PRACTICE QUESTIONS AND PROCEDURE for Quiz #9 Here is the basic format of the first three parts of the upcoming quiz dealing with tautological and FO relations, along with a few practice questions and some explanation of a procedure for tackling these parts of the quiz.Part 1: For each of the following sentences, convert the sentence into both First-Order Checking Form and Truth-Functional Form. (You can use capital letters P, Q, R, etc. for First-Order Checking Form and capital letters A, B, C, etc. for Truth-Functional Form.) Then, state whether the sentence is . . .(1) not a logical truth, (2) a logical truth, (3) more specifically, a first-order validity, or (4) even more specifically, a tautology.There’s a basic procedure to follow for answering this sort of question:First, since I’m asking you for these questions to write out both the FO Checking Form and the Truth-Functional Form for each sentence, you might as well start by doing just that. Remember that to convert a sentence of FOL into FO Checking Form, you ‘blind’ yourself to every predicate in the sentence other than the identity (=) predicate. So every ‘Cube’, ‘Smaller’, ‘SameSize’, etc. gets replaced by a capital letter like P, Q, or R. But the arguments of those predicates (i.e., the individual constants and the variables), as well as everything else in the sentence (quantifiers, Boolean connectives) remain. (See the examples below and in the lecture notes for lesson 30.) Then, to convert the sentence into Truth-Functional Form, you need to put on more drastic ‘blinders’: replace every atomic sentence (i.e., predicate + its arguments) and every quantified phrase (i.e., quantifier + everything within the scope of the quantifier) with a capital letter like A, B, or C. (Again, there are examples below and in the notes.)Once you’ve written out the above conversions, it’s time to read and think about the sentence of FOL itself, first just as it’s originally presented—without any ‘blinders’ on. Considering the meaning of the sentence, can you think of any way that the sentence could ever be false in some world? If you can, then it must be merely a logical possibility, and you know that it is “not a logical truth” (i.e., sentence type 1). If, on the other hand, you convince yourself that this sentence could never be false, then you know that it will be one of the sentence types 2-4 (i.e., the three varieties of logical truths). In that case, you’ll need to move on to the next step in order to narrow down which type of logical truth it is.Next step (i.e., for those sentences that you believe to be logical truths): Look at the FO Checking Form of the sentence and ask yourself whether, with those particular ‘blinders’ on, you can still see that the sentence must always be true. If in FO Checking Form you can no longer see that it is a logical truth (i.e., the necessity of the sentence has ‘evaporated’ with your FO-Checking-Form blinders on), then you know that this sentence is of sentence type 2 (i.e., a logical truth that is not more specifically an FO validity or tautology). If, on the other hand, thenecessity of the sentence is still apparent even in FO Checking Form, then you know that you are dealing with one of the sentence types 3-4, and you’ll need to continue on to the final step below.Finally, look at the Truth Functional Form of the sentence. If you could see the necessity of the sentence when it was in FO Checking Form but now you can no longer see that necessity in the Truth Functional Form of the sentence, then you know that the sentence is of sentence type 3 (i.e., it’s an FO validity that is not more specifically a tautology). If, on the other hand, you can still see even in its Truth Functional Form that the sentence must always be true (i.e., even when you’ve blinded yourself to everything except how the Boolean connectives structure the sentence), then you know that you’re dealing with sentence type 4 (i.e., a tautology).First-Order Checking Form Truth-Funct. Form1. ∀x x=x ᴧ ∀y y=y ___ ∀ x x=x ᴧ ∀ y y=y______ ___A ᴧ B__________Sentence type? __3___ The quantifiers and the identity predicate are preserved in FO Checking Form, and you can still see the necessity involved. Thus, this sentence is a FO validity. It is not more specifically a tautology, however, as seen when you convert it to Truth-Functional Form. There, each of the entire quantified phrases is replacedby a capital letter, and the necessity is no longer visible.2. SameShape(a,b) SameShape(b,a) ____P(a,b) P(b,a)______ ___A B_________Sentence type? __2___ You can (hopefully) see that the original sentence is a logical truth, but without knowing what the ‘blinder’ predicate P means in the FO Checking Form, you can no longer tell in that form whether the objects b and a are in a necessarily symmetrical relation to each other, so this sentence is a logical truth but not a FO validity (and therefore not a tautology either).Part 2: For each of the following arguments, state whether the conclusion of the argument is (1) not a logical consequence of the premises (i.e., the argument is invalid), (2) a logical consequence of the premises, (3) more specifically, a first-order consequence of the premises, or (4) even more specifically, a tautological consequence of the premises. If you answered (1), then in the space provided beside the argument draw a counterexample world that proves the argument invalid. Be sure everything in your world is clearly marked and unambiguous, and that it is specific enough to truly provide a counterexample to the argument.If you answered (2), then in the space provided beside the argument convert the entire argument into FO Checking Form (for your ‘blinders’ use alternative predicates, not capital letters) so as to clearlydemonstrate that the consequence relation is no longer visible in FO Checking Form.If you answered (3), then in the space provided beside the argument convert the entire argument intoBOTH FO Checking Form and Truth-Functional Form (label which is which) in order to demonstrate that the consequence relation is still visible in FO Checking Form but NOT visible in Truth-Functional Form. (Again, use alternative predicates for FO Checking Form here, but use capital letters A, B, C, etc. for Truth-Functional Form.)If you answered (4), then in the space provided beside the argument convert the
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