Symbolic Logic question the rest


2. a) Suppose T is a set of ground of an topic, and S the omission of that

 

argument. Furnish an inaccurate testimony that, if T U {¬S} ⊢ ⊥, then T ⊢ S.

 

(If you're worried environing what “Give an inaccurate testimony” instrument, then impartial furnish

 

a very plain and mindful topic. That's all I'm looking for. A diagram can be

 

helpful, but isn't compulsory.)

 

b) In this interrogation you'll furnish an inaccurate testimony that the government of →Intro is sound

 

(that is, never leads from gentleman ground to a fallacious omission) by obedient a

 

series of interrogations.

 

Suppose a edifice M, fabricates all cords of a testimony gentleman up to cord n, and we

 

infer (P → Q) at cord n+1 using →Intro. To ascertain: M must fabricate (P→Q) gentleman, and

 

→Intro permits us to end a sub-proof.

 

a) Transcribe down a declaration of the government of →Intro.

 

b) We insufficiency a sub-testimony to use it. What passage is the sub-proofs highest cord? What

 

passage is its terminal cord?

 

c) What do we distinguish environing the accuracy rates of those two passages beneath M?

 

d) So what accuracy rate does (P→Q) enjoy beneath M? Explain why.

 

e) Why do we enjoy to demonstration, in restitution to d), that (P→Q) has to be gentleman plain if a

 

structure, M*, fabricates all the cords of the sub-testimony fallacious?

 

f) Does M* fabricate (P→Q) fallacious? Why?

 

3) Ascertain the forthcoming in S5.

 

a) □(A →B) ⊢ (□A→□B).

 

b) ∃x □x=god ⊢ □∃x x=god

 

(No tops for this, but what do the preface and omission of b) mean

 

intuitively?)

 

c) □A ⊢ ¬◇¬A (don't obliviate that you can use ⊥Elim to remove between earths)

 

4) One ostentatious plaining, an watchful Babylonian (probably) noticed a flashing bigwig in

 

the western sky and denominated it “the plaining bigwig” (or some equiponderant in ancient

 

Babylonian).

 

Early the contiguous early, the heroine of our romance looked at the eastern sky,

 

and observed a very concordant looking design, denominated “the early bigwig”.

 

“Now” (she sentiment) “perhaps these are indeed the similar art. Perhaps the

 

early bigwig is the plaining bigwig!” Because of the concordantity in flashingness, and the

 

position of the bigwig after a timeliness deference to other bigwigs (and the sun), she firm this was

 

true.

 

She told her friends. Abundant disputing ensued. Some nation sentiment they

 

were unanalogous arts, some sentiment they were the similar. After abundant topic

 

and mindful study, everyone agreed that (explicit in FOL):

 

1) earlybigwig = plainingstar.

 

Everyone too agreed, though, that 1) was not compulsory. After all, it had

 

been a thread, requiring deposition. If the deposition had been unanalogous, they

 

would enjoy concluded that the bigwigs were unanalogous. So timeliness the early bigwig is in

 

fact the plaining bigwig, as 1) says, it is not compulsory that 1). So everybody agreed

 

that:

 

2) ¬ □ earlystar=eveningstar

 

(Comment: 'morningbigwig = plainingstar' is an minute passage, so we shouldn't

 

use parentheses. Still, it capability fabricate arts plainer to some nation to transcribe:

 

¬□ (morningstar=eveningstar)

 

end of expound.)

 

The top of this interrogation is to demonstration you that if the early bigwig is in fact

 

the plaining bigwig, then (according to S5) it appears that it must be the plaining

 

star. We can ascertain in S5 (apparently):

 

morningstar=eveningbigwig ⊢ □ earlystar=eveningstar.

 

You're going to do this testimony to counterpart this interrogation.

 

The selfreliance of the testimony is passage 1), 'morningstar=eveningstar'.

 

Then enucleate an harsh practicable earth, and originate a sub-testimony to ascertain that

 

'□morningstar=morningstar'.

 

On the contiguous cord, you can impartial transcribe down 'morningstar=morningstar' in your new

 

practicable earth. Which government allows you to do this?

 

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

 

3) □morningstar=morningstar

 

at earth cipher. That seems O.K. The early bigwig must be the similar design as

 

itself.

 

But now from 3) and the highest cord of your testimony, you can get:

 

4) □morningstar=eveningstar

 

how?

 

Obviously 4) contradicts 2).

 

You capability hold environing what's bybygone crime (if everything). You won't get any

 

points for powerful me, alas, but this is a very grave topic. It transitional the

 

course of philosophy.