if we have two quantities, say $a$ and $b$ and take their product, $ab$, then are they proportional?

hello is somebody here

@WilliamOliver are you ther

@RajendraBalliwal Yah

thanks

i want to ask about logic

Idk if I'll be able to answer

its just foundational

But as they say "Don't ask to ask, just ask"

i would try to tell you what i understand because in that case you could tell where i am not ok

firstly , i understand logic as a way to tell when a given statement by a person is right or wrong

now to tell when the statement is wrong or right we have many circumstances , but the one that decides that he is wrong or right is the current moment

say somebody says " today is monday" to tell when he is wrong i look the current day in newspaper and say he is telling wrong

generally we use compound statement - which are made up of more than one statement

but to classify all possible statement ( argument more preferaably - this is one problem that i will ask further ) we have come down to have onyl five logical operators that could be used to connect all possible compund statements ( not, and , or , implication and double implicaiton )

@WilliamOliver so am i right till here ?

i take it as yes ( if not ping me )

I am no expert but sure. Assuming the law of the excluded middle.

Now somebody says , " if a man is tall , then he is powerful " .

"i understand logic as a way to tell when a given statement by a person is right or wrong" should also be "A way to tell when a given statement by a person is right or wrong given that another set of statements is true"

ok , i will tak that . now we take this statement , and deduce the possible way in which it can behave depending upon the connective and the atomic statements its made up of hence we draw its truth table

by looking truth table we can tell when the statement will be true

after analysing we look at the current situation and then look at the truth table and accordingly we say is it true or false

Now here my confusion starts

@WilliamOliver i need you to tell me here where iam wrong

is a statement is primarily an argument ?

Truth table of what?

truth table of the compund statement

ok before going to argument , i want to ask one thing

A statement is a statement. An "argument" or "proof" would be transformations on that statement based on the rules of the formal language. Or something like that

so how we tell when we a person speakes argument or statement ?

An argument I guess would be a bunch of statements

With the rules of the formal language listed I suppose

Again I am not an expert

ok i will take argument problem for sometime ,

so if you look at " if a man is tall , then he is strong " - by logical connective " If .. then" we mean that the statement is false when a case like " a person is tall and he is not strong occurs "

?

Yes

ok

one more thing ,

If that is the case then , we know that we can find " a person that is tall but not strong" and this contradicts the statement " if a man is tall , then he is strong " because this statement says it can not happen ( false ) that " a man is tall and he is not strong " hence both contradicts thus we says that ,

the statement " if a man is tall , then he is strong " is not a right logical impication that is of we can not combine these two by logical connective "implication " and thus we need some other connective to show relationship between these two ?

and since this impication does not hold we say this is statement is wrong

Yes, but there might not exist anything connecting them.

ok , that is not the problem , i want to ask that am i thinking right ?

yes

so whenever somebody says to me that " if a person is tall , then he is strong " , i say you are wrong because in my mind i try to formulate total momentarily conditions for this statement and then try to analyse how the statement behaves with actual conditions and if i find something that makes statement hold different value for some condition which in real life have different value i say your statement is false ?

The way to show that it is wrong is either to find a counter example or prove its negation

ok ,

if you could hold for just a moment i have one more thing

an argument is just a conclusion that is followed by statements ( premises)

so the connective that justifies the math of an argument is implication

i.e if premised are correct then argument can not be wrong "

Yes

which when translated have a connection of implicaiton

The way to think of it is really in terms of a formal language. You have rules which visually transform statements from one to another.

This is implication typically.

you know theory of automata ?

Vaguely

yes, yes, yesssssssssssssss

i also have doubts among that m but small

but coming to argument s

What?

like doubts in atutomata ?

please for the moment hold on to argument , like iam coming close to resolve my doubts

so when somebody says an statement we take that as conclusion that has been made over momentarily circumstances

which makes that the statemetn is argument

If someone claims the statement is true then yes. Unless it is an axiom

but when i am asked in question that , should i assume it as true ?

