The construction 'since A , then B' seems fine to me.
'Since' is being used as a shorthand for a modens ponens type argument.
Premise 1. The Pythagorean relationship holds for all triangles.
Premise 2. (since) this is a right triangle.
Conclusion: the Pythagorean relationship holds (for this triangle)
Premise 1 is not explicitly stated , it is implicitly assumed.
"Since this is a right triangle, its sides satisfy the Pythagorean relationship "
"Since I am at least 18 years of age and a citizen of United states, I can vote."
Here again we have an implicit syllogism.
- All people who are 18+ and citizen can vote.
- I am a person that is 18+ and a citizen.
Conclusion : I can vote
I don't see how 'since' can mean the same thing as the hypothetical indicator 'if' used in conditional 'if-then' statements. 'Since' has a meaning that something in fact did happen. The 'if' part of a conditional does not in fact need to occur or be true. But when you say, 'since x, then y' you mean since x did in fact happen, y has to occur.
We can be even more explicit. Any 'since x then y' statement can be rewritten as
"since x is the case in fact, then by logical necessity y is the case"
"Because it is a right triangle, then by logical necessity this triangle satisfies the Pythagorean relationship a^2 + b^2 = c^2"
I don't think we need to complicate issues by bringing in causality here, because showing cause and effect is a different matter. A right triangle does not 'cause' a theorem to be true in the sense of an agent cause, like my kicking a ball causes it to move. It is more like a matter of logical necessity; e.g. as a result of how we define right triangles on a Euclidean plane the Pythagorean relationship holds for all right triangles. Because x is an even integer, then x^2 is an even integer because of how we define even integers and the properties of algebra. Not because integers physically cause their squares to become even by some sort of phenomenon.
So to summarize
'since x, then y ' is equivalent in meaning to
'As a result of x being in fact true, then by logical necessity y must be true as well, (because we know that y is true whenever x is true)'. The last part in parentheses is often omitted.