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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | stabnot 801 | Every negated formula is stable. (Contributed by David A. Wheeler, 13-Aug-2018.) |
STAB | ||
Syntax | wdc 802 | Extend wff definition to include decidability. |
DECID | ||
Definition | df-dc 803 |
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 819, exmiddc 804, peircedc 880, or notnotrdc 811, any of which would correspond to the assertion that all propositions are decidable. (Contributed by Jim Kingdon, 11-Mar-2018.) |
DECID | ||
Theorem | exmiddc 804 | 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 | ||
Theorem | pm2.1dc 805 | 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 | ||
Theorem | dcbid 806 | Equivalence property for decidability. Deduction form. (Contributed by Jim Kingdon, 7-Sep-2019.) |
DECID DECID | ||
Theorem | dcbiit 807 | Equivalence property for decidability. Closed form. (Contributed by BJ, 27-Jan-2020.) |
DECID DECID | ||
Theorem | dcbii 808 | Equivalence property for decidability. Inference form. (Contributed by Jim Kingdon, 28-Mar-2018.) |
DECID DECID | ||
Theorem | dcim 809 | An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.) |
DECID DECID DECID | ||
Theorem | dcn 810 | The negation of a decidable proposition is decidable. The converse need not hold, but does hold for negated propositions, see dcnn 816. (Contributed by Jim Kingdon, 25-Mar-2018.) |
DECID DECID | ||
Theorem | notnotrdc 811 | Double negation elimination for a decidable proposition. The converse, notnot 601, 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 | ||
Theorem | dcstab 812 | Decidability implies stability. The converse need not hold. (Contributed by David A. Wheeler, 13-Aug-2018.) |
DECID STAB | ||
Theorem | stdcndc 813 | 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 | ||
Theorem | stdcndcOLD 814 | Obsolete version of stdcndc 813 as of 28-Oct-2023. (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
STAB DECID DECID | ||
Theorem | stdcn 815 | 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 810. (Contributed by BJ, 18-Nov-2023.) |
STAB DECID DECID | ||
Theorem | dcnn 816 | Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 810. The relation between dcn 810 and dcnn 816 is analogous to that between notnot 601 and notnotnot 606 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 816 means that a proposition is testable if and only if its negation is testable, and dcn 810 means that decidability implies testability. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof shortened by BJ, 25-Nov-2023.) |
DECID DECID | ||
Theorem | dcnnOLD 817 | Obsolete proof of dcnnOLD 817 as of 25-Nov-2023. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
DECID DECID | ||
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 803), double negation elimination (notnotrdc 811), or contraposition (condc 819). 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 799). Decidable propositions are stable (dcstab 812), but the converse need not hold. | ||
Theorem | const 818 | Contraposition of a stable proposition. See comment of condc 819. (Contributed by BJ, 18-Nov-2023.) |
STAB | ||
Theorem | condc 819 |
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 | ||
Theorem | condcOLD 820 | Obsolete proof of condc 819 as of 18-Nov-2023. (Contributed by Jim Kingdon, 13-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
DECID | ||
Theorem | pm2.18dc 821 | Proof by contradiction for a decidable proposition. Based on Theorem *2.18 of [WhiteheadRussell] p. 103 (also called the Law of Clavius). Intuitionistically it requires a decidability assumption, but compare with pm2.01 588 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.) |
DECID | ||
Theorem | con1dc 822 | Contraposition for a decidable proposition. Based on theorem *2.15 of [WhiteheadRussell] p. 102. (Contributed by Jim Kingdon, 29-Mar-2018.) |
DECID | ||
Theorem | con4biddc 823 | A contraposition deduction. (Contributed by Jim Kingdon, 18-May-2018.) |
DECID DECID DECID DECID | ||
Theorem | impidc 824 | An importation inference for a decidable consequent. (Contributed by Jim Kingdon, 30-Apr-2018.) |
DECID DECID | ||
Theorem | simprimdc 825 | Simplification given a decidable proposition. Similar to Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by Jim Kingdon, 30-Apr-2018.) |
DECID | ||
Theorem | simplimdc 826 | Simplification for a decidable proposition. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by Jim Kingdon, 29-Mar-2018.) |
DECID | ||
Theorem | pm2.61ddc 827 | Deduction eliminating a decidable antecedent. (Contributed by Jim Kingdon, 4-May-2018.) |
