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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | orim2d 701 | Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.) |
Theorem | orim2 702 | Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.) |
Theorem | orbi2d 703 | Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Theorem | orbi1d 704 | Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | orbi1 705 | Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) |
Theorem | orbi12d 706 | Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 5-Aug-1993.) |
Theorem | pm5.61 707 | Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.) |
Theorem | jaoian 708 | Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.) |
Theorem | jaodan 709 | Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.) |
Theorem | mpjaodan 710 | Eliminate a disjunction in a deduction. A translation of natural deduction rule E ( elimination). (Contributed by Mario Carneiro, 29-May-2016.) |
Theorem | pm4.77 711 | Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.63 712 | Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.64 713 | Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm5.53 714 | Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.38 715 | Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
Theorem | pm2.36 716 | Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
Theorem | pm2.37 717 | Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
Theorem | pm2.73 718 | Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.74 719 | Theorem *2.74 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2015.) |
Theorem | pm2.76 720 | Theorem *2.76 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Theorem | pm2.75 721 | Theorem *2.75 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Jan-2013.) |
Theorem | pm2.8 722 | Theorem *2.8 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2015.) |
Theorem | pm2.81 723 | Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.82 724 | Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm3.2ni 725 | Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.) |
Theorem | orabs 726 | 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.) |
Theorem | oranabs 727 | Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton, 23-Jun-2005.) (Proof shortened by Wolf Lammen, 10-Nov-2013.) |
Theorem | ordi 728 | Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Theorem | ordir 729 | Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) |
Theorem | andi 730 | 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.) |
Theorem | andir 731 | Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) |
Theorem | orddi 732 | Double distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) |
Theorem | anddi 733 | Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) |
Theorem | pm4.39 734 | Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm4.72 735 | 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.) |
Theorem | pm5.16 736 | Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Theorem | biort 737 | A wff is disjoined with truth is true. (Contributed by NM, 23-May-1999.) |
Syntax | wstab 738 | Extend wff definition to include stability. |
STAB | ||
Definition | df-stab 739 |
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 "x = y is stable". (Contributed by David A. Wheeler, 13-Aug-2018.) |
STAB | ||
Theorem | stabnot 740 | Every formula of the form is stable. Uses notnotnot 627. (Contributed by David A. Wheeler, 13-Aug-2018.) |
STAB | ||
Syntax | wdc 741 | Extend wff definition to include decidability. |
DECID | ||
Definition | df-dc 742 |
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 "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.) |
DECID | ||
Theorem | exmiddc 743 | 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 744 | 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 | dcn 745 | A decidable proposition is decidable when negated. (Contributed by Jim Kingdon, 25-Mar-2018.) |
DECID DECID | ||
Theorem | dcbii 746 | The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.) |
DECID DECID | ||
Theorem | dcbid 747 | The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.) |
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 742), double negation elimination (notnotdc 765), or contraposition (condc 748). 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. | ||
Theorem | condc 748 |
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.) |
DECID | ||
Theorem | pm2.18dc 749 | 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 pm2.01 546 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.) |
DECID | ||
Theorem | notnot2dc 750 | Double negation elimination for a decidable proposition. The converse, notnot1 559, 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 | dcimpstab 751 | Decidability implies stability. The converse is not necessarily true. (Contributed by David A. Wheeler, 13-Aug-2018.) |
DECID STAB | ||
Theorem | con1dc 752 | Contraposition for a decidable proposition. Based on theorem *2.15 of [WhiteheadRussell] p. 102. (Contributed by Jim Kingdon, 29-Mar-2018.) |
DECID | ||
Theorem | con4biddc 753 | A contraposition deduction. (Contributed by Jim Kingdon, 18-May-2018.) |
DECID DECID DECID DECID | ||
Theorem | impidc 754 | An importation inference for a decidable consequent. (Contributed by Jim Kingdon, 30-Apr-2018.) |
DECID DECID | ||
Theorem | simprimdc 755 | 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 756 | 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 757 | Deduction eliminating a decidable antecedent. (Contributed by Jim Kingdon, 4-May-2018.) |
