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Theorem List for Intuitionistic Logic Explorer - 801-900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempm2.81 801 Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)

Theorempm2.82 802 Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)

Theorempm3.2ni 803 Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.)

Theoremorabs 804 Absorption of redundant internal disjunct. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 28-Feb-2014.)

Theoremoranabs 805 Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton, 23-Jun-2005.) (Proof shortened by Wolf Lammen, 10-Nov-2013.)

Theoremordi 806 Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)

Theoremordir 807 Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)

Theoremandi 808 Distributive law for conjunction. Theorem *4.4 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)

Theoremandir 809 Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)

Theoremorddi 810 Double distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)

Theoremanddi 811 Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)

Theorempm4.39 812 Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)

Theorempm4.72 813 Implication in terms of biconditional and disjunction. Theorem *4.72 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Jan-2013.)

Theorempm5.16 814 Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)

Theorembiort 815 A disjunction with a true formula is equivalent to that true formula. (Contributed by NM, 23-May-1999.)

1.2.7  Stable propositions

Syntaxwstab 816 Extend wff definition to include stability.
STAB

Definitiondf-stab 817 Propositions where a double-negative can be removed are called stable. See Chapter 2 [Moschovakis] p. 2.

Our notation for stability is a connective STAB which we place before the formula in question. For example, STAB corresponds to " is stable".

(Contributed by David A. Wheeler, 13-Aug-2018.)

STAB

Theoremstbid 818 The equivalent of a stable proposition is stable. (Contributed by Jim Kingdon, 12-Aug-2022.)
STAB STAB

Theoremstabnot 819 Every negated formula is stable. (Contributed by David A. Wheeler, 13-Aug-2018.)
STAB

1.2.8  Decidable propositions

Syntaxwdc 820 Extend wff definition to include decidability.
DECID

Definitiondf-dc 821 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 corresponds to " is decidable".

We could transform intuitionistic logic to classical logic by adding unconditional forms of condc 839, exmiddc 822, peircedc 900, or notnotrdc 829, any of which would correspond to the assertion that all propositions are decidable.

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

DECID

Theoremexmiddc 822 Law of excluded middle, for a decidable proposition. The law of the excluded middle is also called the principle of tertium non datur. Theorem *2.11 of [WhiteheadRussell] p. 101. It says that something is either true or not true; there are no in-between values of truth. The key way in which intuitionistic logic differs from classical logic is that intuitionistic logic says that excluded middle only holds for some propositions, and classical logic says that it holds for all propositions. (Contributed by Jim Kingdon, 12-May-2018.)
DECID

Theorempm2.1dc 823 Commuted law of the excluded middle for a decidable proposition. Based on theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by Jim Kingdon, 25-Mar-2018.)
DECID

Theoremdcbid 824 Equivalence property for decidability. Deduction form. (Contributed by Jim Kingdon, 7-Sep-2019.)
DECID DECID

Theoremdcbiit 825 Equivalence property for decidability. Closed form. (Contributed by BJ, 27-Jan-2020.)
DECID DECID

Theoremdcbii 826 Equivalence property for decidability. Inference form. (Contributed by Jim Kingdon, 28-Mar-2018.)
DECID DECID

Theoremdcim 827 An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
DECID DECID DECID

Theoremdcn 828 The negation of a decidable proposition is decidable. The converse need not hold, but does hold for negated propositions, see dcnn 834. (Contributed by Jim Kingdon, 25-Mar-2018.)
DECID DECID

Theoremnotnotrdc 829 Double negation elimination for a decidable proposition. The converse, notnot 619, holds for all propositions, not just decidable ones. This is Theorem *2.14 of [WhiteheadRussell] p. 102, but with a decidability condition added. (Contributed by Jim Kingdon, 11-Mar-2018.)
DECID

Theoremdcstab 830 Decidability implies stability. The converse need not hold. (Contributed by David A. Wheeler, 13-Aug-2018.)
DECID STAB

Theoremstdcndc 831 A formula is decidable if and only if its negation is decidable and it is stable (that is, it is testable and stable). (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof shortened by BJ, 28-Oct-2023.)
STAB DECID DECID

TheoremstdcndcOLD 832 Obsolete version of stdcndc 831 as of 28-Oct-2023. (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
STAB DECID DECID

