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Type | Label | Description |
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Statement | ||
Theorem | dummylink 1 |
(Note: This inference rule and the next one, idi 2, will
normally
never appear in a completed proof. It can be ignored if you are using
this database to assist learning logic - please start with the statement
wn 3 instead.)
This is a technical inference to assist proof development. It provides a temporary way to add an independent subproof to a proof under development, for later assignment to a normal proof step. The metamath program's Proof Assistant requires proofs to be developed backwards from the conclusion with no gaps, and it has no mechanism that lets the user to work on isolated subproofs. This inference provides a workaround for this limitation. It can be inserted at any point in a proof to allow an independent subproof to be developed on the side, for later use as part of the final proof. Instructions: (1) Assign this inference to any unknown step in the proof. Typically, the last unknown step is the most convenient, since 'assign last' can be used. This step will be replicated in hypothesis dummylink.1, from where the development of the main proof can continue. (2) Develop the independent subproof backwards from hypothesis dummylink.2. If desired, use a 'let' command to pre-assign the conclusion of the independent subproof to dummylink.2. (3) After the independent subproof is complete, use 'improve all' to assign it automatically to an unknown step in the main proof that matches it. (4) After the entire proof is complete, use 'minimize *' to clean up (discard) all dummylink references automatically. This inference was originally designed to assist importing partially completed Proof Worksheets from the mmj2 Proof Assistant GUI, but it can also be useful on its own. Interestingly, no axioms are required for its proof. |
Theorem | idi 2 | Inference form of id 18. This inference rule, which requires no axioms for its proof, is useful as a copy-paste mechanism during proof development in mmj2. It is normally not referenced in the final version of a proof, since it is always redundant and can be removed using the 'minimize *' command in the metamath program's Proof Assistant. (Contributed by Alan Sare, 31-Dec-2011.) |
Syntax | wn 3 | If is a wff, so is or "not ." Part of the recursive definition of a wff (well-formed formula). In classical logic (which is our logic), a wff is interpreted as either true or false. So if is true, then is false; if is false, then is true. Traditionally, Greek letters are used to represent wffs, and we follow this convention. In propositional calculus, we define only wffs built up from other wffs, i.e. there is no starting or "atomic" wff. Later, in predicate calculus, we will extend the basic wff definition by including atomic wffs (weq 1384 and wel 1386). |
Syntax | wi 4 | If and are wff's, so is or " implies ." Part of the recursive definition of a wff. The resulting wff is (interpreted as) false when is true and is false; it is true otherwise. (Think of the truth table for an OR gate with input connected through an inverter.) The left-hand wff is called the antecedent, and the right-hand wff is called the consequent. In the case of , the middle may be informally called either an antecedent or part of the consequent depending on context. |
Axiom | ax-1 5 |
Axiom Simp. Axiom A1 of [Margaris] p.
49. One of the 3 axioms of
propositional calculus. The 3 axioms are also given as Definition 2.1 of
[Hamilton] p. 28. This axiom is called
Simp or "the principle of
simplification" in Principia Mathematica (Theorem *2.02 of
[WhiteheadRussell] p. 100)
because "it enables us to pass from the joint
assertion of and to
the assertion of simply."
General remarks: Propositional calculus (axioms ax-1 5 through ax-3 7 and rule ax-mp 8) can be thought of as asserting formulas that are universally "true" when their variables are replaced by any combination of "true" and "false." Propositional calculus was first formalized by Frege in 1879, using as his axioms (in addition to rule ax-mp 8) the wffs ax-1 5, ax-2 6, pm2.04 75, con3 550, notnot2 718, and notnot1 541. Around 1930, Lukasiewicz simplified the system by eliminating the third (which follows from the first two, as you can see by looking at the proof of pm2.04 75) and replacing the last three with our ax-3 7. (Thanks to Ted Ulrich for this information.) The theorems of propositional calculus are also called tautologies. Tautologies can be proved very simply using truth tables, based on the true/false interpretation of propositional calculus. To do this, we assign all possible combinations of true and false to the wff variables and verify that the result (using the rules described in wi 4 and wn 3) always evaluates to true. This is called the semantic approach. Our approach is called the syntactic approach, in which everything is derived from axioms. A metatheorem called the Completeness Theorem for Propositional Calculus shows that the two approaches are equivalent and even provides an algorithm for automatically generating syntactic proofs from a truth table. Those proofs, however, tend to be long, since truth tables grow exponentially with the number of variables, and the much shorter proofs that we show here were found manually. |
Axiom | ax-2 6 | Axiom Frege. Axiom A2 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. It "distributes" an antecedent over two consequents. This axiom was part of Frege's original system and is known as Frege in the literature. It is also proved as Theorem *2.77 of [WhiteheadRussell] p. 108. The other direction of this axiom also turns out to be true, as demonstrated by pm5.41 239. |
Axiom | ax-3 7 | Axiom Transp. Axiom A3 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. It 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 axiom is called Transp or "the principle of transposition" in Principia Mathematica (Theorem *2.17 of [WhiteheadRussell] p. 103). 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. |
Axiom | ax-mp 8 |
Rule of Modus Ponens. The postulated inference rule of propositional
calculus. See e.g. Rule 1 of [Hamilton] p. 73. The rule says, "if
is true,
and implies
, then must also be
true." This rule is sometimes called "detachment," since
it detaches
the minor premise from the major premise.
