Description:
Here are typical natural deduction (ND) rules in the style of Gentzen
and Jaśkowski, along with MPE translations of them. This also
shows the recommended theorems when you find yourself needing these
rules (the recommendations encourage a slightly different proof style
that works more naturally with metamath). A decent list of the standard
rules of natural deduction can be found beginning with definition /\I in
[Pfenning] p. 18. For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer. Many more citations could be added.
Name | Natural Deduction Rule | Translation |
Recommendation | Comments |
IT |
Γ⊢ 𝜓 => Γ⊢ 𝜓 |
idi 2 |
nothing | Reiteration is always redundant in Metamath.
Definition "new rule" in [Pfenning] p. 18,
definition IT in [Clemente] p. 10. |
∧I |
Γ⊢ 𝜓 & Γ⊢ 𝜒 => Γ⊢ 𝜓 ∧ 𝜒 |
jca 554 |
jca 554, pm3.2i 471 |
Definition ∧I in [Pfenning] p. 18,
definition I∧m,n in [Clemente] p. 10, and
definition ∧I in [Indrzejczak] p. 34
(representing both Gentzen's system NK and Jaśkowski) |
∧EL |
Γ⊢ 𝜓 ∧ 𝜒 => Γ⊢ 𝜓 |
simpld 475 |
simpld 475, adantr 481 |
Definition ∧EL in [Pfenning] p. 18,
definition E∧(1) in [Clemente] p. 11, and
definition ∧E in [Indrzejczak] p. 34
(representing both Gentzen's system NK and Jaśkowski) |
∧ER |
Γ⊢ 𝜓 ∧ 𝜒 => Γ⊢ 𝜒 |
simprd 479 |
simpr 477, adantl 482 |
Definition ∧ER in [Pfenning] p. 18,
definition E∧(2) in [Clemente] p. 11, and
definition ∧E in [Indrzejczak] p. 34
(representing both Gentzen's system NK and Jaśkowski) |
→I |
Γ, 𝜓⊢ 𝜒 => Γ⊢ 𝜓 → 𝜒 |
ex 450 | ex 450 |
Definition →I in [Pfenning] p. 18,
definition I=>m,n in [Clemente] p. 11, and
definition →I in [Indrzejczak] p. 33. |
→E |
Γ⊢ 𝜓 → 𝜒 & Γ⊢ 𝜓 => Γ⊢ 𝜒 |
mpd 15 | ax-mp 5, mpd 15, mpdan 702, imp 445 |
Definition →E in [Pfenning] p. 18,
definition E=>m,n in [Clemente] p. 11, and
definition →E in [Indrzejczak] p. 33. |
∨IL | Γ⊢ 𝜓 =>
Γ⊢ 𝜓 ∨ 𝜒 |
olcd 408 |
olc 399, olci 406, olcd 408 |
Definition ∨I in [Pfenning] p. 18,
definition I∨n(1) in [Clemente] p. 12 |
∨IR | Γ⊢ 𝜒 =>
Γ⊢ 𝜓 ∨ 𝜒 |
orcd 407 |
orc 400, orci 405, orcd 407 |
Definition ∨IR in [Pfenning] p. 18,
definition I∨n(2) in [Clemente] p. 12. |
∨E | Γ⊢ 𝜓 ∨ 𝜒 & Γ, 𝜓⊢ 𝜃 &
Γ, 𝜒⊢ 𝜃 => Γ⊢ 𝜃 |
mpjaodan 827 |
mpjaodan 827, jaodan 826, jaod 395 |
Definition ∨E in [Pfenning] p. 18,
definition E∨m,n,p in [Clemente] p. 12. |
¬I | Γ, 𝜓⊢ ⊥ => Γ⊢ ¬ 𝜓 |
inegd 1503 | pm2.01d 181 |
|
¬I | Γ, 𝜓⊢ 𝜃 & Γ⊢ ¬ 𝜃 =>
Γ⊢ ¬ 𝜓 |
mtand 691 | mtand 691 |
definition I¬m,n,p in [Clemente] p. 13. |
¬I | Γ, 𝜓⊢ 𝜒 & Γ, 𝜓⊢ ¬ 𝜒 =>
Γ⊢ ¬ 𝜓 |
pm2.65da 600 | pm2.65da 600 |
Contradiction. |
¬I |
Γ, 𝜓⊢ ¬ 𝜓 => Γ⊢ ¬ 𝜓 |
pm2.01da 458 | pm2.01d 181, pm2.65da 600, pm2.65d 187 |
For an alternative falsum-free natural deduction ruleset |
¬E |
Γ⊢ 𝜓 & Γ⊢ ¬ 𝜓 => Γ⊢ ⊥ |
pm2.21fal 1505 |
pm2.21dd 186 | |
¬E |
Γ, ¬ 𝜓⊢ ⊥ => Γ⊢ 𝜓 |
|
pm2.21dd 186 |
definition →E in [Indrzejczak] p. 33. |
¬E |
Γ⊢ 𝜓 & Γ⊢ ¬ 𝜓 => Γ⊢ 𝜃 |
pm2.21dd 186 | pm2.21dd 186, pm2.21d 118, pm2.21 120 |
For an alternative falsum-free natural deduction ruleset.
