Logic - 2017-12-12 (page 2 of 3)

David Reed

05:00

I've literally slept 38ish of the last 40 hrs

48 hrs

user21820

Okay let's continue next time!

Have a good sleep!

David Reed

I didn't necessarily mean that I'm going to sleep, just going to be very difficult to process new information.

LastIronStar

@DavidReed get well soon

David Reed

thanks man

LastIronStar

Can a valid argument be made invalid by the addition of a new premise? @user21820

from forallx

user21820

05:05

@LastIronStar No a valid argument is always valid, and if any extra premise contradicts the conclusion, then one of the premises (possibly the extra one) must be wrong.

LastIronStar

@user21820 But if the premise i add is NOT(Conclusion) is true then the argument gets invalidated right?

because in the case that all the premises are true, due to the new one, my conclusion is now false

thus it's an invalid argument

David Reed

hrm

LastIronStar

on a lighter note, is it self-referential to argue about arguing?

user21820

No. "Valid" means "the conclusion is necessarily true if the premises are all true". If originally you have a valid argument, adding premises does not make it invalid.

@LastIronStar Yes.

David Reed

you would have the conclusion be both true and false i think

user21820

05:08

Yes too.

LastIronStar

forallx defines VALID argument as one for which it is impossible for all the premises to be simultaneously true whilst the conclusion is false.

user21820

Yes that is correct.

If you add a premise that contradicts the conclusion of a valid argument, it simply means that the premises now (including the new one) cannot be all true.

LastIronStar

so, i have constructed a possible addition of a clause by construction that makes conclusion false.

whilst the added premise is stilltrue

user21820

I don't get you. Do you have an explicit example?

LastIronStar

i have a meta-example if you're interested

user21820

05:09

If your added premise is true, but your valid argument gives the opposite, then one of your earlier premises must be false, by what I just said.

LastIronStar

why is that the case?

user21820

That is the nature of logical deduction.

LastIronStar

I see where you are going but i'm not quite convinced

you pulled out the BIG guns :P

user21820

It guarantees that if you start with only true premises then you can deduce only true conclusions.

LastIronStar

we are talking about VALID arguments which have a specific definition

user21820

05:11

It guarantees nothing more than a headache if your premises are not all true. (Just joking.)

David Reed

is your question regarding whether an argument is valid if its impossible for all of the premises to be true?

LastIronStar

hahaa

no

I think if even one of your premise is false then argument is automatically VALID

user21820

No.

LastIronStar

ofc

user21820

That is not the case for forallx's definition of "valid".

LastIronStar

05:12

4 mins ago, by LastIronStar

forallx defines VALID argument as one for which it is impossible for all the premises to be simultaneously true whilst the conclusion is false.

user21820

Yes. Using his definition, it is not true that you have a valid argument if one premise is false.

This goes into meta-logic, so no point going there now.

LastIronStar

ok

David Reed

then adding the negation of your conclusion would preserve validity I think

LastIronStar

exactly!

that's the meta-example i was giving

David Reed

I feel like that's 820 already said

LastIronStar

05:14

no he is saying that if we do that, one of the premises HAS to be false now

I don't quite follow that bit

user21820

That just follows from the definition of valid argument.

Because of all the new premises are true, then the argument must only give a true conclusion, but then you have both something and its negation being true.

David Reed

it would impossible for a cogent argument of that form

user21820

That cannot be, and hence not all the new premises are true.

Actually when I teach basic logic I do not teach the notions of premises and conclusions, as they are in my opinion slightly misleading. I instead insist that there are absolutely no assumptions in every proof.

LastIronStar

Ok, let me specify me example more carefully

Premises are as follows: A is true, B is true, C is true; Conclusion D is true.

Now add a new premise NOT(D) is true

then conclusion is false now

user21820

@LastIronStar This last line is not a valid claim.

You will see later in chapters 6−7 that Fitch-style does not have such free-floating premises, just as I prefer it. So I'm going to rewrite your example to make that clear.

You have:

LastIronStar

05:19

please do

user21820

If A and B and C:
  D.

Right?

Notice that I purposely indent to show that D holds only within the context where A and B and C hold.

LastIronStar

well, yeah, the definition is the exact contrapositive of what is defined as VALID

so it works out

user21820

This is the true meaning of the argument you are referring to.

