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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | orim1d 701 | Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.) |
Theorem | orim2d 702 | Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.) |
Theorem | orim2 703 | Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.) |
Theorem | orbi2d 704 | 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 705 | Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | orbi1 706 | Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) |
Theorem | orbi12d 707 | Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 5-Aug-1993.) |
Theorem | pm5.61 708 | Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.) |
Theorem | jaoian 709 | Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.) |
Theorem | jaodan 710 | Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.) |
Theorem | mpjaodan 711 | Eliminate a disjunction in a deduction. A translation of natural deduction rule E ( elimination). (Contributed by Mario Carneiro, 29-May-2016.) |
Theorem | pm4.77 712 | Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.63 713 | Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.64 714 | Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm5.53 715 | Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.38 716 | Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
Theorem | pm2.36 717 | Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
Theorem | pm2.37 718 | Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
Theorem | pm2.73 719 | Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.74 720 | Theorem *2.74 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2015.) |
Theorem | pm2.76 721 | Theorem *2.76 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Theorem | pm2.75 722 | Theorem *2.75 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Jan-2013.) |
Theorem | pm2.8 723 | Theorem *2.8 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2015.) |
Theorem | pm2.81 724 | Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.82 725 | Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm3.2ni 726 | Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.) |
Theorem | orabs 727 | 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 728 | Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton, 23-Jun-2005.) (Proof shortened by Wolf Lammen, 10-Nov-2013.) |
Theorem | ordi 729 | 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 730 | Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) |
Theorem | andi 731 | 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 732 | Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) |
Theorem | orddi 733 | Double distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) |
Theorem | anddi 734 | Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) |
Theorem | pm4.39 735 | Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm4.72 736 | 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 | pm5.16 737 | Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Theorem | biort 738 | A wff is disjoined with truth is true. (Contributed by NM, 23-May-1999.) |
Syntax | wstab 739 | Extend wff definition to include stability. |
STAB | ||
Definition | df-stab 740 |
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 741 | Every formula of the form is stable. Uses notnotnot 628. (Contributed by David A. Wheeler, 13-Aug-2018.) |
STAB | ||
Syntax | wdc 742 | Extend wff definition to include decidability. |
DECID | ||
Definition | df-dc 743 |
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 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.) |
DECID | ||
Theorem | exmiddc 744 | 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 745 | 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 746 | A decidable proposition is decidable when negated. (Contributed by Jim Kingdon, 25-Mar-2018.) |
DECID DECID | ||
Theorem | dcbii 747 | The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.) |
DECID DECID | ||
Theorem | dcbid 748 | 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 743), double negation elimination (notnotrdc 751), or contraposition (condc 749). 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 749 |
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 750 | 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 | notnotrdc 751 | Double negation elimination for a decidable proposition. The converse, notnot 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 752 | Decidability implies stability. The converse is not necessarily true. (Contributed by David A. Wheeler, 13-Aug-2018.) |
DECID STAB | ||
Theorem | con1dc 753 | Contraposition for a decidable proposition. Based on theorem *2.15 of [WhiteheadRussell] p. 102. (Contributed by Jim Kingdon, 29-Mar-2018.) |
DECID | ||
Theorem | con4biddc 754 | A contraposition deduction. (Contributed by Jim Kingdon, 18-May-2018.) |
DECID DECID DECID DECID | ||
Theorem | impidc 755 | An importation inference for a decidable consequent. (Contributed by Jim Kingdon, 30-Apr-2018.) |
DECID DECID | ||
Theorem | simprimdc 756 | 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 757 | 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 758 | Deduction eliminating a decidable antecedent. (Contributed by Jim Kingdon, 4-May-2018.) |
DECID | ||
Theorem | pm2.6dc 759 | 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 760 | 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 761 | 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 762 | 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 763 | 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 764 | Theorem *2.521 of [WhiteheadRussell] p. 107, but with an additional decidability condition. (Contributed by Jim Kingdon, 5-May-2018.) |
DECID | ||
Theorem | con34bdc 765 | Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116, but for a decidable proposition. (Contributed by Jim Kingdon, 24-Apr-2018.) |
DECID | ||
Theorem | notnotbdc 766 | Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot 559, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.) |
DECID | ||
Theorem | con1biimdc 767 | Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.) |
DECID | ||
Theorem | con1bidc 768 | Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.) |
DECID DECID | ||
Theorem | con2bidc 769 | Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.) |
DECID DECID | ||
Theorem | con1biddc 770 | A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.) |
DECID DECID | ||
Theorem | con1biidc 771 | A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.) |
DECID DECID | ||
Theorem | con1bdc 772 | Contraposition. Bidirectional version of con1dc 753. (Contributed by NM, 5-Aug-1993.) |
DECID DECID | ||
Theorem | con2biidc 773 | A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.) |
DECID DECID | ||
Theorem | con2biddc 774 | A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.) |
DECID DECID | ||
Theorem | condandc 775 | 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 pm2.65 585, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.) |
DECID | ||
Theorem | bijadc 776 | Combine antecedents into a single biconditional. This inference is reminiscent of jadc 760. (Contributed by Jim Kingdon, 4-May-2018.) |
DECID | ||
Theorem | pm5.18dc 777 | 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 778 | 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 566. (Contributed by Jim Kingdon, 30-Apr-2018.) |
DECID DECID | ||
Theorem | pm2.13dc 779 | 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 780 | Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.) |
DECID DECID | ||
Theorem | pm4.67dc 781 | Theorem *4.67 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.) |
DECID DECID | ||
Theorem | annimim 782 | 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 845. (Contributed by Jim Kingdon, 24-Dec-2017.) |
Theorem | pm4.65r 783 | 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 784 | An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.) |
DECID DECID DECID | ||
Theorem | imanim 785 | Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 786. (Contributed by Jim Kingdon, 24-Dec-2017.) |
Theorem | imandc 786 | Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 785, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.) |
DECID | ||
Theorem | pm4.14dc 787 | Theorem *4.14 of [WhiteheadRussell] p. 117, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.) |
DECID | ||
Theorem | pm3.37dc 788 | Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.) |
DECID | ||
Theorem | pm4.15 789 | Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.) |
Theorem | pm2.54dc 790 | Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 641, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.) |
DECID | ||
Theorem | dfordc 791 | Definition of 'or' in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 641, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.) |
DECID | ||
Theorem | pm2.25dc 792 | 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 793 | Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 667 and one half of dfor2dc 794. (Contributed by Jim Kingdon, 27-Mar-2018.) |
DECID | ||
Theorem | dfor2dc 794 | 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 795 | Simplify an implication between implications, for a decidable proposition. (Contributed by Jim Kingdon, 18-Mar-2018.) |
DECID | ||
Theorem | imordc 796 | Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 797, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.) |
DECID | ||
Theorem | imorr 797 | 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 796. (Contributed by Jim Kingdon, 21-Jul-2018.) |
Theorem | pm4.62dc 798 | 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 799 | 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, pm3.14 670, holds for all propositions, but the equivalence only holds where one proposition is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.) |
DECID | ||
Theorem | oibabs 800 | Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) |
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