Basic Mathematics - 2021-09-27

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Pax Daga

10:24

@user21820 Why not put everything you can infer from truth tables as the inference laws?Even If some of them are not needed?

3 hours later…

user21820

13:53

@PaxDaga 2 reasons: (1) The kind of truth tables for PL that you are thinking of are not rigorous, and any proper precise system based on truth tables would have similar flavour to a Fitch-style system; (2) People who cannot even use the deductive system I gave for PL cannot possibly understand FOL.

Pax Daga

I mean can't be made rigorous with something like

A;negB

Pax Daga

B implies A

user21820

Yes you can do that, which would have a Fitch-style flavour as I said in (1), but such a system would suffer from exponentially-sized proofs for even simple tautologies unless you know how to use the original rules efficiently in the first place.

In other words, you can add more rules to the system, but they would merely function as a means of skipping steps. They wouldn't improve the system in any essential way.

Pax Daga

   Neg B implies B implies A


      If neg B:

             If B:
                   B or A
                   Neg B
                   A

I guess the last needs justification and can be proven using contradiction

@user21820

user21820

14:09

Firstly, I require you to use brackets for expressions like "Neg B implies ( B implies A )", because there is no reason to omit brackets. Secondly, you do not even need "B or A", but yes you need contradiction.

Pax Daga

Isn't implies read backwards?

user21820

Some people use that convention. I do not allow it, as it is pointless.

Pax Daga

why B or A can be omitted

user21820

In the first place, why are you not doing my exercises? Are you unable to do them?

Pax Daga

14:26

I was doing it but thought of this so decided to prove it

@user21820

user21820

@PaxDaga Ok. It's about as easy as (P3).

Pax Daga

But why can B or A be omitted I don't understand?

user21820

It's just not needed in the final proof at all. If you did up to (P3) it would be obvious how to prove it.

Pax Daga

My proof relies on the fact that if B or A and neg B then A which I proved by contradiction I don't understand how it can just be removed ?

@user21820

user21820

You did not have any proof.

Pax Daga

14:41

I'm pretty sure my proof works what am I doing wrong?

user21820

I'll say for the last time, learn to use the deductive system I gave you. You do not have a proof in that system.

Pax Daga

but you said that the statement is not needed so why would its proof be needed?

user21820

Look, you do not know PL. Therefore I do not accept any claim about proving some PL tautology from you unless you prove it in the system I gave you. If you are unable to do so, then don't claim to have a proof.

Just go and do the exercises.

Pax Daga

   Neg B implies B implies A


      If neg B:

             If B:
                   B or A
                   Neg B
                   If neg A:
                                Neg B and neg A
                                neg(A or B)
                                Contradiction
                   A

@user21820

I can't understand what's worng

user21820

14:57

If you can't understand what's wrong, then it means you didn't even bother to look at the deductive rules that are allowed.

Pax Daga

@user21820 Please tell me what I did wrong

I did look at the deductive rules

Pax Daga