DECID | ||
Theorem | pm2.6dc 828 | Case elimination for a decidable proposition. Based on theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 25-Mar-2018.) |
DECID | ||
Theorem | jadc 829 | 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 | ||
Theorem | jaddc 830 | 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 | ||
Theorem | pm2.61dc 831 | 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 | ||
Theorem | pm2.5gdc 832 | 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 | ||
Theorem | pm2.5dc 833 | 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 | ||
Theorem | pm2.521gdc 834 | A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107, under a decidability condition. (Contributed by BJ, 28-Oct-2023.) |
DECID | ||
Theorem | pm2.521dc 835 | Theorem *2.521 of [WhiteheadRussell] p. 107, but with an additional decidability condition. Note that by replacing in proof pm2.52 628 with conax1k 626, 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 | ||
Theorem | pm2.521dcALT 836 | Alternate proof of pm2.521dc 835. (Contributed by Jim Kingdon, 5-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
DECID | ||
Theorem | con34bdc 837 | Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116, but for a decidable proposition. (Contributed by Jim Kingdon, 24-Apr-2018.) |
DECID | ||
Theorem | notnotbdc 838 | Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot 601, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.) |
DECID | ||
Theorem | con1biimdc 839 | Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.) |
DECID | ||
Theorem | con1bidc 840 | Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.) |
DECID DECID | ||
Theorem | con2bidc 841 | Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.) |
DECID DECID | ||
Theorem | con1biddc 842 | A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.) |
DECID DECID | ||
Theorem | con1biidc 843 | A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.) |
DECID DECID | ||
Theorem | con1bdc 844 | Contraposition. Bidirectional version of con1dc 822. (Contributed by NM, 5-Aug-1993.) |
DECID DECID | ||
Theorem | con2biidc 845 | A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.) |
DECID DECID | ||
Theorem | con2biddc 846 | A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.) |
DECID DECID | ||
Theorem | condandc 847 | 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 pm2.65 631, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.) |
DECID | ||
Theorem | bijadc 848 | Combine antecedents into a single biconditional. This inference is reminiscent of jadc 829. (Contributed by Jim Kingdon, 4-May-2018.) |
DECID | ||
Theorem | pm5.18dc 849 | 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 | ||
Theorem | dfandc 850 | 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 609. (Contributed by Jim Kingdon, 30-Apr-2018.) |
DECID DECID | ||
Theorem | pm2.13dc 851 | 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 | ||
Theorem | pm4.63dc 852 | Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.) |
DECID DECID | ||
Theorem | pm4.67dc 853 | Theorem *4.67 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.) |
DECID DECID | ||
Theorem | imanst 854 | 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 | ||
Theorem | imandc 855 | Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 660, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.) |
DECID | ||
Theorem | pm4.14dc 856 | Theorem *4.14 of [WhiteheadRussell] p. 117, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.) |
DECID | ||
Theorem | pm2.54dc 857 | Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 694, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.) |
DECID | ||
Theorem | dfordc 858 | Definition of disjunction in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 694, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.) |
DECID | ||
Theorem | pm2.25dc 859 | 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 | ||
Theorem | pm2.68dc 860 | Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 720 and one half of dfor2dc 861. (Contributed by Jim Kingdon, 27-Mar-2018.) |
DECID | ||
Theorem | dfor2dc 861 | 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 | ||
Theorem | imimorbdc 862 | Simplify an implication between implications, for a decidable proposition. (Contributed by Jim Kingdon, 18-Mar-2018.) |
DECID | ||
Theorem | imordc 863 | Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 693, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.) |
DECID | ||
Theorem | pm4.62dc 864 | 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 | ||
