@de budding,

de_budding;15402 wrote:does 'if' get a symbol or have any impact on sentences?

Dan.

Yes. Imo, we can name the propositional operators in this way:

(p -> q) is read (if p then q), or (p only if q).

(p <- q) is read (if q then p), or (p if q).

(p <-> q) is read (p is equivalent to q), or (p if and only if q).

(p <-> q) <-> ((p if q) and (p only if q)).

For the positive functors we have:

(tautology)

(p is true) = p

(q is true) = q

(p v q) = (p or q)

(p & q) = (p and q)

(p -> q) = (p only if q).

(p <- q) = (p if q)

(p <-> q) = (p if and only if q).

For the negative functors we have:

(contradiction)

~p = (not p)

~q = (not q)

(p / q) = (p nor q) ...~(p v q)

(p | q) = (p nand q) ...~(p & q)

(p -|-> q) = (p nonly if q) ...~(p -> q)

(p <-|- q) = (p nif q) ...~(p <- q)

(p <-|-> q) = (p xor q) ...~(p <-> q).

eg. ((p if q) and q) only if p, is a theorem.

ie. ((p <- q) & q) -> p.

ie. ((q -> p) & q) -> p.

eg. (p nif q) iff (q nonly if p) ....~(p <- q) <-> ~(q -> p).

etc.