Definition·D109
Notation for formulas and terms
Ordinary symbols - not, and, or, implies, iff, the quantifiers, equals, and function/relation application - written for the object-level formulas and terms themselves, exactly as they are for the logic doing the writing.
For a language
: write, for
:
for
and
:
for
and
:
for
and
:
for
and
with
:
and for
and
with
:
In words
For any term, negation; for any two formulas, conjunction, disjunction, implication, and the biconditional; for any variable and formula, universal and existential quantification; for any two terms, the equality formula they form; and a relation symbol applied to a matching-arity tuple of terms, or a function symbol applied to one, are all written exactly as the ordinary symbols already used everywhere else in this wiki's own reasoning - now naming the specific formula or term of the language that each notation abbreviates.
Rests onno axioms yet
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A02 · A03 · A04 · A05 · A06 · A07 · A08 · A09 (computed from the citation graph, not asserted).
Remarks
Pure notation, no new mathematics: each display is a definitional equation naming an already-existing object (a formula or a term of
, built by admissibility/D081) via the symbol ordinarily used for the corresponding claim in this wiki's own logic. Context alone disambiguates the two readings, as in every standard treatment of first-order logic (Enderton, Mendelson): once
are known to be elements of
, "
" can only name the formula built by the implication clause, not assert a claim - the same symbol, the same shape, at a different level.
for a *specific* tuple
may be written
with the entries spelled out; likewise
. From here on, every definition and theorem about formulas and terms uses this notation instead of spelling out
directly - exactly as formula-admissible sets and term-admissible sets themselves are still defined using the raw construction, since notation has to bottom out somewhere.