Definition df-dc 742

Description: Propositions which are known to be true or false are called decidable. The (classical) Law of the Excluded Middle corresponds to the principle that all propositions are decidable, but even given intuitionistic logic, particular kinds of propositions may be decidable (for example, the proposition that two natural numbers are equal will be decidable under most sets of axioms).

Our notation for decidability is a connective DECID which we place before the formula in question. For example, DECID x = y corresponds to "x = y is decidable".

We could transform intuitionistic logic to classical logic by adding unconditional forms of condc 748, exmiddc 743, peircedc 819, or notnot2dc 750, any of which would correspond to the assertion that all propositions are decidable.

(Contributed by Jim Kingdon, 11-Mar-2018.)

Assertion

Ref	Expression
df-dc	|- (DECID ph <-> (ph \/ -. ph))

Detailed syntax breakdown of Definition df-dc

Step	Hyp	Ref	Expression
1		wph	. . 3 wff ph
2	1	wdc 741	. 2 wff DECID ph
3	1	wn 3	. . 3 wff -. ph
4	1, 3	wo 628	. 2 wff (ph \/ -. ph)
5	2, 4	wb 98	1 wff (DECID ph <-> (ph \/ -. ph))

This definition is referenced by:  exmiddc  743  pm2.1dc  744  dcn  745  dcbii  746  dcbid  747  condc  748  notnot2dc  750  pm2.61ddc  757  pm5.18dc  776  pm2.13dc  778  dcim  783  pm2.25dc  791  pm2.85dc  810  pm5.12dc  815  pm5.14dc  816  pm5.55dc  818  peircedc  819  dftest  821  testbitestn  822  stabtestimpdc  823  pm5.54dc  826  dcan  841  dcor  842  pm5.62dc  851  pm5.63dc  852  pm4.83dc  857  xordc1  1281  biassdc  1283  19.30dc  1515  nfdc  1546  exmodc  1947  moexexdc  1981  dcne  2211  eueq2dc  2708  eueq3dc  2709  abvor0dc  3236  nndceq0  4282  nndceq  6015  nndcel  6016  elni2  6298  indpi  6326  distrlem4prl  6560  distrlem4pru  6561  zdcle  8093  zdclt  8094  uzin  8281  elnn1uz2  8320  eluzdc  8323  fztri3or  8673  fzdcel  8674  fzneuz  8733  fzfig  8887  expival  8911  nndc  9235  dcdc  9236  bddc  9283  bj-dcbi  9383