Computability and Logic

14.1. SEQUENT CALCULUS 171
condition A holds, we ‘give it a name’ and say ‘let c be something for which the
condition A holds’, where c is some previously unused name, and thereafter proceed
to count whatever statements not mentioning c that can be shown to follow from the
assumption that condition A holds for c as following from the original assumption
that there is something for which condition A holds. (R8a, b) correspond to two forms
of ‘substituting equals for equals’.
A couple of trivial examples will serve show how derivations are written.
14.3 Example. The deduction of a disjunction from a disjunct.
(1) A ⇒ A (R0)
(2) A ⇒ A, B (R1), (1)
(3) A ⇒ A ∨ B (R3), (2)
The first thing to note here is that though officially what occur on the left and
right sides of the double arrow in a sequent are sets, and sets have no intrinsic order
among their elements, in writing a sequent, we do have to write those elements in
some order or other. {A, B} and {B, A} and for that matter {A, A, B} are the same set,
and therefore {A}⇒{A, B} and {A}⇒{B, A} and for that matter {A}⇒{A, A, B}
are the same sequent, but we have chosen to write the sequent the first way. Actually,
we have not written the braces at all, nor will they be written in future when writing
out derivations. [For that matter, have also been writing A ∨ B for (A ∨ B), and will
be writing Fx for F(x) below.] An alternative approach would be to have sequences
rather than sets of formulas on both sides of a sequent, and introduce additional
‘structural’ rules allowing one to reorder the sentences in a sequences, and for that
matter, to introduce or eliminate repetitions.
The second thing to note here is that the numbering of the lines on the left, and
the annotations on the right, are not officially part of the derivation. In practice, their
presence makes it easier to check that a purported derivation really is one; but in
principle it can be checked whether a string symbols constituties a derivation even
without such annotation. For there are, after all, at each step only finitely many
rules that could possibly have been applied to get that step from earlier steps, and
only finitely many earlier steps any rule could possibly have been applied to, and in
principle we need only check through these finitely many possibilities to find whether
there is a justification for the given step.
14.4 Example. The deduction of a conjunct from a conjunction
(1) A ⇒ A (R0)
(2) A, B ⇒ A (R1), (1)
(3) B ⇒ A, ∼A (R2a), (2)
(4) ⇒ A, ∼A, ∼B (R2a), (3)
(5) ⇒ A, ∼A ∨∼B (R3), (4)
(6) ∼(∼A ∨∼B) ⇒ A (R2b), (5)
(7) A & B ⇒ A abbreviation, (6)
Here the last step, reminding us that ∼(∼A ∨∼B) is what A & B abbreviates, is
unofficial, so to speak. We omit the word ‘abbreviation’ in such cases in the future. It