Definition df-dc 743

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 749, exmiddc 744, peircedc 820, or notnotrdc 751, 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 742	. 2 wff DECID ph
3	1	wn 3	. . 3 wff -. ph
4	1, 3	wo 629	. 2 wff ( ph \/ -. ph )
5	2, 4	wb 98	1 wff (DECID ph <-> ( ph \/ -. ph ) )

This definition is referenced by:  exmiddc  744  pm2.1dc  745  dcn  746  dcbii  747  dcbid  748  condc  749  notnotrdc  751  pm2.61ddc  758  pm5.18dc  777  pm2.13dc  779  dcim  784  pm2.25dc  792  pm2.85dc  811  pm5.12dc  816  pm5.14dc  817  pm5.55dc  819  peircedc  820  dftest  822  testbitestn  823  stabtestimpdc  824  pm5.54dc  827  dcan  842  dcor  843  pm5.62dc  852  pm5.63dc  853  pm4.83dc  858  xordc1  1284  biassdc  1286  19.30dc  1518  nfdc  1549  exmodc  1950  moexexdc  1984  dcne  2216  eueq2dc  2714  eueq3dc  2715  abvor0dc  3242  ifcldcd  3358  nndceq0  4339  nndceq  6077  nndcel  6078  nndifsnid  6080  fidceq  6330  fidifsnid  6332  elni2  6412  indpi  6440  distrlem4prl  6682  distrlem4pru  6683  zdcle  8317  zdclt  8318  uzin  8505  elnn1uz2  8544  eluzdc  8547  fztri3or  8903  fzdcel  8904  fzneuz  8963  fzfig  9206  expival  9257  nndc  9900  dcdc  9901  bddc  9948  bj-dcbi  10048