PHIL1012 Introductory Logic | USYD悉尼大学 | final exam代考

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PHIL 1012 Introductory Logic
Final Examination
Second Semester 2021
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) Alice likes Paris.
   (ii) Either Alice does not like Paris, or Bob does.
   (iii) Bob likes Alice unless Alice likes Bob.
   (iv) If Bob and Alice are in Paris, then someone is.
   (v) If someone in Paris has been to Shanghai, then someone in Paris is well-travelled.
   (vi) Someone likes Shanghai, but it is not Bob.
   (vii) More than one person likes Shanghai.
   (viii) At least two people apart from Bob like Shanghai only if Bob does.
   (ix) One person is Bob, but no one else is.
   (x) If at least three people pass the logic exam, then everyone will.
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2. [25 marks: 2.5 marks per part]
   Here is a model:
   Domain: {1, 2, 3, 4, 5, 6, 7}
   a : 1, b : 4, c : 6
   V : {1, 2, 3, 4}, W : {1, 2, 6}
   R : {h1, 1i,h1, 3i,h2, 1i,h2, 2i,h2, 7i,h5, 4i,h6, 1i,h6, 6i}
   S : ∅
   Say whether the following propositions are true or false in this model. Explain your
   answers briefly.
   (i) V b → ¬W a
   (ii) ∃xRxx
   (iii) ∀x(Rxx ↔ W x)
   (iv) ∃x∀yRxy ∨ ¬V c
   (v) ∀w((Rww ∧ Ww) → (w = a ∨ w = c))
   (vi) ∃y(Rcy ∧ ∀z(Rcz → z = y))
   (vii) ∀x∀y∀z(Syxz → (∃wRzw ∨ ∃v¬V v))
   (viii) ∀y¬∃zSzby ∧ ∀x((W x ∨ V x) → ∃wRxw)
   (ix) ∃w∃x(V w ∧ V x ∧ W x ∧ ¬w = x)
   (x) ∃y(Ray ∧ Rcy) ↔ ∃y(Ryb ∧ Ryc)
3. [25 marks: 5 marks per part]
   (i) Use a tree to test whether the following wff is a tautology. If it is not, then read
   off from your tree a model on which it is false.
   ∃x(Gax ↔ (Gax ∧ Gxb))
   (ii) Use a tree to test whether the following wffs are jointly satisfiable. If they are,
   then read off from your tree a model on which they are both true.
   ∃x((Kxx → Kxa) ∧ Jbx)
   Jba ∧ (∀yLyy ↔ Kba)
   (iii) Use a tree to test whether the following argument is valid. If it is invalid, then
   read off a countermodel from your tree.
   (b = b → Gba) → (Gcc → Gab)
   ¬¬∀x(Gxx ∧ c = x)
   ∴ b = b ↔ b = a
   (iv) Use a tree to test whether the following wffs are jointly satisfiable. If they are,
   then read off from your tree a model on which they are both true.
   (Hba → a = a) ∨ (∃yy = y → c = c)
   b = a ∧ ∃xx = x
   (v) Use a tree to test whether the following wff is satisfiable. If it is, then read off
   from your tree a model on which it is true.
   ∃x¬∀y∃z((x = y ∧ Ryxy) → Rxyz)
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4. [25 marks – 5 marks per part]
   Short answer section:
   (i) Give an example of a valid argument of the form
   α
   ∴ ¬α
   (ii) What is the shortest wff that is equivalent to (P a → P a) ∧ (∀xSx ∨ Sb)?
   (iii) Consider the proposition ∀x∃y(Lxy → (P z ∧ Cy)). Which variables occur free
   in the scope of the universal quantifier?
   (iv) Give an example of a wff that is false in every model whose domain contains fewer
   than three objects.
   (v) Of the logics that we have learned (PL, MPL, GPL, GPLI), which have atomic/basic
   propositions that are logical truths? Explain.
   End of examination paper
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