Question Mark
1
2
3
4
5
6
7
Bonus: Knights and Knaves
/6
/5
/5
/5
/10
/5
/4
/3
TOTAL /40
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Questions about Language
1. (6 pts.) Suppose we have an FOL with constant symbols/names a, b, c, d, and e, function symbols
f (x), g (x), h (x, y), predicate symbols P (x), Q (x), R (x, y), x = y, and one ternary connective (x, y, z).
For each of the following, indicate whether they are simple terms, complex terms, atomic
sentences, or complex (non-atomic) sentences, or none of these.
a. (f(a), g(b), h(c, d))
b. Q(h(c, d))
c. (b = c, P(f(e)), R(e, g(a)))
d. (Q(d), Q(a), Q(b)) = c
e. a ≠ b
f. h(f(d), h(e, f(a)))
Informal Proof
2. (5 pts.) Prove (informally) whether if S is a tautological consequence of P1, . . . , Pn, then the set of
sentences {S, P1, . . . , Pn} is consistent.
Explanation
3. (5 pts.) Do exercise 7.21 of your textbook.
Formal Proofs (You may not use TautCon or AnaCon for any of these proofs.)
4. (5 pts.) Provide a formal proof in f showing that the sentence (P P) is a tautology.
5. (10 pts.) Determine whether the following argument is tautologically valid. If it is, provide a
formal proof in f. If it is not, then describe a counterexample.
1. (P(a) P(b)) (R(a, c) Q(a))
2. (R(b, c) Q(b)) (P(b) Q(a))
3. R(a, c) R(b, c)
———-
4. P(a) Q(b)
Translations
6. (5 pts.) Using the predicate symbols from Tarski’s World, translate the following English
sentences into FOL.
1. c is to the right of a, provided it (i.e., c) is small.
2. c is to the right of d only if b is to the right of c and left of e.
3. If e is a tetrahedron, then it’s to the right of b if and only if it is also in front of b.
4. e is in front of d unless it (i.e., e) is a large tetrahedron.
5. d is the same shape as b only if they are the same size.
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Equivalence (4 pts.)
Consider the following ternary truth-functional connective:
P Q R (P, Q, R)
T T T F
T T F T
T F T T
T F F T
F T T T
F T F T
F F T T
F F F F
Express this connective with a sentence that only uses the Boolean connectives in its most simplified
form—that is, using a sentence containing no more than two occurrences each of P, Q, and R, and
no more than six occurrences of the Boolean connectives , , and . Briefly explain how you
arrived at this sentence. If you used a chain of equivalences, please include it in your response.
Bonus Question: Knights and Knaves (3pts.)
Harry Potter and Hermione Granger are inhabitants of the island of knights and knaves, where
knights only tell the truth, and knaves only lie. Harry says “I am a knight if and only if Snape is on
the island”. Hermione says that if Harry is a knight, then Snape is not on the island. What are Harry
and Hermione, and is Snape on the island? Explain
Sample Solution
Dante Alighieri played a critical role in the literature world through his poem Divine Comedy that was written in the 14th century. The poem contains Inferno, Purgatorio, and Paradiso. The Inferno is a description of the nine circles of torment that are found on the earth. It depicts the realms of the people that have gone against the spiritual values and who, instead, have chosen bestial appetite, violence, or fraud and malice. The nine circles of hell are limbo, lust, gluttony, greed and wrath. Others are heresy, violence, fraud, and treachery. The purpose of this paper is to examine the Dante’s Inferno in the perspective of its portrayal of God’s image and the justification of hell.
In this epic poem, God is portrayed as a super being guilty of multiple weaknesses including being egotistic, unjust, and hypocritical. Dante, in this poem, depicts God as being more human than divine by challenging God’s omnipotence. Additionally, the manner in which Dante describes Hell is in full contradiction to the morals of God as written in the Bible. When god arranges Hell to flatter Himself, He commits egotism, a sin that is common among human beings (Cheney, 2016). The weakness is depicted in Limbo and on the Gate of Hell where, for instance, God sends those who do not worship Him to Hell. This implies that failure to worship Him is a sin.
God is also depicted as lacking justice in His actions thus removing the godly image. The injustice is portrayed by the manner in which the sodomites and opportunists are treated. The opportunists are subjected to banner chasing in their lives after death followed by being stung by insects and maggots. They are known to having done neither good nor bad during their lifetimes and, therefore, justice could have demanded that they be granted a neutral punishment having lived a neutral life. The sodomites are also punished unfairly by God when Brunetto Lattini is condemned to hell despite being a good leader (Babor, T. F., McGovern, T., & Robaina, K. (2017). While he commited sodomy, God chooses to ignore all the other good deeds that Brunetto did.
Finally, God is also portrayed as being hypocritical in His actions, a sin that further diminishes His godliness and makes Him more human. A case in point is when God condemns the sin of egotism and goes ahead to commit it repeatedly. Proverbs 29:23 states that “arrogance will bring your downfall, but if you are humble, you will be respected.” When Slattery condemns Dante’s human state as being weak, doubtful, and limited, he is proving God’s hypocrisy because He is also human (Verdicchio, 2015). The actions of God in Hell as portrayed by Dante are inconsistent with the Biblical literature. Both Dante and God are prone to making mistakes, something common among human beings thus making God more human.
To wrap it up, Dante portrays God is more human since He commits the same sins that humans commit: egotism, hypocrisy, and injustice. Hell is justified as being a destination for victims of the mistakes committed by God. The Hell is presented as being a totally different place as compared to what is written about it in the Bible. As a result, reading through the text gives an image of God who is prone to the very mistakes common to humans thus ripping Him off His lofty status of divine and, instead, making Him a mere human. Whether or not Dante did it intentionally is subject to debate but one thing is clear in the poem: the misconstrued notion of God is revealed to future generations.
References
Babor, T. F., McGovern, T., & Robaina, K. (2017). Dante’s inferno: Seven deadly sins in scientific publishing and how to avoid them. Addiction Science: A Guide for the Perplexed, 267.
Cheney, L. D. G. (2016). Illustrations for Dante’s Inferno: A Comparative Study of Sandro Botticelli, Giovanni Stradano, and Federico Zuccaro. Cultural and Religious Studies, 4(8), 487.
Verdicchio, M. (2015). Irony and Desire in Dante’s” Inferno” 27. Italica, 285-297.
