Download INTLOGS17 Test 1

INTLOGS17 Test 1
Prof: S Bringsjord • TA: Rini Palamittam
0306170626NY
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,
email 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
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2 Informal Proof
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3 Bonus Problem on HyperGrader: GreenCheeseMoon2
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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?
G
A
9
6
a You should flip G only.
b You should flip A only.
c You should flip A and 9 .
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?
G
A
7
6
a You should flip G only.
b You should flip A only.
c You should flip A and G .
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.
a Yes
b No
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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
b
c
d
e
X
X
X
X
X
:=
:=
:=
:=
:=
Euclid; Y := Boole
Leibniz; Y := Boole
Boole; Y := Leibniz
Pascal; Y := Boole
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
b
c
d
e
X
X
X
X
X
:=
:=
:=
:=
:=
Euclid
Leibniz
Boole
Pascal
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
b
c
d
e
X
X
X
X
X
:=
:=
:=
:=
:=
Aristotle; Y
Aristotle; Y
Aristotle; Y
Plato; Y :=
Plato; Y :=
:= syllogisms; Z1 := Turing; Z2 := FOL
:= syllogisms; Z1 := Frege; Z2 := FOL
:= modus ponens; Z1 := Gödel; Z2 := FOL
syllogisms; Z1 := Post; Z2 := FOL
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
b
c
d
e
Euclid
Good
Boole
Turing
Gödel
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Q8 Let φ be an arbitrary wff in the propositional calculus. Is whether or not ` φ R-decidable?
a Yes
b No
Q9 Let φ be an arbitrary wff in the pure predicate calculus. Is whether or not ` φ R-decidable?
a Yes
b No
Q10 Let φ be an arbitrary wff in first-order logic (FOL). Is whether or not ` φ R-decidable?
a Yes
b No
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)?
a Yes
b No
Q13 Is it true that ` (P → Q) → (¬Q → ¬P)?
a Yes
b No
Q14 Is it true that ` ((P → Q) ∧ Q) → P?
a Yes
b No
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Q15 Is it true that ` P → (Q → P)?
a Yes
b No
Q16 Is it true that ` (ζ ∧ ¬ζ) → R?
a Yes
b No
Q17 Is it true that ` (P → (Q → R)) → ((P → Q) → (P → R))?
a Yes
b No
Q18 Is it true that ` ((P → Q) ∧ ¬P) → ¬Q?
a Yes
b No
Q19 Is it true that {R(a), a = b} ` R(b)?
a Yes
b No
Q20 Is it true that ` ∃x(x = a)?
a Yes
b No
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?
a Yes
b No
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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.
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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
http://www.logicamodernapproach.com/allProblems
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.
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