Now consider what happens when you so-called 'add a new premise':

If A and B and C and not D:
  D.

LastIronStar

well, how do you know there is a context?

I think i disagree with your example actually

it changes the meaning of what it means for an argument to be VALID in the forallx sense i fear

user21820

@LastIronStar This is the underlying idea of logic. You can consider any context you like. Whether or not that context can ever be realized is irrelevant.

@LastIronStar No it doesn't.

As I said, this is exactly what is done in chapters 6−7.

LastIronStar

05:22

I agree VALID arguments are not necessarily SOUND arguments

user21820

The problem with the earlier chapters is that they follow the tradition of "premises ... conclusion" format, which in my opinion is bad.

LastIronStar

ok, so you recommend i can skip ahead safely?

and then come back to these chapters at a later stage?

user21820

There is no need to read the earlier chapters, unless you need motivation for logic. But actually you should think carefully about what I just said, and you will find that there is no disagreement with what he said.

He did not seem to make an error despite following tradition.

Just treat a context as simply some situation where you can stipulate additional conditions hold.

Then if you have a valid argument from premises A,B,C to conclusion D, you must agree with what I wrote in context form.

6 mins ago, by user21820

If A and B and C:
  D.

This one ^.

It says nothing about whether the context (situation) can ever occur.

LastIronStar

'If A && B && C, then D.'

is the argument

user21820

Yes.

LastIronStar

05:27

right?

user21820

Well, more or less.

LastIronStar

Consider If NOT(D), then D - is this a valid argument?

user21820

No.

LastIronStar

exactly

now add any context you want

user21820

You don't get it.

LastIronStar

05:28

like A or B or C or alltogether

user21820

7 mins ago, by user21820

If A and B and C and not D:
  D.

Do you agree that this is valid. Notice that the original valid argument only relies on premises A,B,C to be true to guarantee that D is true.

So adding extra conditions merely restricts the context (situation). D still must hold in a restricted context if it already did without the restriction.

LastIronStar

this is not a valid argument!

user21820

It is valid. You're missing the point.

LastIronStar

I agree

user21820

Think slowly. The situation in which A,B,C,not D all hold is a more restricted situation than that in which A,B,C all hold.

If something is true in the latter situation, it must be true in the former.

LastIronStar

05:31

Where did former and latter come from!?

i.e., temporality

user21820

53 secs ago, by user21820

Think slowly. The situation in which A,B,C,not D all hold is a more restricted situation than that in which A,B,C all hold.

^ refers to a former situation and a latter situation.

LastIronStar

Ok, The situation in which A,B,C, NOT(D) all hold is a more restricted situation than that in which NOT(D) holds.

how does it matter the order in which we add the premises?

user21820

@LastIronStar This is not what I said.

LastIronStar

It is what i'm saying from what you gave

user21820

It's true but irrelevant.

4 mins ago, by user21820

Do you agree that this is valid. Notice that the original valid argument only relies on premises A,B,C to be true to guarantee that D is true.

We can deduce D in any context where A,B,C all hold.

This is the meaning of the validity of the argument.

LastIronStar

05:34

:(

VALID argument - an argument where it is impossible for conclusion to be false given that premises are true. Agree?

user21820

Yes.

As I said, the premise-conclusion format is simply misleading you.

LastIronStar

ok, now the example starts off as a valid argument.

user21820

So I suggest you forget about that format and switch to my format instead.

LastIronStar

i.e., it is known that it is impossible for D to be false given that A, B and C are true.

user21820

Yes.

LastIronStar

05:38

consider the following argument:

it is known that it is impossible for D to be false given that A, B, C and NOT(D) are true.

we need to now establish VALIDITY of this argument given the previous argument

user21820

@LastIronStar This by definition says that the argument is valid.

So what is your point?

LastIronStar

Suppose A,B,C & NOT(D) are true, this means that NOT(D) is true, this means that D is false. Therefore, the only way out is if we can deduce that either A or B or C is false in this case.

user21820

No you missed out another possibility.

LastIronStar

oh?

user21820

As I said, the premise-thing is confusing you, because you think that the premise is true.

LastIronStar

05:42

in any case, this possibility is enough to show that the argument is INVALID

user21820

No.

LastIronStar

no, i'm not saying that premise is absolutely true...

user21820

Just stop saying that the argument is invalid when you don't get it.

