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I am asked to show this following argument is valid.

There is a politician who does not tell the truth. Every politician says we should tell the truth. Therefore, there is a politician who says we should tell the truth, but does not tell the truth.

So, What I've done until now is below

The domain is the set of politicians.

T(x) : x tells the truth

S(x,y) : x says y should tell the truth.

∃x¬T(x) = There is a politician who does not tell the truth. ∀x(x,y) = Every politician says we should tell the truth. ∴∃x(¬T(x) ∧ S(x,y)

1.∃x(¬T(x)) : Pemises

2.c∧¬T(c)) : Existential instantiation 1

3.∀xS(x,y) : Premises

4.S(c,d) : Universal instantiation 3

  1. c : simplification 2

6.¬(T(c)) : Simplification 2

7.¬T(c) ∧ S(c,d) : Addition 4,7

8.∃x(¬T(x) ∧ S(x,y) ) Existential generalization 7

My question is whether I need to define "y" or not && this processing is fine. The reason I use y without definition is I've seen a lot of solutions using y without definition. To be honest, English is not my first language, so there might be something ambiguous you guys can not understand. Let me know. I want you to be clear.

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    I don't see why you are using y at all. No politician says "Bill Bailey should tell the truth". It's All politicians say "we should tell the truth". And actually if you get down to it. What the politiician says has nothing to do with what the politician does. The proof of this would be *exactly* the same as the proof: There is a cow that is green. All cows have horns. Therefore there is a green cow with horns. It's exactly the same proof. – 2017-01-24

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I would not use $S(x,y)$ for '$x$ says $y$ should tell the truth', because in the problem we never specify or work with this $y$; it just says 'we should tell the truth' as a general claim.

So, instead I would simply use: $S(x)$: $x$ says that we should tell the truth.

That will simplify your symbolization and proof, and you don't have to deal with any $y$'s

Also:

On line 2 you say $c \land \neg T(c)$, thus treating $c$ as a claim. In fact, on line 5 you even have $c$ by itself as a claim. But: $c$ is not a claim, but a constant used to denote a politician. And a politician is not a claim!

So: on line 2 you should just get $\neg T(c)$, and you should get rid of lines 5 and 6.

In sum, you get:

  1. $\exists x \neg T(x) \qquad$ Premise

  2. $\forall x S(x) \qquad $ Premise (for organization sake we typically start with all premises on top)

  3. $\neg T(c) \qquad$ Existential Instantiation 1

  4. $S(c)$ \qquad$ Universal Instantiation 2

  5. $\neg T(c) \land S(c) \qquad$ Conjunction 3 ,4 (you said Addition on line 7, but that is typically used to go from $P$ to $P \lor Q$)

  6. $\exists x (\neg T(x) \land S(x))\qquad$ Existential Generalization 5

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    Yeah! There is Conjunction. I totally forgot using that. Thanks really helpful. However, what I'm still curious is about "y", because of the question "All insects have six legs / Dragonflies are insects/ Spider do not have six legs / Spiders eat dragonflies.", which is famous statements in logic. In this case, even though they don't use E(x,y); x eats y; at all for solution, why do they define y ?? I don't know where my question is going. BTW, can I use y as if this solution used without ∃y ∀y or something? – 2017-01-24
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    It's worth noting, I think, that although S is "says we should tell truth" and T is "tells the truth" there is no relation at all between saying we should tell the truth and telling the truth. If you replaced "politician" with "cows" and "tells the truth" with "has horns" and "says we should tell the truth" with "has tails". This proof would be: there is a cow without horns. All cows have tails: therefore there is a cow with a tail be no horns. The argument and proof are exactly the same. – 2017-01-24
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    The statement "Spiders eat dragonflies" refers to two objects; spiders and dragonflies. None of the poiltician statements refer to two objects. – 2017-01-24