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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | imorri 701 | Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Theorem | ioran 702 | Negated disjunction in terms of conjunction. This version of DeMorgan's law is a biconditional for all propositions (not just decidable ones), unlike oranim 843, anordc 900, or ianordc 835. Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Theorem | pm3.14 703 | Theorem *3.14 of [WhiteheadRussell] p. 111. One direction of De Morgan's law). The biconditional holds for decidable propositions as seen at ianordc 835. The converse holds for decidable propositions, as seen at pm3.13dc 903. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Theorem | pm3.1 704 | Theorem *3.1 of [WhiteheadRussell] p. 111. The converse holds for decidable propositions, as seen at anordc 900. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Theorem | jao 705 | Disjunction of antecedents. Compare Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 4-Apr-2013.) |
Theorem | pm1.2 706 | Axiom *1.2 (Taut) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 10-Mar-2013.) |
Theorem | oridm 707 | Idempotent law for disjunction. Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 16-Apr-2011.) (Proof shortened by Wolf Lammen, 10-Mar-2013.) |
Theorem | pm4.25 708 | Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) |
Theorem | orim12i 709 | Disjoin antecedents and consequents of two premises. (Contributed by NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.) |
Theorem | orim1i 710 | Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.) |
Theorem | orim2i 711 | Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.) |
Theorem | orbi2i 712 | Inference adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Dec-2012.) |
Theorem | orbi1i 713 | Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | orbi12i 714 | Infer the disjunction of two equivalences. (Contributed by NM, 5-Aug-1993.) |
Theorem | pm1.5 715 | Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) |
Theorem | or12 716 | Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.) |
Theorem | orass 717 | Associative law for disjunction. Theorem *4.33 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Theorem | pm2.31 718 | Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.32 719 | Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.) |
Theorem | or32 720 | A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Theorem | or4 721 | Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.) |
Theorem | or42 722 | Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.) |
Theorem | orordi 723 | Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.) |
Theorem | orordir 724 | Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.) |
Theorem | pm2.3 725 | Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.41 726 | Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.42 727 | Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.4 728 | Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm4.44 729 | Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) |
Theorem | mtord 730 | A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Theorem | pm4.45 731 | Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm3.48 732 | Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM, 28-Jan-1997.) (Revised by NM, 1-Dec-2012.) |
Theorem | orim12d 733 | Disjoin antecedents and consequents in a deduction. (Contributed by NM, 10-May-1994.) |
Theorem | orim1d 734 | Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.) |
Theorem | orim2d 735 | Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.) |
Theorem | orim2 736 | Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.) |
Theorem | orbi2d 737 | 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 738 | Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | orbi1 739 | Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) |
Theorem | orbi12d 740 | Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 5-Aug-1993.) |
Theorem | pm5.61 741 | Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.) |
Theorem | jaoian 742 | Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.) |
Theorem | jao1i 743 | Add a disjunct in the antecedent of an implication. (Contributed by Rodolfo Medina, 24-Sep-2010.) |
Theorem | jaodan 744 | Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.) |
Theorem | mpjaodan 745 | Eliminate a disjunction in a deduction. A translation of natural deduction rule E ( elimination). (Contributed by Mario Carneiro, 29-May-2016.) |
Theorem | pm4.77 746 | Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.63 747 | Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.64 748 | Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm5.53 749 | Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.38 750 | Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
Theorem | pm2.36 751 | Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
Theorem | pm2.37 752 | Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
Theorem | pm2.73 753 | Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.74 754 | Theorem *2.74 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2015.) |
Theorem | pm2.76 755 | Theorem *2.76 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Theorem | pm2.75 756 | Theorem *2.75 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Jan-2013.) |
Theorem | pm2.8 757 | Theorem *2.8 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2015.) |
Theorem | pm2.81 758 | Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.82 759 | Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm3.2ni 760 | Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.) |
Theorem | orabs 761 | 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 762 | Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton, 23-Jun-2005.) (Proof shortened by Wolf Lammen, 10-Nov-2013.) |
Theorem | ordi 763 | 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 764 | Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) |
Theorem | andi 765 | 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 766 | Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) |
Theorem | orddi 767 | Double distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) |
Theorem | anddi 768 | Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) |
Theorem | pm4.39 769 | Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm4.72 770 | 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 | pm5.16 771 | Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Theorem | biort 772 | A wff is disjoined with truth is true. (Contributed by NM, 23-May-1999.) |
Syntax | wstab 773 | Extend wff definition to include stability. |
STAB | ||
Definition | df-stab 774 |
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 | stbid 775 | The equivalent of a stable proposition is stable. (Contributed by Jim Kingdon, 12-Aug-2022.) |
STAB STAB | ||
Theorem | stabnot 776 | Every formula of the form is stable. Uses notnotnot 661. (Contributed by David A. Wheeler, 13-Aug-2018.) |
STAB | ||
Theorem | imanst 777 | 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 | ||
Syntax | wdc 778 | Extend wff definition to include decidability. |
DECID | ||
Definition | df-dc 779 |
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 785, exmiddc 780, peircedc 856, or notnotrdc 787, any of which would correspond to the assertion that all propositions are decidable. (Contributed by Jim Kingdon, 11-Mar-2018.) |
DECID | ||
Theorem | exmiddc 780 | 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 781 | 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 782 | A decidable proposition is decidable when negated. (Contributed by Jim Kingdon, 25-Mar-2018.) |
DECID DECID | ||
Theorem | dcbii 783 | The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.) |
DECID DECID | ||
Theorem | dcbid 784 | 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 779), double negation elimination (notnotrdc 787), or contraposition (condc 785). 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 785 |
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 786 | 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 pm2.01 579 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.) |
DECID | ||
Theorem | notnotrdc 787 | Double negation elimination for a decidable proposition. The converse, notnot 592, 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 788 | Decidability implies stability. The converse is not necessarily true. (Contributed by David A. Wheeler, 13-Aug-2018.) |
DECID STAB | ||
Theorem | con1dc 789 | Contraposition for a decidable proposition. Based on theorem *2.15 of [WhiteheadRussell] p. 102. (Contributed by Jim Kingdon, 29-Mar-2018.) |
DECID | ||
Theorem | con4biddc 790 | A contraposition deduction. (Contributed by Jim Kingdon, 18-May-2018.) |
DECID DECID DECID DECID | ||
Theorem | impidc 791 | An importation inference for a decidable consequent. (Contributed by Jim Kingdon, 30-Apr-2018.) |
DECID DECID | ||
Theorem | simprimdc 792 | 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 793 | 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 794 | Deduction eliminating a decidable antecedent. (Contributed by Jim Kingdon, 4-May-2018.) |
DECID | ||
Theorem | pm2.6dc 795 | 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 796 | 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 797 | 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 798 | 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 799 | 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 800 | Theorem *2.521 of [WhiteheadRussell] p. 107, but with an additional decidability condition. (Contributed by Jim Kingdon, 5-May-2018.) |
DECID |
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