Theorem | ianordc 865 | 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, pm3.14 725, holds for all propositions, but the equivalence only holds where one proposition is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.) |
DECID | ||
Theorem | pm4.64dc 866 | Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability condition. The reverse direction, pm2.53 694, holds for all propositions. (Contributed by Jim Kingdon, 2-May-2018.) |
DECID | ||
Theorem | pm4.66dc 867 | Theorem *4.66 of [WhiteheadRussell] p. 120, given a decidability condition. (Contributed by Jim Kingdon, 2-May-2018.) |
DECID | ||
Theorem | pm4.54dc 868 | 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 | ||
Theorem | pm4.79dc 869 | 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 | ||
Theorem | pm5.17dc 870 | 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 | ||
Theorem | pm2.85dc 871 | 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 | ||
Theorem | orimdidc 872 | Disjunction distributes over implication. The forward direction, pm2.76 780, is valid intuitionistically. The reverse direction holds if is decidable, as can be seen at pm2.85dc 871. (Contributed by Jim Kingdon, 1-Apr-2018.) |
DECID | ||
Theorem | pm2.26dc 873 | Decidable proposition version of theorem *2.26 of [WhiteheadRussell] p. 104. (Contributed by Jim Kingdon, 20-Apr-2018.) |
DECID | ||
Theorem | pm4.81dc 874 | Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions. This one needs a decidability condition, but compare with pm4.8 679 which holds for all propositions. (Contributed by Jim Kingdon, 4-Jul-2018.) |
DECID | ||
Theorem | pm5.11dc 875 | 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 | ||
Theorem | pm5.12dc 876 | 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 | ||
Theorem | pm5.14dc 877 | 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 | ||
Theorem | pm5.13dc 878 | 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 | ||
Theorem | pm5.55dc 879 | 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 | ||
Theorem | peircedc 880 | Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 804, condc 819, or notnotrdc 811 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 | ||
Theorem | looinvdc 881 | The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 861, we can see that this expresses "disjunction commutes." Theorem *2.69 of [WhiteheadRussell] p. 108 (plus the decidability condition). (Contributed by NM, 12-Aug-2004.) |
DECID | ||
Theorem | pm5.21nd 882 | Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.) |
Theorem | pm5.35 883 | Theorem *5.35 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm5.54dc 884 | A conjunction is equivalent to one of its conjuncts, given a decidable conjunct. Based on theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.) |
DECID | ||
Theorem | baib 885 | Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.) |
Theorem | baibr 886 | Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.) |
Theorem | rbaib 887 | Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Theorem | rbaibr 888 | Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Theorem | baibd 889 | Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Theorem | rbaibd 890 | Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Theorem | pm5.44 891 | Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm5.6dc 892 | Conjunction in antecedent versus disjunction in consequent, for a decidable proposition. Theorem *5.6 of [WhiteheadRussell] p. 125, with decidability condition added. The reverse implication holds for all propositions (see pm5.6r 893). (Contributed by Jim Kingdon, 2-Apr-2018.) |
DECID | ||
Theorem | pm5.6r 893 | Conjunction in antecedent versus disjunction in consequent. One direction of Theorem *5.6 of [WhiteheadRussell] p. 125. If is decidable, the converse also holds (see pm5.6dc 892). (Contributed by Jim Kingdon, 4-Aug-2018.) |
Theorem | orcanai 894 | Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.) |
Theorem | intnan 895 | Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.) |
Theorem | intnanr 896 | Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.) |
Theorem | intnand 897 | Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.) |
Theorem | intnanrd 898 | Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.) |
Theorem | dcan 899 | A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.) |
DECID DECID DECID | ||
Theorem | dcor 900 | A disjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.) |
DECID DECID DECID |
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