triclino

If f(x) = 1 if x is rational and f(x) =0 if x is irrational ,then f(x)dx is the differential of which function?? Is that an exact or inexact?

You do not seem to have read my original post carefully .We are not examining [math]W = \int_C \vec F \cdot d\vec l[/math] ,but the scalar product :[math]\int Fdl[/math] as the OP points out. But let me ask you a question: Suppose we are given the differential f(x)dx. How do we know whether this differential is excact or inexact ??

from mathematics we have : W=[math]\int F.dl[/math] => [math]\frac{dw}{dl} = F[/math] => dw = [math]\frac{dw}{dl}.dl[/math] => dw = F.dl. Isn't dw an exact differential and thus F.dl ,since dw/dl = F??

Sorry it was a typo i meant to say that δW is an inexact differential. I corrected my post

F.dl is an exact differential . If you consider δW as an inexact differential (which you may) ,then we have an inexact differential equal to an exact differential. And again my problem is how do we mathematically prove that : if [math] W =\int F.dl[/math] then δW = F.dl

Charlatan ,is Table mountain still there??

In a certain book i read and i quote : "As in mechanics this work is defined by the integral. W = [math]\int F.dl[/math]. where F is the component of the force acting in the direction of the displacement dl.In a differential form this equation is written. δW = F.dl. where δW represents a differential quantity of work" Is that correct??

So the sqrt(4) = 2 ,because this the only value that makes sense?

This is not the subject of our discussion now. We discussed that in another thread ,let it rest.

First of all this is not an argument and thus cannot be valid or invalid .It is simply an implication and can only be true or false. It is true because we have false implying false Merged post follows: Consecutive posts merged I simply asked Remember in this forum we have not come up with a theorem or axiom or definition deciding for the validity or invalidity of an argument As i said examples do not make up a theory

So the argument : if the moon is made of cheese then i am a superman.But since the moon is not made of cheese ,hence i am not a superman . Is valid??

In defining implication we say that: p => q is false, if p is true and q false, and is true for all other cases. The question is if the definition could be other wised. For example instead of F =>T defined as T ,why not defined as F

I don't know what do you mean when you say: "there is no division" ,but since dy = f'(x)dx ,if you divide by dx ,then dy/dx = f'(x) = [math]\frac{dy}{dx}[/math]. Hence the ratio of dy to dx is equal to the derivative of y w.r.t x

You do not disagree with me ,but with the book of logic.

If you do not want to learn the difference between implication and logical implication,that is not my problem. Does not your book of logic have a section explaining logical implication?? M.ponens and M. tollens has nothing to do whether our arguments are valid or non valid. If an argument does not fit the general form of another valid argument that does not mean that it is invalid Does not your book of logic have a theorem deciding whether an argument is valid or non valid??

That definition does justify the equation: dy = (x+dx)^2 ,but only the equation : [math](x+dx)^2 = x^2+2xdx[/math]

What is the exact definition in mathematics of nilpotent ??

In mathematics every statement equation e.t.c is justified by an axiom ,theorem,definition,or a law of logic ,otherwise mathematics would not be mathematics , but a composition of ideas resulting in a chaotic situation. Yes a change in x will respectively produce a change in y since y =y(x). Now since by by definition ,dy = ([math]x^2[/math])'dx = 2xdx ,then y + dy = [math]x^2 + 2xdx[/math] and not equal to [math](x+dx)^2[/math]. Unless you have a different definition for differentials . Further more there is a difference between infinitesimals and differentials

This the truth table for the definition for implication and not for logical implication. There is a great difference between implication and logical implication. Read your logic book again carefully. In a simple implication the value of p =>q depends on the values (T,F) of p and q as the above table shows . In a logical implication the value of P=>Q is always true irrespective of the values of P and Q. Merged post follows: Consecutive posts merged oh ,yes you can You can prove whether an arguments is valid or non valid thru the axioms or theorems of logic, otherwise we would have a situation similar to the one in this thread. Merged post follows: Consecutive posts merged Little you know about logic.

Let logic decide that with its axioms and theorems , non of which you have cited so far

Here is an example where we deny the antecedent and the argument is still valid : we say : If i was rich i would have bought Kennedy Airport ,but since i am not rich i cannot bye Kennedy Airport. Is it not that argument valid?? There might be other cases that we do not know where although we deny the antecedent or affirm the consequent are valid

O.K then ,why when denying the antecedent or affirming the consequent the conclusions of an argument do not logically follow??

hobz ,how do you know that: [math] y + dy = (x + dx)^2 [/math]

Why when "denying the antecedent"or when "affirming the consequent" as you say the arguments are invalid? When is an argument invalid ??

1) An example or examples cannot support a general statement . 2) In my post #3 i showed that the conclusions of both arguments are logically implied