LastIronStar

I said suppose if it were then etc,.

haha sure, i can oblige that

user21820

The argument's validity is a totally separate matter from whether any sentences are true or false. For now I think it is better to use my format first to understand this issue before you continue on.

In my format you are not allowed to write any premise at all!

LastIronStar

05:44

btw we can put a pause and continue later if you need to go.

user21820

You can only create contexts where more conditions hold inside than outside.

For example:

If A:
  A.

The rule is that besides creating subcontexts, you can only write down a sentence that is true within its context.

LastIronStar

where you are using IF and :, I'm using suppose and then...

user21820

No do not think "suppose".

That's where the mistake is coming from

LastIronStar

whats the difference?

user21820

The above says that in the subcontext where A is true, we make a statement that A is true.

The difference is that we shall never suppose anything.

LastIronStar

05:46

isn't If the same thing

If something is true then ...

user21820

You will soon see that it's not the same thing.

First do you agree with this

2 mins ago, by user21820

If A:
  A.

Does it satisfy my rule, namely that every single sentence I write is true in its context.

LastIronStar

If they are the same, then what's there to disagree?

user21820

17 secs ago, by user21820

Does it satisfy my rule, namely that every single sentence I write is true in its context.

LastIronStar

@user21820 Ok, i think it is best if I take you on faith and read ahead...what are your thoughts on this?

user21820

It's important to follow my questions. Do you agree that I wrote only true statements in their context?

LastIronStar

05:49

cos frankly i'm missing the essence of the approach you are advocating is what i feel not that you are right or i am right or anything like that.

@user21820 please restart the example.

user21820

In the following:

5 mins ago, by user21820

If A:
  A.

LastIronStar

Can you verbally explain this?

user21820

Is it true that every sentence I wrote is true in its context?

We indent to indicate what context is governing each sentence.

LastIronStar

I don't know since i'm not aware of what context you are speaking about

user21820

"If A:" defines a subcontext, where A holds.

LastIronStar

05:52

basically you are scoping a block kind of deal?

user21820

The indented "A" indicates that it is stated within the subcontext defined by "If A:".

LastIronStar

you can use parentheses, i think it will be clearer

user21820

I'm using Python notation, and it's easier than using something else like C/Java braces.

Just follow my notation now. You can use whatever you like later.

LastIronStar

OK

user21820

So I repeat my question. Based on the meaning I've explained for my notation, in what I just wrote, is it true that every sentence I wrote is true in its context?

Yes/no.

LastIronStar

05:55

Yes, provided that context is supplied apriori

i.e., before indenting

user21820

Yes we are allowed to write context headers like "If A:" as well as write sentences/statements within them.

The essence of logic is that I shall ensure that this remains true throughout every proof.

So let's see another example:

LastIronStar

yes that would help

user21820

If A:
  If B:
    A.

LastIronStar

the If B: is moot?

user21820

I will call a proof valid as long as it satisfies my rule, namely that every sentence in it is true in its context.

So the question is, is the above valid?

@LastIronStar Yes it is intuitively moot, but is the proof valid?

LastIronStar

05:59

No

user21820

Why not? I had 2 context-headers and only one sentence.

Are you saying that sentence is not necessarily true in its context?

Remember the meaning of subcontexts; it merely restricts the current context by imposing additional constraints.

So if you agreed with:

LastIronStar

yes, in the context given by B, the proof fails if B is false since we fail to conclude A is true even though A is true from teh beginning

user21820

Then you missed the point of contexts.

LastIronStar

scratch that

whenever A is false, A & B can't be both true - is this the verbal equivalent of what you are saying?

user21820

If A:
  [In here we have the constraint that A holds.]
  If B:
    [In here we have the additional constraint that B holds.]

In that innermost context, both A and B hold. So it is perfectly valid to state "A" in there.

Do you get this?

LastIronStar

06:04

I think so, each additional inner layer is an additional constraint

user21820

Exactly.

That is the meaning of contexts.

Now another example:

If A:
  If not A:
    A.
    not A.

LastIronStar

This argument is valid

oops

user21820

It is valid!

Notice that the fact that it is valid (we only write statements that are true in its context) implies that the innermost context can never occur!

LastIronStar

:@

yeah that was what i thought first

user21820

This will happen in proofs in general.

