INTLOGS17 Test 1. Prof: S Bringsjord TA: Rini Palamittam NY

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1 INTLOGS17 Test 1 Prof: S Bringsjord TA: Rini Palamittam NY Immediate Action Items: Please now, before you do anything else, write down the following details on the Scantron sheets as well as on the exam booklet in print: Name, your CD/HyperGrader Code, address, and RIN. In addition: Make sure you label your answer to Q22 in your blue answer booklets with the label Q22. Please also make sure you announce which of the two possible theorems you are going to try to proceed to establish. And please strive to write as legibly as possible when giving your written answer. Thank you. Contents 1 Multiple Choice Questions 1 2 Informal Proof 5 3 Bonus Problem on HyperGrader: GreenCheeseMoon2 5

2 1 Multiple Choice Questions Q1 Suppose that you are presented with a version of the Wason Selection Task exactly the same as the one we considered in class, except the four cards in front of you have the following appearance. Which card or cards should you turn over? a You should flip G only. b You should flip A only. c You should flip A and 9. G A 9 6 d You should flip 9 and 6. e You should turn over all the cards. Q2 Suppose that you are presented with a version of the Wason Selection Task exactly the same as the one we considered in class, except for two things: viz., (i) the four cards in front of you have the appearance of what follows the present paragraph; and (ii), the rule from before is supplanted with this new one: If there s either a vowel or a consonant on one side, then there is a prime number on the other. Which card or cards should you turn over? a You should flip G only. b You should flip A only. c You should flip A and G. G A 7 6 d You should flip A and G and 7. e You should turn over all the cards. Q3 The following four statements are either all true, or all false. Given this, does there exist a valid, oracle-free Slate proof (based on some sensible symbolization in the propositional calculus) that Lola lies? 1. If Lucy lies, then so does Larry. 2. If Larry lies, then so does Linda. 3. If Linda lies, then Lola does as well. 4. Lucy lies. 1

3 Q4 While X is traditionally credited with inventing the proposiitonal calculus, the truth of the matter is that Y already had the propositional calculus over a century before the relevant work by X. Which of the following five options is the best instantiation of X and Y in the previous sentence? a X := Euclid; Y := Boole b X := Leibniz; Y := Boole c X := Boole; Y := Leibniz d X := Pascal; Y := Boole e X := Frege; Y := Boole Q5 X is traditionally credited with inaugurating the systematic study of probability, and inductive logic. Which of the following five options is the best instantiation for X in the previous sentence? a X := Euclid b X := Leibniz c X := Boole d X := Pascal e X := Frege Q6 While X claimed that what made Euclid s remarkable reasoning thoroughly compelling was that that reasoning was based on Y, we had to wait until the 20th century for Z1 to provide the correct answer, which was that this reasoning was fundamentally deduction in Z2. Which of the following five options is the best instantiation of X, Y, Z1, Z2 in the previous sentence? a X := Aristotle; Y := syllogisms; Z1 := Turing; Z2 := FOL b X := Aristotle; Y := syllogisms; Z1 := Frege; Z2 := FOL c X := Aristotle; Y := modus ponens; Z1 := Gödel; Z2 := FOL d X := Plato; Y := syllogisms; Z1 := Post; Z2 := FOL e X := Plato; Y := modus ponens; Z1 := Pascal; Z2 := PC Q7 Which of the following thinkers originally gave the deductive argument for The Singularity that we considered in class? a Euclid b Good c Boole d Turing e Gödel 2

4 Q8 Let φ be an arbitrary wff in the propositional calculus. Is whether or not φ R-decidable? Q9 Let φ be an arbitrary wff in the pure predicate calculus. Is whether or not φ R-decidable? Q10 Let φ be an arbitrary wff in first-order logic (FOL). Is whether or not φ R-decidable? Q11 The somewhat modernized proof shown and discussed in class of Euclid s Theorem that φ makes use of the two proof techniques of X and Y. Which of the following five options is the best instantiation to φ, X, and Y in the previous sentence? a φ := there are infinitely many integers; X = reductio ad absurdum; Y = proof by cases b φ := there are infinitely many reals; X = mathematical induction; Y = proof by cases c φ := there are infinitely many primes; X = mathematical induction; Y = proof by cases d φ := there are infinitely many reals; X = proof by contradiction; Y = proof by cases e φ := there are infinitely many primes; X = indirect proof; Y = proof by cases Q12 Is it true that P (P Q)? Q13 Is it true that (P Q) ( Q P)? Q14 Is it true that ((P Q) Q) P? 3

5 Q15 Is it true that P (Q P)? Q16 Is it true that (ζ ζ) R? Q17 Is it true that (P (Q R)) ((P Q) (P R))? Q18 Is it true that ((P Q) P) Q? Q19 Is it true that {R(a), a = b} R(b)? Q20 Is it true that x(x = a)? Q21 Is it a theorem in FOL that there exists at least one thing x which is such that: if x is a llama, then everything is a llama? 4

6 2 Informal Proof Q22 Prove that that the answer you gave above to question Q9 is correct. (This is of course to be an informal proof, not a formal proof in Slate. By informal proof we mean the style of proof that is expressed in a mixture of English and formal symbols, such as what we saw in the case of Euclid s Theorem.) The theorem you will prove is thus either that theoremhood in the pure predicate calculus is R-decidable, or the opposite, that is that theoremhood in the pure predicate calculus is not R-decidable. Finally, again, please strive to write as legibly as possible when giving your written answer. 3 Bonus Problem on HyperGrader: GreenCheeseMoon2 A new problem for solving through HyperGrader has just been posted, or will momentarily be posted; it s called GreenCheeseMoon2 (not to be confused with GreenCheeseMoon1, which presumably you ve already solved and is a Required problem). To earn bonus points on the present test, go to and obtain the underlying Slate file, create a proof in it that obtains the GOAL from the lone GIVEN without any remaining use of an oracle, submit your solution file, and earn a trophy (which signifies that you have earned the bonus points at stake). Only if you earn a trophy can you be sure that your bonus points have been earned. 5