Theoremstdcn 833 A formula is stable if and only if the decidability of its negation implies its decidability. Note that the right-hand side of this biconditional is the converse of dcn 828. (Contributed by BJ, 18-Nov-2023.)
STAB DECID DECID

Theoremdcnn 834 Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 828. The relation between dcn 828 and dcnn 834 is analogous to that between notnot 619 and notnotnot 624 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 834 means that a proposition is testable if and only if its negation is testable, and dcn 828 means that decidability implies testability. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof shortened by BJ, 25-Nov-2023.)
DECID DECID

TheoremdcnnOLD 835 Obsolete proof of dcnnOLD 835 as of 25-Nov-2023. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
DECID DECID

Theoremnnexmid 836 Double negation of decidability of a formula. See also comment of nndc 837 to avoid a pitfall that could come from the label "nnexmid". This theorem can also be proved from bj-nnor 13153 as in bj-nndcALT 13173. (Contributed by BJ, 9-Oct-2019.)

Theoremnndc 837 Double negation of decidability of a formula. Intuitionistic logic refutes the negation of decidability (but does not prove decidability) of any formula.

This should not trick the reader into thinking that EXMID is provable in intuitionistic logic. Indeed, if we could quantify over formula metavariables, then generalizing nnexmid 836 over would give " DECID ", but EXMID is "DECID ", so proving EXMID would amount to proving " DECID ", which is not implied by the above theorem. Indeed, the converse of nnal 1629 does not hold. Since our system does not allow quantification over formula metavariables, we can reproduce this argument by representing formulas as subsets of , like we do in our definition of EXMID (df-exmid 4129): then, we can prove DECID but we cannot prove DECID because the converse of nnral 2430 does not hold.

Actually, EXMID is not provable in intuitionistic logic since intuitionistic logic has models satisfying EXMID and noncontradiction holds (pm3.24 683). (Contributed by BJ, 9-Oct-2019.) Add explanation on non-provability of EXMID. (Revised by BJ, 11-Aug-2024.)

DECID

1.2.9  Theorems of decidable propositions

Many theorems of logic hold in intuitionistic logic just as they do in classical (non-inuitionistic) logic, for all propositions. Other theorems only hold for decidable propositions, such as the law of the excluded middle (df-dc 821), double negation elimination (notnotrdc 829), or contraposition (condc 839). Our goal is to prove all well-known or important classical theorems, but with suitable decidability conditions so that the proofs follow from intuitionistic axioms. This section is focused on such proofs, given decidability conditions.

Many theorems of this section actually hold for stable propositions (see df-stab 817). Decidable propositions are stable (dcstab 830), but the converse need not hold.

Theoremconst 838 Contraposition of a stable proposition. See comment of condc 839. (Contributed by BJ, 18-Nov-2023.)
STAB

Theoremcondc 839 Contraposition of a decidable proposition.

This theorem swaps or "transposes" the order of the consequents when negation is removed. An informal example is that the statement "if there are no clouds in the sky, it is not raining" implies the statement "if it is raining, there are clouds in the sky." This theorem (without the decidability condition, of course) is called Transp or "the principle of transposition" in Principia Mathematica (Theorem *2.17 of [WhiteheadRussell] p. 103) and is Axiom A3 of [Margaris] p. 49. We will also use the term "contraposition" for this principle, although the reader is advised that in the field of philosophical logic, "contraposition" has a different technical meaning.

(Contributed by Jim Kingdon, 13-Mar-2018.) (Proof shortened by BJ, 18-Nov-2023.)

DECID

TheoremcondcOLD 840 Obsolete proof of condc 839 as of 18-Nov-2023. (Contributed by Jim Kingdon, 13-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
DECID

Theorempm2.18dc 841 Proof by contradiction for a decidable proposition. Based on Theorem *2.18 of [WhiteheadRussell] p. 103 (also called Clavius law). Intuitionistically it requires a decidability assumption, but compare with pm2.01 606 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.)
DECID

Theoremcon1dc 842 Contraposition for a decidable proposition. Based on theorem *2.15 of [WhiteheadRussell] p. 102. (Contributed by Jim Kingdon, 29-Mar-2018.)
DECID

Theoremcon4biddc 843 A contraposition deduction. (Contributed by Jim Kingdon, 18-May-2018.)
DECID DECID        DECID DECID

Theoremimpidc 844 An importation inference for a decidable consequent. (Contributed by Jim Kingdon, 30-Apr-2018.)
DECID        DECID

