PHIL1012 Introductory Logic · Final Assignment · Stuvac
1.
(i) Give an example of a proposition in GPLI that is false in any model with more thanthree 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 GPLIAdd :
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 thetree, 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, andsay 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 o↵).
9x8yDyx
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
10. Define a model with exactly two objects in its domain in which all of the followingpropositions 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.
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 (inGPL) to it. Demonstrate that the argument with the additional premise is valid by presenting a tree.
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