Definition ¬E in [Pfenning] p. 18. |
⊤I | Γ⊢ ⊤ |
a1tru 1500 | tru 1487, a1tru 1500, trud 1493 |
Definition ⊤I in [Pfenning] p. 18. |
⊥E | Γ, ⊥⊢ 𝜃 |
falimd 1499 | falim 1498 |
Definition ⊥E in [Pfenning] p. 18. |
∀I |
Γ⊢ [𝑎 / 𝑥]𝜓 => Γ⊢ ∀𝑥𝜓 |
alrimiv 1855 | alrimiv 1855, ralrimiva 2966 |
Definition ∀Ia in [Pfenning] p. 18,
definition I∀n in [Clemente] p. 32. |
∀E |
Γ⊢ ∀𝑥𝜓 => Γ⊢ [𝑡 / 𝑥]𝜓 |
spsbcd 3449 | spcv 3299, rspcv 3305 |
Definition ∀E in [Pfenning] p. 18,
definition E∀n,t in [Clemente] p. 32. |
∃I |
Γ⊢ [𝑡 / 𝑥]𝜓 => Γ⊢ ∃𝑥𝜓 |
spesbcd 3522 | spcev 3300, rspcev 3309 |
Definition ∃I in [Pfenning] p. 18,
definition I∃n,t in [Clemente] p. 32. |
∃E |
Γ⊢ ∃𝑥𝜓 & Γ, [𝑎 / 𝑥]𝜓⊢ 𝜃 =>
Γ⊢ 𝜃 |
exlimddv 1863 | exlimddv 1863, exlimdd 2088,
exlimdv 1861, rexlimdva 3031 |
Definition ∃Ea,u in [Pfenning] p. 18,
definition E∃m,n,p,a in [Clemente] p. 32. |
⊥C |
Γ, ¬ 𝜓⊢ ⊥ => Γ⊢ 𝜓 |
efald 1504 | efald 1504 |
Proof by contradiction (classical logic),
definition ⊥C in [Pfenning] p. 17. |
⊥C |
Γ, ¬ 𝜓⊢ 𝜓 => Γ⊢ 𝜓 |
pm2.18da 459 | pm2.18da 459, pm2.18d 124, pm2.18 122 |
For an alternative falsum-free natural deduction ruleset |
¬ ¬C |
Γ⊢ ¬ ¬ 𝜓 => Γ⊢ 𝜓 |
notnotrd 128 | notnotrd 128, notnotr 125 |
Double negation rule (classical logic),
definition NNC in [Pfenning] p. 17,
definition E¬n in [Clemente] p. 14. |
EM | Γ⊢ 𝜓 ∨ ¬ 𝜓 |
exmidd 432 | exmid 431 |
Excluded middle (classical logic),
definition XM in [Pfenning] p. 17,
proof 5.11 in [Clemente] p. 14. |
=I | Γ⊢ 𝐴 = 𝐴 |
eqidd 2623 | eqid 2622, eqidd 2623 |
Introduce equality,
definition =I in [Pfenning] p. 127. |
=E | Γ⊢ 𝐴 = 𝐵 & Γ[𝐴 / 𝑥]𝜓 =>
Γ⊢ [𝐵 / 𝑥]𝜓 |
sbceq1dd 3441 | sbceq1d 3440, equality theorems |
Eliminate equality,
definition =E in [Pfenning] p. 127. (Both E1 and E2.) |
Note that MPE uses classical logic, not intuitionist logic. As is
conventional, the "I" rules are introduction rules, "E" rules are
elimination rules, the "C" rules are conversion rules, and Γ
represents the set of (current) hypotheses. We use wff variable names
beginning with 𝜓 to provide a closer representation
of the Metamath
equivalents (which typically use the antedent 𝜑 to represent the
context Γ).
Most of this information was developed by Mario Carneiro and posted on
3-Feb-2017. For more information, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
For annotated examples where some traditional ND rules
are directly applied in MPE, see ex-natded5.2 27261, ex-natded5.3 27264,
ex-natded5.5 27267, ex-natded5.7 27268, ex-natded5.8 27270, ex-natded5.13 27272,
ex-natded9.20 27274, and ex-natded9.26 27276.
(Contributed by DAW, 4-Feb-2017.) (New usage is
discouraged.) |