Instructions: Answer all parts of all questions. Read each question carefully. The mark value of each question is shown below, and 118 marks are available. The examination as a whole is worth 50% of your final mark for the course. Please clearly label your answers, so that it is always clear exactly which question you are answering at any given point in your submission.

Questions:

1. Translate the following into GPLI:

[32 marks: 4 marks per part]

(i) Nora Jones isn’t a drummer. (ii) Either the Sun is shining or it isn’t, and if it is, I won’t need an umbrella. (iii) Only if Peter is OK with it will Paul start his engine. (iv) Neither Miriam nor Julie are sitting between Fisher Library and The Quad. (v) The Sun is the only star in the Solar System. (vi) Julius can see everything that anyone writes. (vii) Everyone except Miriam is on Facebook. (viii) I know exactly one person who has visited Chile.

2. Here is a model: Domain: {2, 7, 10, 11} Referents: a : 7 c : 2 g : 10 h : 2 Extensions: D : {10, 11} Q : {7, 10} S : Ø E : Ø L : {2, 11} M : {⟨11, 11⟩, ⟨7, 11⟩, ⟨10, 2⟩} P : {⟨7, 2, 10⟩} R : {⟨7, 7, 10, 7⟩}

For each proposition below, say whether it is true or false on the above model. Explain your answers with reference to the semantics (i.e. the truth rules) of the relevant operators. [30 marks: 5 marks per part] (i) (Sa → Mca ) ∧ ㄱLc (ii) ∀x (ㄱQx → Lx ) (iii) ∃y (Ly ∧ ∃zMzy ) (iv) ∀x ∃yMyx ↔ ∀x (ㄱRaaxa ∨ x = g ) (v) Pacg → ∀x (Sx → ∃y (Mhy ∨ ㄱㄱRyaba )) (vi) ∃x ∃y ∃z (∃wMwx ∧ ∃wMwy ∧ ∃wMwz ∧ y ≠ z)

3. Use the tree method to answer the following questions. Justify your answers. [30 marks: 6 marks per part]

(i) Is the following proposition logically true? If it is not, read off a model on which it is false.

∀x (Px → Qx ) ∧ ∃y (Py ↔ Py )

(ii) Are the following propositions equivalent? If they are not, read off a model on which their truth values differ.

a = b ∧ ㄱ∀xRabx ∃y ㄱRbay ∧ b = a

(iii) Is the following set of propositions satisfiable? If so, read off a model on which they are all true.

{Rab , ∃x (Rxb ∧ x = c ), ㄱc ≠ a }

(iv) Is the following argument valid? If it is not, read off a model on which its premises are true and its conclusion is false.

∃x ∀y (Py → Ryx ) ∀x (Sax → Px ) Sab ∴ ∃xRbx

(v) Is the following argument valid? If it is not, read off a model on which its premises are true and its conclusion is false

∃x (Bx ∧ ㄱSxa ) ∀x (Bx → x = a ) ∴ ㄱSaa

4. Questions involving a new concept:

[10 marks: 2 marks per part]

Let us say that a formula of GPLI is fickle if it is possible to take a model on which it is true, and add objects to the domain (changing nothing else in the model) so that the formula is false on the resulting model.

By ‘changing nothing else’, I mean that these extra objects must not be given names or put into the extensions of any predicates, and that existing referents and extensions on the model remain unchanged.

To explain this idea of a fickle formula in other words: Say that a model M′ of GPLI is an outgrowth of a model M of GPLI iff:

- M′ and M assign the same referents to the same names and the same extensions to the same predicates. - Every object in the domain of M is in the domain of M′.

Now, a formula of GPLI is fickle iff there is a model M on which it is true and there is an outgrowth of M on which is it false.

For each of the following formulas, say whether or not it is fickle and justify your answer:

(i) ∃z (Lz ∧ ㄱLz ) (ii) ∀xFx (iii) ∀x (Fx → Gx ) (iv) ∀x (x = a ∨ x = b ∨ x = c ) (v) ∃x ∃yx ≠ y

5. Short answer questions:

[16 marks: 4 marks per part]

(i) What is wrong with the expression ‘This argument is invalid on the following model’? (For the purposes of the question, it doesn’t matter what the argument is or what the model is.)

(ii) Give, in English, a valid argument whose validity cannot be captured in PL but can be captured in MPL. Explain why its validity cannot be captured in PL.

(iii) Give, in English, an argument which is necessarily truth-preserving but not valid, and explain why it is not valid.

(iv) In logic, we translate ‘All humans are mortal’ (and similar propositions) in a particular way. The way we translate it, it could be true even if there are no humans. With reference to the semantics of MPL (or GPL, or GPLI), explain how it is possible for the translation to be true even if there are no humans.

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