"I have a proof that...."

This is about an observation of a type that I don't particularly like to make, but once the observation has been made, it had better be recorded. In the following, A and B stand for propositions for which I may have a proof.

Consider statements a0 and a1:

(a0) I have a proof that true (holds)
(a1) true (holds)   .

Then, a0 and a1 are equivalent. (Since —by definition— the proof that true (holds) is empty, it is impossible not to have it.)

Consider statements b0 and b1:

(b0) I have a proof that false (holds)
(b1) false (holds)   .

Then b0 and b1 are equivalent. (Since —by definition— the proof that false (holds) does not exist, it is impossible to have it.)

From the above we conclude by case analysis the equivalence of c0 and c1:

(c0) I have a proof that I have a proof that A (holds)
(c1) I have a proof that A (holds)   .

Consider statements d0 and d1:

(d0) I have a proof that A∧B (holds)
(d1) I have a proof that A (holds) and I have a proof that B (holds)   .

Then d0 and d1 are equivalent. (Well, that is what "∧" (="and") means.)

This last law can be generalized to universal quantification. Consider statements e0 and e1 (in which the range for n is implicitly understood):

(e0) I have a proof that, for all n, A.n (holds)
(e1) For all n, I have a proof that A.n (holds)   .

Then e0 and e1 are equivalent.

Remark: As a result it is semantically irrelevant that the sentence "I have a proof of A.n for all n" is syntactically ambiguous. (End of remark.)

Consider the statements f0 and f1:

(f0) If I have a proof that A (holds) then I have a proof that B (holds)
(f1) I have a proof that, if I have a proof that A holds, then B (holds)

Then f0 and f1 are equivalent. (If I don't have a proof that A (holds), f0 and f1 are both "vacuously" true; if I do have a proof that A (holds), both f0 and f1 reduce to "I have a proof that B (holds)".)

Remark: As a result it is semantically irrelevant that the sentence "I have a proof that B holds if I have a proof of A." is syntactically ambiguous. (End of remark.)

But consider now statements g0 and g1:

(g0) I have a proof that A ∨ B (holds)
(g1) I have a proof that A (holds) or I have a proof that B (holds) or I have both proofs.

In this case, the two statements are not equivalent: g1 implies g0, but it is in general not the other way round.

Let us now do away with all the above verbosity and abbreviate "I have a proof that A (holds)" to "[A]". Our laws about having proofs can then be summa rized as follows:

(a) [true] ≡ true

(b) [false] ≡ false

(c) [[A]] ≡ [A]

(d) [A ∧ B] ≡ [A] ∧ [B]

(e) [〈∀n :: A.n〉] ≡ 〈∀n :: [A.n]〉

(f) [A] ⇒ [B] ≡ [[A] ⇒ B]

(g) [A ∨ B] ⇐ [A] ∨ [B]

The moral of the story is that "I have a proof that..." has all the algebraic properties of the "everywhere" operator, i.e. of universal quantification over a non-empty domain (see [DS90]).

[DS90] Dijkstra, Edsger W. and Scholten, Carel S., Predicate Calculus and Program Semantics, Springer-Verlag, New York, 1990.