Symbolic Logic question the rest

2. a) Suppose T is a set of premises of an argument, and S the conclusion of that

argument. Give an informal proof that, if T U {¬S} ⊢ ⊥, then T ⊢ S.

(If you're worried about what “Give an informal proof” means, then just give

a very clear and careful argument. That's all I'm looking for. A diagram can be

helpful, but isn't necessary.)

b) In this question you'll give an informal proof that the rule of →Intro is sound

(that is, never leads from true premises to a false conclusion) by answering a

series of questions.

Suppose a structure M, makes all lines of a proof true up to line n, and we

infer (P → Q) at line n+1 using →Intro. To prove: M must make (P→Q) true, and

→Intro permits us to end a sub-proof.

a) Write down a statement of the rule of →Intro.

b) We need a sub-proof to use it. What sentence is the sub-proofs first line? What

sentence is its last line?

c) What do we know about the truth values of those two sentences under M?

d) So what truth value does (P→Q) have under M? Explain why.

e) Why do we have to show, in addition to d), that (P→Q) has to be true even if a

structure, M*, makes all the lines of the sub-proof false?

f) Does M* make (P→Q) false? Why?

3) Prove the following in S5.

a) □(A →B) ⊢ (□A→□B).

b) ∃x □x=god ⊢ □∃x x=god

(No points for this, but what do the premise and conclusion of b) mean

intuitively?)

c) □A ⊢ ¬◇¬A (don't forget that you can use ⊥Elim to shift between worlds)

4) One fine evening, an observant Babylonian (probably) noticed a bright star in

the western sky and called it “the evening star” (or some equivalent in ancient

Babylonian).

Early the next morning, the heroine of our story looked at the eastern sky,

and observed a very similar looking object, called “the morning star”.

“Now” (she thought) “perhaps these are really the same thing. Perhaps the

morning star is the evening star!” Because of the similarity in brightness, and the

position of the star with respect to other stars (and the sun), she decided this was

true.

She told her friends. Much bickering ensued. Some people thought they

were different things, some thought they were the same. After much argument

and careful observation, everyone agreed that (expressed in FOL):

1) morningstar = eveningstar.

Everyone also agreed, though, that 1) was not necessary. After all, it had

been a discovery, requiring evidence. If the evidence had been different, they

would have concluded that the stars were different. So while the morning star is in

fact the evening star, as 1) says, it is not necessary that 1). So everybody agreed

that:

2) ¬ □ morningstar=eveningstar

(Comment: 'morningstar = eveningstar' is an atomic sentence, so we shouldn't

use parentheses. Still, it might make things clearer to some people to write:

¬□ (morningstar=eveningstar)

end of comment.)

The point of this question is to show you that if the morning star is in fact

the evening star, then (according to S5) it appears that it must be the evening

star. We can prove in S5 (apparently):

morningstar=eveningstar ⊢ □ morningstar=eveningstar.

You're going to do this proof to answer this question.

The assumption of the proof is sentence 1), 'morningstar=eveningstar'.

Then pick an arbitrary possible world, and begin a sub-proof to prove that

'□morningstar=morningstar'.

On the next line, you can just write down 'morningstar=morningstar' in your new

possible world. Which rule allows you to do this?

So you can end your sub-proof, and get:

3) □morningstar=morningstar

at world zero. That seems O.K. The morning star must be the same object as

itself.

But now from 3) and the first line of your proof, you can get:

4) □morningstar=eveningstar

how?

Obviously 4) contradicts 2).

You might think about what's gone wrong (if anything). You won't get any

points for telling me, alas, but this is a very important argument. It changed the

course of philosophy.