Sorry, I couldn't quite understand that question, could you rephrase it?

likw when in questions a statement is written then we assume it as true

and then by reasoning we deduce whether it holds true in every cricumstance

You mean like when a math book asks you to prove a statement?

yeah

Oh well, that scenario is not really realistic in actual logic. You aren't supposed to assume its true or false really. But in the case of a math book, someone has already proven it.

Technically, you don't assume a statement is true or false until you prove it

thanks , i think i have reduced one confusion about statement but for argument i need to self analyse to come up with doubt because i have an itch but dont know where in arguments hence i need to find it

and then i will ask you

but i dont know how long will you be here

I dont either. Generally in these chats you just ask your question to the general public and whoever answers answers.

in that case , i genuinely thank you for your time and patience for helping me as it has helped very much

No problem!

@WilliamOliver

here iam and still the same place

i asked it somewhere else but since you are here its ok

Like when somebody says " if a man is tall , then he is strong " then what comes to your ming

*mind

like what you do ? like do you start analysing it that whether the statement is right or wrong

and if you do so then how would you do it

like do you start taking cases when the statement is true or false by looking around facts that related to this statement

@Noob The whole "truth table" thing I think is tripping you up. The idea is just to look at all of the rules of the formal language that are assumed in your axiomatic system. And then either prove the statement true by using them to morph the axioms into the statement "If a man is tall, then he is strong", or you can prove it false by morphing them into the statement "There exists a man who is tall, but not strong" or morphing them into the statement "If a mean is tall, then he is not strong".

yeah i think i had a long talk , as you know and i know better what i did sometime yeaterday

@WilliamOliver i did not get the reply of " death occur = heart stops " is true ( = denote biconditional ) and this denotes much more precisely

$P = Q$ is the same as $P \iff Q$

yeah i know this

ans that is why i suggested this

can you show some thoughts over this

and does this find appealing to you that

heart stops => death occur

is false beacause " heart stops but person does not died " if true

*is true as opposed to false ( because we used implication )

sorry soryy

its false as suggested " " heart stops but person does not died " and that is what implicaiton tell s

hence in terms of how we evalutate implication this is true

but it specifically becomes false of " heart does not stops but person died " which is false ( f implies true is true according to implication )

thus this implication is false not because it does not hold for "true implies false " but for " false implies true"

thus checking whether an implication is false just by checking for " true implies false " does no talways hold true

?

@WilliamOliver

I'm sorry, I am having trouble following what you are saying. Are you a non native english speaker?

Lets take a step back, tell me first what you are assuming is true under this axiomatic system

no i understand english perfectly

ok keep fresh i will write it again

heart stops => death occur

i will make it fresd

If you are a native english speaker, try and take a little more time with your words. Its often hard to understand what you are saying.

ok i think i ate some words there .

← 119 messages moved from Logic

ok

wo we are here

@Noob Yup. The other room is (as the room description says) for the study of mathematical logic, not (the use of) basic logic.

So feel free to continue here.

the implication " heart stops => person died " is false if " heart stops and person does not dies " occurs

which is possible to happen

thus the implication is fale

@Noob Err, no, possible does not mean true.

i think i does

* i think it does

false means not correct

You're wrong.

@Noob What @user21820 is trying to say is that "possible" generally means that "you don't know for sure"

Let's use simpler examples.

Is "x=1" true?

Is it false?

Or is its truth value dependent on the context which ought to specify what x is?

oh oh , yeah that is correct and as " innocence untill proven guilty"

No that is wrong too. "Innocent until proven guilty" is not a fact.

If you don't provide a context that specifies x, then "x=1" is meaningless. If you provide a context that specifies x, then "x=1" is either true or false.

but that is used to deduce implications?

No.

@Noob The point is that every statement in logic is either assumed or derived. You have to state what you are assuming to be true first.

oh oh

@Noob In logic there is no a priori truth that exists in the real world. That is the realm of empirical science, not logic

so when somebody says to me " heart stops => death occurs " what should i use to tell whether the person is telling lie or truth

@Noob You have to decide that.

Anyway I'm sorry I don't have time this few days, but just remember that a statement only has a truth value within a context that is sufficiently precise to make everything in that statement well-defined.

@Noob So your quoted statement is meaningless unless you provide a sufficiently precise context.

