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Final exam (The Laws of Truth Smith by Nicholas J. J. and Logic Textbook Problems and Solutions included)
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PHIL 1012 Introductory Logic
Final Examination
Second Semester 2020
Time Allowed:
Three (3) hours.
Instructions:
Label your answers clearly. Make sure that it is always clear exactly which question you are answering at any given point in your answers. Upload your answers to the PHIL1012
Examination Canvas page.
Questions:
1. [25 marks: 2.5 marks per part]
Translate the following into GPLI:
(i) Tom likes chocolate.
(ii) Tom either likes chocolate or it likes him.
(iii) If Alice likes chocolate, then either she or Tom are happy.
(iv) Someone likes chocolate and does not like porridge.
(v) If everyone likes chocolate then everything is okay.
(vi) Every alien on Mars keeps bees.
(vii) There are at least two mice who are friends.
(viii) Anyone who likes someone is liked by someone.
(ix) With the exception of Alex, everyone likes either chocolate or porridge.
(x) Some alien is friends with a mouse who is not friends with any other mouse.
2. [25 marks: 2.5 marks per part]
Here is a model:
Domain: {1, 2, 3, 4}
Referents: a: 1 b: 2 c: 3 d: 4
Extensions: P : {1, 2}
Q : {}
R : {, , }
S : {, , , }
Say whether each of the following propositions is true or false on this model. Explain your answers in your own words briefly.
(i) Sba
(ii) ¬Sab → ¬Pa
(iii) Qc ˅ Qd ˅ ¬Pb
(iv) ¬∃xQx ↔ Sad
(v) (Raa ˅ Saa) Λ ∀x(Px ˅ Qx)
(vi) ¬∃x(Rxx Λ Sxx)
(vii) ∀x(Qx → ∃y(Ryx ˅ Rxy))
(viii) ∃x(x = a Λ (Rxx ˅ Sxx))
(ix) Sad ↔ ∃x(x ≠ a Λ Sax)
(x) ∀x(Sxx → ∃yRyx)
3. [25 marks: 5 marks per part]
Using trees, determine whether the following arguments are valid. For any argument that is not valid, read o↵ from your tree a model in which the premises are true and the conclusion false.
(i) ∃y∀xRxy
∴ ∀x∃yRxy
(ii) Sa
∴ ∀x(x=a→Sx)
(iii) ∀x∀y(Rxy→¬Ryx)
∴ ∀x¬Rxx
(iv) ∀x(Px→∃yRyx)
∀x(Rxa→Sxa)
∀x¬Sxa
∴ ¬Pa
(v) ∀x((PxΛQx)→Sx)
PaΛQa
¬Sb
∴ a=b
4. [25 marks - 5 marks per part]
Short answer section: Answer each of the following in your own words in a short paragraph.
(i) What effect might the addition of premises to a valid argument have on the validity of that argument?
(ii) Why can we not use truth-tables to give the semantics of predicate logic?
(iii) Why must we use a new name when replacing existentially quantified variables along a path for a tree in predicate logic?
(iv) In PHIL1012 this semester, we have followed a general instruction that states that when we are constructing models in predicate logic, we place objects, and not names, in the extensions of non-empty predicates. Why has this instruction been in place?
(v) Does there exists an invalid argument for which a domain with only 1 object in it would be an insufficiently populated domain with regard to demonstrating the invalidity of that argument? If so, then give an example of such an argument and explain how it is that the example satisfies the description above. If not, then explain why there cannot be such an argument.
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