Note: In some web page displays such as the Statement List, the symbols "&" and "=>" informally indicate the relationship between the hypotheses and the assertion (conclusion), abbreviating the English words "and" and "implies." They are not part of the formal language. |
Theorem | mp2b 9 | A double modus ponens inference. (Contributed by Mario Carneiro, 24-Jan-2013.) |
Theorem | a1i 10 | Inference derived from axiom ax-1 5. See a1d 21 for an explanation of our informal use of the terms "inference" and "deduction." See also the comment in syld 39. |
Theorem | a2i 11 | Inference derived from axiom ax-2 6. |
Theorem | imim2i 12 | Inference adding common antecedents in an implication. |
Theorem | mpd 13 | A modus ponens deduction. |
Theorem | syl 14 | An inference version of the transitive laws for implication imim2 48 and imim1 69, which Russell and Whitehead call "the principle of the syllogism...because...the syllogism in Barbara is derived from them" (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Some authors call this law a "hypothetical syllogism." (The proof was shortened by O'Cat, 20-Oct-2011.) (The proof was shortened by Wolf Lammen, 26-Jul-2012.) |
Theorem | mpi 15 | A nested modus ponens inference. (The proof was shortened by Stefan Allan, 20-Mar-2006.) |
Theorem | mp2 16 | A double modus ponens inference. (The proof was shortened by Wolf Lammen, 23-Jul-2013.) |
Theorem | 3syl 17 | Inference chaining two syllogisms. |
Theorem | id 18 | Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. For another version of the proof directly from axioms, see id1 19. (The proof was shortened by Stefan Allan, 20-Mar-2006.) |
Theorem | id1 19 | Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. This version is proved directly from the axioms for demonstration purposes. This proof is a popular example in the literature and is identical, step for step, to the proofs of Theorem 1 of [Margaris] p. 51, Example 2.7(a) of [Hamilton] p. 31, Lemma 10.3 of [BellMachover] p. 36, and Lemma 1.8 of [Mendelson] p. 36. It is also "Our first proof" in Hirst and Hirst's A Primer for Logic and Proof p. 17 (PDF p. 23) at http://www.mathsci.appstate.edu/~jlh/primer/hirst.pdf. For a shorter version of the proof that takes advantage of previously proved theorems, see id 18. |
Theorem | idd 20 | Principle of identity with antecedent. |
Theorem | a1d 21 |
Deduction introducing an embedded antecedent. (The proof was revised by
Stefan Allan, 20-Mar-2006.)
Naming convention: We often call a theorem a "deduction" and suffix its label with "d" whenever the hypotheses and conclusion are each prefixed with the same antecedent. This allows us to use the theorem in places where (in traditional textbook formalizations) the standard Deduction Theorem would be used; here would be replaced with a conjunction (df-an 775) of the hypotheses of the would-be deduction. By contrast, we tend to call the simpler version with no common antecedent an "inference" and suffix its label with "i"; compare theorem a1i 10. Finally, a "theorem" would be the form with no hypotheses; in this case the "theorem" form would be the original axiom ax-1 5. We usually show the theorem form without a suffix on its label (e.g. pm2.43 46 vs. pm2.43i 42 vs. pm2.43d 43). |
Theorem | a2d 22 | Deduction distributing an embedded antecedent. |
Theorem | a1ii 23 | Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.) (The proof was shortened by Wolf Lammen, 23-Jul-2013.) |
Theorem | sylcom 24 | Syllogism inference with commutation of antecedents. (The proof was shortened by O'Cat, 2-Feb-2006.) (The proof was shortened by Stefan Allan, 23-Feb-2006.) |
Theorem | syl5com 25 | Syllogism inference with commuted antecedents. |
Theorem | com12 26 | Inference that swaps (commutes) antecedents in an implication. (The proof was shortened by Wolf Lammen, 4-Aug-2012.) |
Theorem | syl5 27 | A syllogism rule of inference. The second premise is used to replace the second antecedent of the first premise. (The proof was shortened by Wolf Lammen, 25-May-2013.) |
Theorem | syl6 28 | A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (The proof was shortened by Wolf Lammen, 30-Jul-2012.) |
Theorem | syl56 29 | Combine syl5 27 and syl6 28. |
Theorem | syl6com 30 | Syllogism inference with commuted antecedents. |
Theorem | mpcom 31 | Modus ponens inference with commutation of antecedents. |
Theorem | syli 32 | Syllogism inference with common nested antecedent. |
Theorem | syl2im 33 | Replace two antecedents. Implication-only version of syl2an 272. (Contributed by Wolf Lammen, 14-May-2013.) |
Theorem | pm2.27 34 | This theorem, called "Assertion," can be thought of as closed form of modus ponens ax-mp 8. Theorem *2.27 of [WhiteheadRussell] p. 104. |
Theorem | mpdd 35 | A nested modus ponens deduction. |
Theorem | mpid 36 | A nested modus ponens deduction. |
Theorem | mpdi 37 | A nested modus ponens deduction. (The proof was shortened by O'Cat, 15-Jan-2008.) |
Theorem | mpii 38 | A doubly nested modus ponens inference. (The proof was shortened by Wolf Lammen, 31-Jul-2012.) |
Theorem | syld 39 |
Syllogism deduction. (The proof was shortened by O'Cat, 19-Feb-2008.)