It's not even grammatical, by the way.

It's analogous to "If x = 1 then y = 2.".

Which is only meaningful in a context that specifies x and y.

what about " if it rains , soil becomes wet "

" if water rains , soil on the earth becomes wet "

@Noob Please put in more effort. You didn't specify a time, and which soil.

@Noob The point is, you always need to assume some statement to be true without proof. In logic, you have to take some statements on "faith" for lack of a better word.

ok

I hope you get the point, because precision is key to everything in logic and mathematics.

And because I really have to go now...

but some statement which are tautology does not need any statement

@Noob A tautology is simply a statement that is true in every context.

Such as "1=1".

@user21820 That is wrong

they are just a conneciton of logical connectives which is true irrespective of its atomic statement

You need to assume that too

You need to explicitly state what you are considering a tautology

@WilliamOliver No you're wrong... What I said applies to any (desirable) formal system. What you want to say only applies to a particular idiosyncratic system.

@user21820 The concept of a "formal system" in itself has axioms. "a = a" is one of them.

@WilliamOliver No that's wrong.

You're only talking about a system for FOL.

I'm not.

@user21820 i think i am getting your point, but in logic we state a proposition now that proposition can be true or false depending upon the momentarily facts the are presented

but to tell whether a given proposition is true or false is a general idea to connect simple atomic statement in such away that it align with the current facts

@Noob Um if you're arguing that "1=1" isn't a tautology just because you don't accept equality (just like that), then I should make clear that I consider classical logic completely correct (when used correctly).

and if the connection of simple atomic statement fails to address current facts then we say that statement is false

Wait I mixed up who's talking.

yeah

@user21820 These nitpicks are more confusing than helpful I think. Even in classical logic, "a = a" was assumed and explicitly stated. Its the first postulate of Euclid's elements for example.

@WilliamOliver It's necessary to distinguish it. There are purely logical tautologies that are true in every reasonable formal system.

The non-logical axioms are needed to describe various structures, and only those are explicitly stated as axioms in every modern logic text.

Anyway I'll come back another day, otherwise I'll end up spending an hour here haha..

@user21820 when somebody says to you a statement how do you deduce it

in normal procedure

@user21820 But what counts as a tautology is a matter of opinion and changes over time. Tautologies shouldn't really be entering this conversation IMO.

yeah so please help me

@Noob Lets look at an example from history, Euclid's elements

i have been kinda left out among two mighty forces

ok

@WilliamOliver i think what you are saying is that when somebody or somewhere if a statement is given without any supportive statement then it is assumed that all the things that you have seen are the information you need

like " if it rains , then it gets wet " is true because i know by my experience that when " it rains the ground does not get wet "is false

Kind of, but not really

thus i say the implication holds

Actually, lets use Peano arithmetic, its simpler to explain

ok

Lets say Peano wants to prove the statement 2 + 2 = 4

k

gyazo.com/ea5e03bac9f6de085bbdf0111b212a6f Before he does anything, he assumes the following statements are true a priori, without proof

that is kinda enlightling

so to tell whether something is right or wrong supportive assumption are taken

So, he starts by defining what "2" is in this axiomatic system. He defines 2 to be "S(S(0))"

in logic we need some prior argument to deduce whether something is right or wrong

And he defines 4 to be "S(S(S(S(0))))"

@Noob yes

i got your point

Note that these don't necessarily have to correspond to anything in the real world

As long as they don't contradict each other

that also means that sometime we may came across some absurd statement but given the assumption they might me correct

@Noob yes, and this has actually happened.

For example, Euclid assumes the "Parallel Postulate" which states that, for given a line L and a point P, there is only one line passing through P which is parallel to L.

In reality, this postulate turned out to be false. But Euclid's Elements is still logically consistent.

so when a statement in logic how i am supposed to to know the assumptions that it has made

You use wikipedia haha

also how " parallel postulate " is fale it looks fine untill you consider 3D

@Noob Thats a complicated question, but basically spacetime is bent, so lines are different.

For example, Lines on a sphere which are parallel actually intersect each-other eventually.