Consider the following:

LastIronStar

06:06

8 mins ago, by user21820

I will call a proof valid as long as it satisfies my rule, namely that every sentence in it is true in its context.

what about this then?

user21820

Okay let's just go through this slowly to check that it works.

LastIronStar

are you saying sentences are the inner most assertions alone and not the constraints as such

user21820

In the innermost context, both "A" and "not A" hold, and so it is indeed the case that either of them is true in that innermost context.

The point is that the rule does not care whether the context can actually occur.

It only cares that if the context occurs, then what is stated inside is true.

LastIronStar

OK

user21820

Anyway I got to go a while. But I want to show you how it occurs in a proof by contradiction.

LastIronStar

06:10

conditioned on the constraint(s) are the assertions true?

essentially

if the answer is yes, then it is valid

user21820

If ( not B ) and ( A implies B ):
  If A:
    A.
    A implies B.
    B.
    not B.
    Contradiction!
  not A.

@LastIronStar Essentially, a sentence only makes sense when taken together with its context.

Consider the above. It is a valid proof. It goes by contradiction.

I'll be back later!

LastIronStar

OK

you can ping me once you're back, interested to get this idea properly

ttyl

Secret

06:42

@DavidReed yup

David Reed

interesting

Ive heard from reliable authorities all of the following:

they always do

they never do

sometimes they do and sometimes they dont

user21820

07:27

@LastIronStar I'm back for now.

LastIronStar

@user21820 Hi, I will be here for 10 mins then back after some time.

user21820

Well so did you get the proof above, which is often called a proof by contradiction?

LastIronStar

@user21820 I didn't get a chance to read it yet

user21820

Let's move the chem discussion to Secret's room, since Secret also likes Chem, and I like tidy rooms. =)

LastIronStar

sure

user21820

07:39

→ 6 messages moved to Mathworks (Not the main chat!)

→ 1 message moved to Mathworks (Not the main chat!)

@LastIronStar: Okay take a look at the proof.

LastIronStar

@user21820 yeah i'll do that. Leaving now to meet my advisor.

user21820

Oh okay.

Just ping me after you've looked at it then.

See you!

LastIronStar

yes!

ttyl

LastIronStar

13:06

@user21820 I'm looking at this.

user21820

13:18

@LastIronStar Okay so check line by line whether it is a valid proof.

If you are not sure about any line, just say so.

Sorry got to go. Basically, I think you should have no problem with all except perhaps the last two lines. The line "Contradiction" can be ignored. The next line "not A" is still inside the outermost context but outside the innermost context. The "B" and "not B" inside the innermost context under "If A:" show that the innermost context is impossible.

We are dealing with classical logic, where in any context "A" is either true or false, but not both. Therefore the final line is valid. Let me comment inside the proof itself.

If ( not B ) and ( A implies B ):
  [In this context either "A" is true or "not A" is true.]
  If A:
    A.
    A implies B.
    B.
    not B.
    Contradiction!
    [In this context we deduced that "B" and "not B" are both true, which is impossible.]
  not A.
  [Since the subcontext where "A" is true is impossible, "not A" must be true.]

@LastIronStar: Okay I got to go. See you!

LastIronStar

16:27

hey sorry, hectic day @user21820

Ovi

17:34

Hello @user21820

Time for me to try that sum you gave me :D let me look it up

David Reed

18:31

@user21820 Thank you for your feedback! All atoms in a benzene ring undergo hybridization. I asked secret because he is a chemist. I'm sorry regarding off-topic nature of the question. I assumed it was ok as you had asked me about quantum the other day. I will endeavor to keep conversation here strictly on logic

Leaky Nun

21:34

@user21820 I must be missing something obvious

16 hours ago, by LastIronStar

forallx defines VALID argument as one for which it is impossible for all the premises to be simultaneously true whilst the conclusion is false.

16 hours ago, by user21820

Yes. Using his definition, it is not true that you have a valid argument if one premise is false.

16 hours ago, by LastIronStar

Premises are as follows: A is true, B is true, C is true; Conclusion D is true.