(The proof was shortened by Wolf Lammen, 3-Aug-2012.)
Notice that syld 39 has the same form as syl 14 with added in front of each hypothesis and conclusion. When all theorems referenced in a proof are converted in this way, we can replace with a hypothesis of the proof, allowing the hypothesis to be eliminated with id 18 and become an antecedent. The Deduction Theorem for propositional calculus, e.g. Theorem 3 in [Margaris] p. 56, tells us that this procedure is always possible. |
Theorem | mp2d 40 | A double modus ponens deduction. (The proof was shortened by Wolf Lammen, 23-Jul-2013.) |
Theorem | a1dd 41 | Deduction introducing a nested embedded antecedent. (The proof was shortened by O'Cat, 15-Jan-2008.) |
Theorem | pm2.43i 42 | Inference absorbing redundant antecedent. (The proof was shortened by O'Cat, 28-Nov-2008.) |
Theorem | pm2.43d 43 | Deduction absorbing redundant antecedent. (The proof was shortened by O'Cat, 28-Nov-2008.) |
Theorem | pm2.43a 44 | Inference absorbing redundant antecedent. (The proof was shortened by O'Cat, 28-Nov-2008.) |
Theorem | pm2.43b 45 | Inference absorbing redundant antecedent. |
Theorem | pm2.43 46 | Absorption of redundant antecedent. Also called the "Contraction" or "Hilbert" axiom. Theorem *2.43 of [WhiteheadRussell] p. 106. (The proof was shortened by O'Cat, 15-Aug-2004.) |
Theorem | imim2d 47 | Deduction adding nested antecedents. |
Theorem | imim2 48 | A closed form of syllogism (see syl 14). Theorem *2.05 of [WhiteheadRussell] p. 100. (The proof was shortened by Wolf Lammen, 6-Sep-2012.) |
Theorem | embantd 49 | Deduction embedding an antecedent. (Contributed by Wolf Lammen, 4-Oct-2013.) |
Theorem | 3syld 50 | Triple syllogism deduction. (Contributed by Jeff Hankins, 4-Aug-2009.) |
Theorem | sylsyld 51 | Virtual deduction rule. (Contributed by Alan Sare, 20-Apr-2011.) |
Theorem | imim12i 52 | Inference joining two implications. (The proof was shortened by O'Cat, 29-Oct-2011.) |
Theorem | imim1i 53 | Inference adding common consequents in an implication, thereby interchanging the original antecedent and consequent. (The proof was shortened by Wolf Lammen, 4-Aug-2012.) |
Theorem | imim3i 54 | Inference adding three nested antecedents. |
Theorem | sylc 55 | A syllogism inference combined with contraction. |
Theorem | syl3c 56 | A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011.) |
Theorem | syl6mpi 57 | syl6 28 combined with mpi 15. (Contributed by Alan Sare, 8-Jul-2011.) (The proof was shortened by Wolf Lammen, 13-Sep-2012.) |
Theorem | mpsyl 58 | Modus ponens combined with a syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.) |
Theorem | syl6c 59 | Inference combining syl6 28 with contraction. (Contributed by Alan Sare, 2-May-2011.) |
Theorem | syldd 60 | Nested syllogism deduction. (The proof was shortened by Wolf Lammen, 11-May-2013.) |
Theorem | syl5d 61 | A nested syllogism deduction. (The proof was shortened by Josh Purinton, 29-Dec-2000 and shortened further by O'Cat, 2-Feb-2006.) |
Theorem | syl7 62 | A syllogism rule of inference. The second premise is used to replace the third antecedent of the first premise. (The proof was shortened by Wolf Lammen, 3-Aug-2012.) |
Theorem | syl6d 63 | A nested syllogism deduction. (The proof was shortened by Josh Purinton, 29-Dec-2000 and shortened further by O'Cat, 2-Feb-2006.) |
Theorem | syl8 64 | A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (The proof was shortened by Wolf Lammen, 3-Aug-2012.) |
