One of the ordered field, F, property is the following (i):(adsbygoogle = window.adsbygoogle || []).push({});

(i) for every x, y, and z, if both x, y, z in F and y < z, then x + y < x + z.

Now please think about (i'):

(i') for every x, y, and z, if x, y, z in F implies y < z, then x + y < x + z.

I cannot prove that (i) and (ii) are equivalent simply using some simple logical rules. But when I inspect (i') in a intuitive sense, it seems (i') does not have a difference in terms of meaning with (i). Moreover, (i') is equivalent to (i'') which is the following:

(i'') for every x, y, z in F, if y < z, then x + y < x + z.

Especially when I think about this (i''), it seems really similar in terms of meaning to (i''). Yet, I cannot make (i) in the same form as (i'').

Thus, my question are these:

(1) Is (i) equivalent to (i')?

(2) If they are not, why does my intuitive understanding of those two sentences' meaning makes error? Any opinion?

My plausible guess for the question (2) is that it is because (i') implies (i). But anyway, I'm not sure whether it is a right anwer.

Further question:

What is wrong with the following proof, or is the following proof correct:

(->). Suppose (X and Y) implies Z. Suppose X. Suppose Y. Since X and Y, Z is true. Thus, Y implies Z. Thus X implies (Y -> Z).

(<-). Suppose X implies (Y implies Z). Suppose X and Y. Since X is true, Y implies Z. Since Y is true, Z is true too. Thus (X and Y) implies Z.

In conclusion, we proved that that (X and Y) implies Z is equivalnet to that X implies (Y -> Z).

Remark.

When proving the above in Fitch form style, I didn't find any contradiction. Please try that too and comment whatever you think.

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# Error from a intuitive understanding of first-order logic with 'and' and 'if-then'

Can you offer guidance or do you also need help?

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