It's an interesting exercise, because it renders most sentences containing 'equal' (or equality) as very obviously false.
Having rejected 'the same'/ identical as valid synonyms, the next step in the exercise is to find another synonym which makes sense in the context.
This can be difficult...
Therefore, equal/ity is revealed as a fake concept, an obfuscator of thought.
e.g. "Liberty! Sameness! Fraternity!" - ahem...
e.g. "Equal opportunities" goes to Identical opportunities; "Equality of outcomes for men and women" goes to the same outcomes for men and women.
(But how can we make things identical or the same? Only one way: By making them identical or the same - by coercively imposing identicality and sameness.)
e.g. "We hold these truths to be self-evident, that all men are created identical..." um, not really, so how about: "all men are created the same"... Nah. OK then: "all men are created with... indistinguishably-similar... wotsits..." - Oh, I give up.
"Equal" and "the same" have different meanings. 2+2 equals 4, but 2+2 is not the same as 4: one is a sum, the other is a number. I think "equal" can mean the same or identical in a specific attribute, not necessarily identical overall. But this doesn't really contradict what you're saying, because we still have to ask: in what sense are all men created equal? Which characteristic is identical among them?
@PW - in a sense there is no such thing as a synonym: all words have subtly different meanings.
But these 'magic' words like equality, democracy, justice, freedom have the particular property that they cannot be replaced in a sentence without making it obviously absurd.
e.g. "*Democracy* is the only acceptable political system" - "*Replacing the government by some kind of mass vote* is the only acceptable political system."
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