Theoremsimprimdc 845 Simplification given a decidable proposition. Similar to Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by Jim Kingdon, 30-Apr-2018.)
DECID

Theoremsimplimdc 846 Simplification for a decidable proposition. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by Jim Kingdon, 29-Mar-2018.)
DECID

Theorempm2.61ddc 847 Deduction eliminating a decidable antecedent. (Contributed by Jim Kingdon, 4-May-2018.)
DECID

Theorempm2.6dc 848 Case elimination for a decidable proposition. Based on theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 25-Mar-2018.)
DECID

Theoremjadc 849 Inference forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 25-Mar-2018.)
DECID               DECID

Theoremjaddc 850 Deduction forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 26-Mar-2018.)
DECID               DECID

Theorempm2.61dc 851 Case elimination for a decidable proposition. Theorem *2.61 of [WhiteheadRussell] p. 107 under a decidability condition. (Contributed by Jim Kingdon, 29-Mar-2018.)
DECID

Theorempm2.5gdc 852 Negating an implication for a decidable antecedent. General instance of Theorem *2.5 of [WhiteheadRussell] p. 107 under a decidability condition. (Contributed by Jim Kingdon, 29-Mar-2018.)
DECID

Theorempm2.5dc 853 Negating an implication for a decidable antecedent. Theorem *2.5 of [WhiteheadRussell] p. 107 under a decidability condition. (Contributed by Jim Kingdon, 29-Mar-2018.)
DECID

Theorempm2.521gdc 854 A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107, under a decidability condition. (Contributed by BJ, 28-Oct-2023.)
DECID

Theorempm2.521dc 855 Theorem *2.521 of [WhiteheadRussell] p. 107, but with an additional decidability condition. Note that by replacing in proof pm2.52 646 with conax1k 644, we obtain a proof of the more general instance where the last occurrence of is replaced with any . (Contributed by Jim Kingdon, 5-May-2018.)
DECID

Theorempm2.521dcALT 856 Alternate proof of pm2.521dc 855. (Contributed by Jim Kingdon, 5-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
DECID

Theoremcon34bdc 857 Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116, but for a decidable proposition. (Contributed by Jim Kingdon, 24-Apr-2018.)
DECID

Theoremnotnotbdc 858 Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot 619, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.)
DECID

Theoremcon1biimdc 859 Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.)
DECID

Theoremcon1bidc 860 Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
DECID DECID

Theoremcon2bidc 861 Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
DECID DECID

Theoremcon1biddc 862 A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.)
DECID        DECID

Theoremcon1biidc 863 A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
DECID        DECID

Theoremcon1bdc 864 Contraposition. Bidirectional version of con1dc 842. (Contributed by NM, 5-Aug-1993.)
DECID DECID

Theoremcon2biidc 865 A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
DECID        DECID

Theoremcon2biddc 866 A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.)
DECID        DECID

Theoremcondandc 867 Proof by contradiction. This only holds for decidable propositions, as it is part of the family of theorems which assume , derive a contradiction, and therefore conclude . By contrast, assuming , deriving a contradiction, and therefore concluding , as in pm2.65 649, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.)
DECID

Theorembijadc 868 Combine antecedents into a single biconditional. This inference is reminiscent of jadc 849. (Contributed by Jim Kingdon, 4-May-2018.)
DECID

Theorempm5.18dc 869 Relationship between an equivalence and an equivalence with some negation, for decidable propositions. Based on theorem *5.18 of [WhiteheadRussell] p. 124. Given decidability, we can consider to represent "negated exclusive-or". (Contributed by Jim Kingdon, 4-Apr-2018.)
DECID DECID

Theoremdfandc 870 Definition of 'and' in terms of negation and implication, for decidable propositions. The forward direction holds for all propositions, and can (basically) be found at pm3.2im 627. (Contributed by Jim Kingdon, 30-Apr-2018.)
DECID DECID

Theorempm2.13dc 871 A decidable proposition or its triple negation is true. Theorem *2.13 of [WhiteheadRussell] p. 101 with decidability condition added. (Contributed by Jim Kingdon, 13-May-2018.)
DECID

Theorempm4.63dc 872 Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.)
DECID DECID

Theorempm4.67dc 873 Theorem *4.67 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.)
DECID DECID

Theoremimanst 874 Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2012.)
STAB

