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.

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads - Error intuitive understanding | Date |
---|---|

A Error estimation in linear regression | Mar 12, 2018 |

B Least / Smallest / Minimum Detectable Difference | Jan 21, 2018 |

B Trueness and Bias | Jan 21, 2018 |

I Overlapping error ellipses? | Dec 13, 2017 |

A Improving intuition on applying likelihood ratio test | Dec 12, 2016 |

**Physics Forums - The Fusion of Science and Community**