ‘b’, or (else) ‘a’ isn’t larger than ‘b’—and given that any sentence of FOL has one of only two truth-values, true or false, then there’s no escaping the fact that this sentence will necessarily be true in every world (i.e., it either is or it isn’t the case that ‘a’ is larger that ‘b’; there is no third option). Notice that the atomic sentences making up each ofthe disjuncts are exactly the same (Larger(a,b)), so we can test more specifically for tautological necessity by replacing each occurrence of this atomic sentence with the same capital letter ‘A’ soas to ‘blind’ ourselves to any meaning-content that is internal to the atomic sentences. We make the substitution like this: A ᴠ ¬A Notice that when we do this, we can still ‘see’ the necessity involved in the overall sentence (i.e., that this sentence must be true in any world) even though the only aspect of the sentence that we can still ‘see’ is the work that the Boolean connectives ᴠand ¬ are doing to structure the overall sentence. It turns out that this sentence is a case of what classical logicians called the “law of the excluded middle”, and it is probably the most famous tautology of all. 6. Larger(a,b) ᴠ Smaller(a,b)logically possibleGiven two objects ‘a’ and ‘b’, this disjunction expresses two possibilities about their size-relation to each other: (1) ‘a’ is larger than ‘b’ or, conversely, (2) ‘a’ is smaller than ‘b’. Either of those options are possibly realized in some worlds, no doubt, but notice that there is a third option (3) not addressed by this sentence: ‘a’ and ‘b’ might be the same size. In such a case, the sentence here will be false. So then, the sentence will sometimes be true (in worlds instantiating either ofthe first two options) but sometimes false (in worlds instantiating the third option). The sentenceis only possibly true, not necessarily true.7. ¬Tet(a) ᴠ ¬Cube(b) ᴠ a ≠ blogically necessaryWe discussed this one in class. Just trying to read and understand it can be headache-inducing, so you’ll probably have to approach it like a puzzle to solve. Notice that the entire sentence is a disjunction, meaning that it will be true in any world where one or more of the disjuncts is
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