DECID | ||
Theorem | pm2.6dc 758 | 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 759 | 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 760 | 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 761 | Case elimination for a decidable proposition. Based on theorem *2.61 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 29-Mar-2018.) |
DECID | ||
Theorem | pm2.5dc 762 | Negating an implication for a decidable antecedent. Based on theorem *2.5 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 29-Mar-2018.) |
DECID | ||
Theorem | pm2.521dc 763 | Theorem *2.521 of [WhiteheadRussell] p. 107, but with an additional decidability condition. (Contributed by Jim Kingdon, 5-May-2018.) |
DECID | ||
Theorem | con34bdc 764 | Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116, but for a decidable proposition. (Contributed by Jim Kingdon, 24-Apr-2018.) |
DECID | ||
Theorem | notnotdc 765 | Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot1 559, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.) |
DECID | ||
Theorem | con1biimdc 766 | Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.) |
DECID | ||
Theorem | con1bidc 767 | Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.) |
DECID DECID | ||
Theorem | con2bidc 768 | Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.) |
DECID DECID | ||
Theorem | con1biddc 769 | A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.) |
DECID DECID | ||
Theorem | con1biidc 770 | A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.) |
DECID DECID | ||
Theorem | con1bdc 771 | Contraposition. Bidirectional version of con1dc 752. (Contributed by NM, 5-Aug-1993.) |
DECID DECID | ||
Theorem | con2biidc 772 | A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.) |
DECID DECID | ||
Theorem | con2biddc 773 | A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.) |
DECID DECID | ||
Theorem | condandc 774 | 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 pm2.65 584, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.) |
DECID | ||
Theorem | bijadc 775 | Combine antecedents into a single biconditional. This inference is reminiscent of jadc 759. (Contributed by Jim Kingdon, 4-May-2018.) |
DECID | ||
Theorem | pm5.18dc 776 | 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 777 | 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 565. (Contributed by Jim Kingdon, 30-Apr-2018.) |
DECID DECID | ||
Theorem | pm2.13dc 778 | 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 779 | Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.) |
DECID DECID | ||
Theorem | pm4.67dc 780 | Theorem *4.67 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.) |
DECID DECID | ||
Theorem | annimim 781 | Express conjunction in terms of implication. One direction of Theorem *4.61 of [WhiteheadRussell] p. 120. The converse holds for decidable propositions, as can be seen at annimdc 844. (Contributed by Jim Kingdon, 24-Dec-2017.) |
Theorem | pm4.65r 782 | One direction of Theorem *4.65 of [WhiteheadRussell] p. 120. The converse holds in classical logic. (Contributed by Jim Kingdon, 28-Jul-2018.) |
Theorem | dcim 783 | An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.) |
DECID DECID DECID | ||
Theorem | imanim 784 | Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 785. (Contributed by Jim Kingdon, 24-Dec-2017.) |
Theorem | imandc 785 | Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 784, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.) |
DECID | ||
Theorem | pm4.14dc 786 | Theorem *4.14 of [WhiteheadRussell] p. 117, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.) |
DECID | ||
Theorem | pm3.37dc 787 | Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.) |
DECID | ||
Theorem | pm4.15 788 | Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.) |
Theorem | pm2.54dc 789 | Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 640, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.) |
DECID | ||
Theorem | dfordc 790 | Definition of 'or' in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 640, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.) |
DECID | ||
Theorem | pm2.25dc 791 | 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 792 | Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 666 and one half of dfor2dc 793. (Contributed by Jim Kingdon, 27-Mar-2018.) |
DECID | ||
Theorem | dfor2dc 793 | Logical 'or' 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 794 | Simplify an implication between implications, for a decidable proposition. (Contributed by Jim Kingdon, 18-Mar-2018.) |
DECID | ||
Theorem | imordc 795 | Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 796, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.) |
DECID | ||
Theorem | imorr 796 | Implication in terms of disjunction. One direction of theorem *4.6 of [WhiteheadRussell] p. 120. The converse holds for decidable propositions, as seen at imordc 795. (Contributed by Jim Kingdon, 21-Jul-2018.) |
Theorem | pm4.62dc 797 | 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 798 | 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, pm3.14 669, holds for all propositions, but the equivalence only holds where one proposition is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.) |
DECID | ||
Theorem | oibabs 799 | Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) |
Theorem | pm4.64dc 800 | Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability condition. The reverse direction, pm2.53 640, holds for all propositions. (Contributed by Jim Kingdon, 2-May-2018.) |
DECID |
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