But thats not the point

i think i got it

@WilliamOliver which also leads to one question that implication is defined in some way but still why we take it true then " p = false "

now is it a possibilty that in a implication " p=false , q= true " leads to falsification of statement

to support it look at this statemtn " death occur => heart stops"

It is possible that you can assume axioms which are not consistent with each-other if thats what you are asking.

"death does not occur but heart stops " P= false , q = true

by implicaiton it is true but using wikipedia we know that its false

thus we have made an implication false by look " p = false , q=true " = false

Well thats not exactly what I meant by "use wikipedia"

rather than conventional " p= true , q = false " as true

In showing the statement "death occurs implies heart stops" is false, you would have to come up with your own set of axioms.

And then using those axioms construct a counter example or show that a counter example exists.

which means iam right here

ok i am wiping out any thing , i know

?

now tell me how to tell when a statement is true or false

At this point I am talking in circles

Hey guys. Let's say you want to measure the rate at which a space is filled

look give me a statement and i can tell you that your effort is not gone in vein

but what you want me to tell is very deep and that is why i fall apart sometime but i will catch up iam trying my best

@Noob To show that "death does not occur but heart stops" you first need to do what Peano did, or Euclid did, and come up with with a reasonable set of axioms that can let you define "death" and "heart" and "stops"

How would one go about doing this?

@Ultradark actually this is a good example of what I am talking about with @Noob. You need to give more information about what you are assuming. You need to define what you mean by "measure" and "space" and "fill" and "rate".

Otherwise I can't answer your question

ok , i got it actually i got it when you talked first time

but the very statement stuck with me and the statement also seemed to be ambigous thus it stucked very badly

@WilliamOliver I posted a question about it but it may be too advanced for this chatroom

@WilliamOliver thanks and loved you thanks thanks

the basic idea is filling a unit sphere with congruent shapes and trying to measure the rate at which the space is filled

but do not go , as i will have more problems

@Ultradark congurent shapes of what

american footballs

You would measure the rate its filled in footballs per second? Haha

I guess, assuming you add $1$ football into the sphere per second

I think you still need to define the problem more, what are you looking for? How are the balls filling the space? Is it the optimal space filling?

Well, the two points of the football lie on opposite ends of the sphere. The balls are evenly spaced apart

I don't think it's optimal space filling

Whats the question you posted

I only asked about the volume enclosed in this question

not the rate at which the space is filled

So you have footballs centered at a point, with a width $2R$?

yeah, and the centers of the footballs would be the center of the sphere

might be a problem for a computer honestly

It's certainly an interesting question. How are you thinking of the footballs?

Do you mean the equation for the footballs?

Yeah, something like $((-u^2 + r)sin(t), (-u^2 + r)sin(t), Ru)$?

I'm just taking a quarter arc of a unit circle and rotating it around

Oh so just taking the unit sphere and stretching it/rotating it essentially?

Well, the problem seems to be equivalent to finding the volume enclosed by $k$ spheres that intersect

This is 2D but see the two football shapes?

Oh I see

Oh so the footballs aren't all centered at the same point?

No they are all centered at the same point, I'm looking at the two intersecting footballs in the center

not the ones outside

Ooh I see

so actually i should have said $2k$ spheres that intersect

because for every $1$ football there are $2$ spheres

So say the football rotated $\theta_0, \phi_0, \gamma_0$ is parametrized by $s_0(t, u)$ such that $s_0(t, u) = M(\theta_0, \phi_0, \gamma_0)(f(u)cos(t), f(u)sin(t), u)$ and $-R < u < R$, I am pretty sure the volume of $s_0$ is $\int_{-R}^{R} \pi f(u)^2 du$.

Oh wait ugh the approach I was trying to use won't work I think

Oh, why not?

Oh wait actually maybe

The idea is to then integrate the volume of $s_n$ over the range $x, y, z \notin s_{n - 1} \cup s_{n - 2} \cup ..., \cup s_0$

Recursively

But this seems like a lot of work

Does it make sense what I was going for?

Its tedious, but I think (?) doable

I think I follow for the most part

I guess the lower the value of $k$ the easier it is to calculate

If they are "evenly spaced" then there are probably symettries you can exploit