16 hours ago, by LastIronStar

Now add a new premise NOT(D) is true

David Reed

The general definition of validity in logic is independent of the truth of the premises

you can have all false premises and still have a valid argument

Leaky Nun

@DavidReed please let me type before you reply

David Reed

a valid argument in which all the premises are true is called a cogent argument

sure

Leaky Nun

this isn't the first time

David Reed

I'm sry

Leaky Nun

21:37

Dec 9 at 4:07, by Leaky Nun

Patience, @DavidReed

@user21820 @LastIronStar

"Yes. Using his definition, it is not true that you have a valid argument if one premise is false."
this is wrong. if a premise is false, then the premises can never be simultaneously true, so the criterion of validity is vacuously satisfied, so it is true that you have a valid argument.

"Premises are as follows: A is true, B is true, C is true; Conclusion D is true."
this isn't a valid deduction to start with, because it is possible for the premises to be true and the conclusion to be false. so this nullifies your example

in addition, the criterion "it is impossible for all the premises to be simultaneously true whilst the conclusion is false" follows from the semantic-completeness theorem

it says that if A ⊢ B, i.e. from A you can deduce B logically, then you have A ⊨ B, i.e. B is true whenever A is true, or i.e. it is not possible for A to be true and B to be false

so that definition of validity is valid

@DavidReed now you can reply

David Reed

Sure, what is it that you are not following?

I haven't read forallx

I'm speaking about the principle of validity of an argument in logic in general, not an argument expressed in a logical system

LastIronStar

@LeakyNun ofc, we assume that the given argument is valid in the beginning since "Can a valid argument be made invalid by the addition of a new premise?" is the question under discussion

@DavidReed it's pretty neat

Leaky Nun

@LastIronStar my answer is no, and that your alleged counter-example is wrong because your argument isn't valid to start with

David Reed

If my name is Jacob, then it is currently august. My name is Jakob, therefore it is currently august

this is a valid argument

but not cogent

it's not possible for the premises to be true and the conclusion false

simply because its not possible for the premises to be true at all

LastIronStar

@LeakyNun what issue do you take with my example's valid argument that is being assumed?

Leaky Nun

21:48

> "Premises are as follows: A is true, B is true, C is true; Conclusion D is true."
this isn't a valid deduction to start with, because it is possible for the premises to be true and the conclusion to be false. so this nullifies your example

your argument isn't valid to start with

LastIronStar

IF it is valid is the starting point.

Leaky Nun

you're saying you have an argument that is valid, but that becomes invalid once you add a premise

i'm saying your argument does not satisfy the property that you claim it satisfies

LastIronStar

can you be more specific?

Let me repeat my example:

David Reed

Leaky just add he premise "if A and B and C are true, then D is true"

i mean ironstar not leaky

your argument then becomes valid

LastIronStar

oh i got it now

thanks david

Leaky Nun

21:52

@LastIronStar are we in agreement?

LastIronStar

@LeakyNun Not yet i think

Leaky Nun

great

go on

17 hours ago, by LastIronStar

forallx defines VALID argument as one for which it is impossible for all the premises to be simultaneously true whilst the conclusion is false.

LastIronStar

Assume that "if A and B and C are true, then D is true" is a valid argument. Can you tell me which part is premise and how many of them are there?

David Reed

I think his question is would the following be valid:

If A,B,C are true then D is true. A,B,C are true, D i false, therefore D is true

Leaky Nun

@LastIronStar why should I assume that it is a valid argument?

David Reed

21:55

prior to the introduction of not D there are 4 premises

LastIronStar

For the sake of argument

David Reed

"If A and B and C are true, then D is true"

A is true

B is true

C is true

Leaky Nun

@LastIronStar you're trying to show that the following is wrong: "you can't add a premise to a valid argument to make it invalid"

LastIronStar

because we are interested in the following question: Can a valid argument be made invalid by the addition of a new premise?

Leaky Nun

if you want to show that, then you need to show me an actual valid argument that you can make invalid by adding a new premise

instead you showed me one that is actually invalid and ask me to pretend that it's valid

LastIronStar

21:56

@LeakyNun ofc not!

Leaky Nun

@LastIronStar then what is "For the sake of argument"?

LastIronStar

@LeakyNun Let me finish

So this is the question: Can a valid argument be made invalid by the addition of a new premise?

2 mins ago, by Leaky Nun

@LastIronStar you're trying to show that the following is wrong: "you can't add a premise to a valid argument to make it invalid"

note that it is not the same as just above

in that it is a different phrasing

Leaky Nun

tell me when I can reply

Transcript for

♦ $Mathematics$ Logic

Transcript for

♦ Logic

♦ $Mathematics$ Logic