Theoremimandc 875 Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 678, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.)
DECID

Theorempm4.14dc 876 Theorem *4.14 of [WhiteheadRussell] p. 117, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.)
DECID

Theorempm2.54dc 877 Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 712, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.)
DECID

Theoremdfordc 878 Definition of disjunction in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 712, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.)
DECID

Theorempm2.25dc 879 Elimination of disjunction based on a disjunction, for a decidable proposition. Based on theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
DECID

Theorempm2.68dc 880 Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 738 and one half of dfor2dc 881. (Contributed by Jim Kingdon, 27-Mar-2018.)
DECID

Theoremdfor2dc 881 Disjunction expressed in terms of implication only, for a decidable proposition. Based on theorem *5.25 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 27-Mar-2018.)
DECID

Theoremimimorbdc 882 Simplify an implication between implications, for a decidable proposition. (Contributed by Jim Kingdon, 18-Mar-2018.)
DECID

Theoremimordc 883 Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 711, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.)
DECID

Theorempm4.62dc 884 Implication in terms of disjunction. Like Theorem *4.62 of [WhiteheadRussell] p. 120, but for a decidable antecedent. (Contributed by Jim Kingdon, 21-Apr-2018.)
DECID

Theoremianordc 885 Negated conjunction in terms of disjunction (DeMorgan's law). Theorem *4.51 of [WhiteheadRussell] p. 120, but where one proposition is decidable. The reverse direction, pm3.14 743, holds for all propositions, but the equivalence only holds where one proposition is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)
DECID

Theorempm4.64dc 886 Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability condition. The reverse direction, pm2.53 712, holds for all propositions. (Contributed by Jim Kingdon, 2-May-2018.)
DECID

Theorempm4.66dc 887 Theorem *4.66 of [WhiteheadRussell] p. 120, given a decidability condition. (Contributed by Jim Kingdon, 2-May-2018.)
DECID

Theorempm4.54dc 888 Theorem *4.54 of [WhiteheadRussell] p. 120, for decidable propositions. One form of DeMorgan's law. (Contributed by Jim Kingdon, 2-May-2018.)
DECID DECID

Theorempm4.79dc 889 Equivalence between a disjunction of two implications, and a conjunction and an implication. Based on theorem *4.79 of [WhiteheadRussell] p. 121 but with additional decidability antecedents. (Contributed by Jim Kingdon, 28-Mar-2018.)
DECID DECID

Theorempm5.17dc 890 Two ways of stating exclusive-or which are equivalent for a decidable proposition. Based on theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 16-Apr-2018.)
DECID

Theorempm2.85dc 891 Reverse distribution of disjunction over implication, given decidability. Based on theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by Jim Kingdon, 1-Apr-2018.)
DECID

Theoremorimdidc 892 Disjunction distributes over implication. The forward direction, pm2.76 798, is valid intuitionistically. The reverse direction holds if is decidable, as can be seen at pm2.85dc 891. (Contributed by Jim Kingdon, 1-Apr-2018.)
DECID

Theorempm2.26dc 893 Decidable proposition version of theorem *2.26 of [WhiteheadRussell] p. 104. (Contributed by Jim Kingdon, 20-Apr-2018.)
DECID

Theorempm4.81dc 894 Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions. This one needs a decidability condition, but compare with pm4.8 697 which holds for all propositions. (Contributed by Jim Kingdon, 4-Jul-2018.)
DECID

Theorempm5.11dc 895 A decidable proposition or its negation implies a second proposition. Based on theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 29-Mar-2018.)
DECID DECID

Theorempm5.12dc 896 Excluded middle with antecedents for a decidable consequent. Based on theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)
DECID

Theorempm5.14dc 897 A decidable proposition is implied by or implies other propositions. Based on theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)
DECID

Theorempm5.13dc 898 An implication holds in at least one direction, where one proposition is decidable. Based on theorem *5.13 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)
DECID

Theorempm5.55dc 899 A disjunction is equivalent to one of its disjuncts, given a decidable disjunct. Based on theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)
DECID

Theorempeircedc 900 Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 822, condc 839, or notnotrdc 829 in the sense of expressing the "difference" between an intuitionistic system of propositional calculus and a classical system. In intuitionistic logic, it only holds for decidable propositions. (Contributed by Jim Kingdon, 3-Jul-2018.)
DECID

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