# Logical difference between 'equivalence' and 'an absence of differences'

It has been a while since my logic classes in philosophy. I was wondering if equality and an absence of difference amount to the same thing, logically.

An example would be the baseline table in reports of randomised controlled trials. These tables are presented to demonstrate that the experimental groups had no statistical differences prior to the experiment beginning, and thus that any observed differences during testing are not an artifact of pre-existing differences. Traditional null-hypothesis significance tests are used in this context to demonstrate the absence of differences (e.g. p=.35 therefore no significant difference between groups on variable x). However it seems to me that this is taken to mean 'these groups are equivalent'. I was wondering therefore if the statements 'there are no observed differences on any of the variables measured' and 'the two groups were the same with regards to the variables measured' amount (semantically or logically) the same thing. I was hoping to leave statistics out of the discussion and to confine it to logic and/or semantics.

• Please expand on what you mean by absence of difference. Do you have any clear cut examples of what an absence of difference? If you are using the phrase as a synonym then there usually are distinctions amongst the different terms. In other words there is a class that both terms can be members of but there are distinct qualities that make one unique and also one of the terms may belong to a sub category that the other term does not belong. Apr 11 '18 at 22:59
• If there is no difference between A and B, then we say A=B, so let's say "an absence of differences" is the same as "equality". Then one logical difference is that logical equality is a logical operator while logical equivalence is a "semantic concept".
– NWR
Apr 11 '18 at 23:09
• equivalence means that two "objects" are interchangeable in a specific context: this does not mean equal. E.g. in propositional logic, where the only "values" are the truth-values T and F, two tautologies are equivalent: but they can be very different formulas. Apr 12 '18 at 6:49

Equivalence differs by context. Two letters 'A' side by side are equivalent in some senses, and yet not in others. They are the same symbol, but occupy different positions in space.

The absolute absence of differences is never useful, as by that standard each thing is equivalent only to itself as referred to by exactly the same existing references. After all, having been thought about by another person than the last one that considered it, is a real difference, just not one that can be ascertained or used.

Therefore, in some important sense, equality or equivalence is always defined by some sort of equivalence relationship that extracts what is significant in the given context for comparison. That means that equivalence is itself always an abstraction.

• @Logikal I have expanded my question. Apr 11 '18 at 23:32
• @llewmills I assume you did not mean to attach that comment to this answer.
– user9166
Apr 11 '18 at 23:37
• You are right jobermark. I apologise Apr 12 '18 at 4:39
• @llewmills I didn't mean for that to be a complaint. I meant you might want to cut it and paste it above, under the main comment stream.
– user9166
Apr 12 '18 at 16:10