@TheGreatSloth
OP, what is propositional calculus? Do I need to dig out my old (very old) logic text books?
(And I agree with your earlier point: because this is an irrational belief akin to a religion, trying to demolish it by means of reason is difficult if not impossible. Its adherents are in search of emotional validation, not accuracy.)
Propositional calculus is the formal basis of logic dealing with the notion and usage of words such as "NOT," "OR," "AND," and "implies."
[This taken from Propositional Calculus, in Eric Weisstein's wonderful encyclopedia of maths (bought by Steven Wolfram a while ago, long story).]
Some people call it ' sentential calculus'; I suppose it depends on whether you think of sentences as themselves being true, or more carefully, perhaps, that it is what sentences say or express ('propositions', in other words) that can be true (or false of course).
I think the latter, but it is easier to explain if I pretend to think the former. So, here are three sentences (which express three propositions; I will stop saying that):
A: The cat sat on the mat.
B: The dog ate the bone.
C: The pigeon flew out of the window.
Each of those could be true or false. Think of these as atomic sentences or atoms of meaning. OK, now, we can combine these to get molecular sentences as follows: (examples)
A=>C ('If the cat sat on the mat, then the pigeon flew out of the window' or '(the truth of) "the cat sat on the mat" implies (the truth of) "the pigeon flew out of the window"'
(A and B) => C ('If the cat sat on the mat and the dog ate the bone, then the pigeon flew out of the window')
not (A or B) ('It is not the case that the cat sat on the mat and the dog ate the bone')
OK? Now, the truth of a molecular sentence depends on the truth of its constituent atoms and how these words 'not', 'and', 'or', 'implies' work. For instance, for any two sentences X and Y, ( X and Y ) will be true whenever X is true and Y is true, but false otherwise ... and so on. (The thing most students find tricky here is how 'implies' works: ( X implies Y ) is true whenever it is not the case that X is true and Y is false. Think about it.)
( 'not', 'and', 'or', 'implies' are called ' logical connectives ': they connect sentences to get other sentences. They are interestingly related.)
There. Propositional calculus. As I said, this is structurally the same as ('isomorphic to') basic set theory (think 'set union' and 'set intersection' as 'or' and 'and' ... 'subset of' will get you 'implies': can you see how?). Also there is an isomorphism between each of these two structures and the structure of certain basic electrical connections in circuits (connectives are now 'in parallel'/'or'/'set union', 'in series'/'and'/'set intersection' ...) -- Hence computers.
This, with all its As, Bs, Xs and Ys, looks a bit like algebra. The algebraic structure exemplified by the logic, the set theory, the computer circuits and so on is called Boolean Algebra, after George Boole (1815-1864). Also important, ... well, look it up: Stanford, a good place to start.
[Sorry, you probably did not want that much. My teacher-mode took over. There's a start, anyway.]