S.O.S. Mathematics CyberBoard

Yes, A is a dummy object which is part of the statement.

E.g. If X is an NBA player, X is a basketball player.

Now I replace X by "Michael Jordan"

If Michael Jordan is an NBA player, Michael Jordan is a basketball player.

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Technically, you need to bind the dummy variables. What you can do is from the statement "for all A, if A is a field, then A is a ring" to "B is a field => B is a ring" (forall-elimination), then from "B is a field => B is a ring" and "B is a field", MP gives "B is a ring" as your desired conclusion. (Or, add enough things to your language and just invoke "B is a field => B is a ring" as an instance of a suitable axiom, then MP as before.)

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Then we need the use of two laws :

1) Universal Elimination ,and then

2) Modus Ponens.

in the following way:

for all ,x ( if x is a field ,then x is a ring) [this is a fact in abstract alebra]

Now by using the law of Universal Elimination ,by puting x=A, we have:

If A is a field ,then A is a ring .



Then we have to add but, A is a field , to conlcude A is a ring

The same thing we can say for B

But certainly :



Tell us nothing.

But to come back to our problem ,apart from the law of transitivity ,what other theorems or axioms are involved in the proof??

Also Shadow mentioned the use of M.Tolens .Where is that law involved in the proof??

No, that is condensing -Elimination and Modus Ponens into a single line.

You don't have Modus Tollens in the proof you gave for (1) (except: you should prove, or at least examine the proof of, 1>0, where you might encounter it).

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So ,which are the theorems or axioms and the laws of logic involved in the proof?

So far we have M.ponens M. tolens Universal Elimination and the theorem of transitivity.

Are there any others??

Also i cannot make out yet how these laws are used

Again, you should look at your conclusions. You wrote down things like:



If you don't know how you concluded them, how do you know they're correct? Learning what techniques you've used involves looking at what you've written, and examining each step to see what you've done. Have you tried this yet?

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Why ,how do you know that my proof is correct??