PHIL1012 Introductory Logic · Final Assignment
(i) Give an example of a proposition in GPLI that is false in any model with more than
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three objects in its domain.
(ii) Recall that two propositions ↵ and � are called contrary if there is no model in
which both ↵ and � are true and there is at least one model in which both ↵ and � are
false. Give an example of two contrary propositions in GPLI without using any logical
operators other than negation.
2. Suppose you have three propositions ↵, �, �, and you want to use a single tree to find
out whether there is a model in which � is true and ↵ and � have opposite truth values.
What propositions should you place at the top of your tree? Explain your answer.
3. Translate the following into MPL:
Nice dogs don’t bite unless they are scared.
4. Translate the following into GPLI:
Mary stole something from Alfred that is more valuable than at least two other things
that he owns.
5. Translate the following into GPLI:
Alfred is smarter than every logician except Ruth, who is smarter than Alfred.
6. Use a tree to test whether the following proposition is a logical truth. (Present the
tree, and say whether the proposition is a logical truth or not.) If the proposition is not
a logical truth, read o↵ a model from your tree in which the proposition is false (and
indicate the path from which you are reading it o↵).
8x8y8z((x = y ^ x = z) ! y = z)
7. Use a tree to test whether the following argument is valid. (Present the tree, and
say whether the argument is valid or invalid.) If the argument is invalid, read o↵ a
counterexample from your tree (and indicate the path from which you are reading it
a = b _ b = c
) ¬Dab ! c = b
8. Here is a model:
Domain: {1, 2, 3, 4, …}
a: 1 b: 9 c: 77
E : {2, 4, 6, 8, …}, F : ;
R : {h9, 1, 4i, h1, 6, 99i, h34, 77, 2i, h9, 77, 88i}
State whether the following propositions are true or false in the model, and briefly explain
your answers.
(i) 8x((¬Ex ^ 9zRxcz) ! x = b)
(ii) 8x9y9zRxyz $ (9yFy ! ¬Rbac)
9. Here is a model M :
Domain: {Agatha, Betsy, Carlos, David}
C : {Betsy, David}
G : {hBetsy, Agatha, Carlosi, hDavid, David, Betsyi, hCarlos, Agatha, Agathai}
State two propositions of GPL such that:
(i) both propositions are false in M ; and
(ii) both propositions are true in a model that is exactly like M except the extension of G
has a single additional 3-tuple in it. State what 3-tuple you are adding to the extension
of G in order to make both of your two propositions true. (Please note that if you do not
explicitly state what 3-tuple you are adding to the extension of G, then you will not get
any marks for this part of the problem.)
10. Define a model with exactly two objects in its domain in which all of the following
propositions are true.
8y9z(y 6= z ^Ryz)
Ma $ ¬9xRxa
8w(Mw ! Rww)
11. Suppose ↵ is an atomic proposition in GPLI and � is a contradiction. Consider the
argument form
(i) Are there any valid arguments of this form? Explain your answer.
(ii) Are there any sound arguments of this form? Explain your answer.
12. The following argument is an enthymeme (see section 12.4 in the textbook or refer
to the second lecture from week 10):
Carlos thanked everyone who helped him.
Ruth helped Carlos.
) Carlos thanked Ruth.
(i) Translate the argument into GPL.
(ii) In one or two sentences of English, explain why your translated argument from part
(i) is invalid.
(iii) Make your translated argument from part (i) valid by adding a single premise (in
GPL) to it. Demonstrate that the argument with the additional premise is valid by
presenting a tree.
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