Theorem | syl9 65 | A nested syllogism inference with different antecedents. (The proof was shortened by Josh Purinton, 29-Dec-2000.) |
Theorem | syl9r 66 | A nested syllogism inference with different antecedents. |
Theorem | imim12d 67 | Deduction combining antecedents and consequents. (The proof was shortened by O'Cat, 30-Oct-2011.) |
Theorem | imim1d 68 | Deduction adding nested consequents. (The proof was shortened by Wolf Lammen, 12-Sep-2012.) |
Theorem | imim1 69 | A closed form of syllogism (see syl 14). Theorem *2.06 of [WhiteheadRussell] p. 100. (The proof was shortened by Wolf Lammen, 25-May-2013.) |
Theorem | pm2.83 70 | Theorem *2.83 of [WhiteheadRussell] p. 108. |
Theorem | com23 71 | Commutation of antecedents. Swap 2nd and 3rd. (The proof was shortened by Wolf Lammen, 4-Aug-2012.) |
Theorem | com3r 72 | Commutation of antecedents. Rotate right. |
Theorem | com13 73 | Commutation of antecedents. Swap 1st and 3rd. (The proof was shortened by Wolf Lammen, 28-Jul-2012.) |
Theorem | com3l 74 | Commutation of antecedents. Rotate left. (The proof was shortened by Wolf Lammen, 28-Jul-2012.) |
Theorem | pm2.04 75 | Swap antecedents. Theorem *2.04 of [WhiteheadRussell] p. 100. (The proof was shortened by Wolf Lammen, 12-Sep-2012.) |
Theorem | com34 76 | Commutation of antecedents. Swap 3rd and 4th. |
Theorem | com4l 77 | Commutation of antecedents. Rotate left. (The proof was shortened by O'Cat, 15-Aug-2004.) |
Theorem | com4t 78 | Commutation of antecedents. Rotate twice. |
Theorem | com4r 79 | Commutation of antecedents. Rotate right. |
Theorem | com24 80 | Commutation of antecedents. Swap 2nd and 4th. (The proof was shortened by Wolf Lammen, 28-Jul-2012.) |
Theorem | com14 81 | Commutation of antecedents. Swap 1st and 4th. (The proof was shortened by Wolf Lammen, 28-Jul-2012.) |
Theorem | com45 82 | Commutation of antecedents. Swap 4th and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.) |
Theorem | com35 83 | Commutation of antecedents. Swap 3rd and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.) |
Theorem | com25 84 | Commutation of antecedents. Swap 2nd and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.) |
Theorem | com5l 85 | Commutation of antecedents. Rotate left. (Contributed by Jeff Hankins, 28-Jun-2009.) (The proof was shortened by Wolf Lammen, 29-Jul-2012.) |
Theorem | com15 86 | Commutation of antecedents. Swap 1st and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.) (The proof was shortened by Wolf Lammen, 29-Jul-2012.) |
Theorem | com52l 87 | Commutation of antecedents. Rotate left twice. (Contributed by Jeff Hankins, 28-Jun-2009.) |
Theorem | com52r 88 | Commutation of antecedents. Rotate right twice. (Contributed by Jeff Hankins, 28-Jun-2009.) |
Theorem | com5r 89 | Commutation of antecedents. Rotate right. (Contributed by Wolf Lammen, 29-Jul-2012.) |
Theorem | jarr 90 | Elimination of a nested antecedent as a kind of reversal of inference ja 738. (Contributed by Wolf Lammen, 9-May-2013.) |
Theorem | pm2.86i 91 | Inference based on pm2.86 93. (The proof was shortened by Wolf Lammen, 3-Apr-2013.) |
Theorem | pm2.86d 92 | Deduction based on pm2.86 93. (The proof was shortened by Wolf Lammen, 3-Apr-2013.) |
Theorem | pm2.86 93 | Converse of axiom ax-2 6. Theorem *2.86 of [WhiteheadRussell] p. 108. (The proof was shortened by Wolf Lammen, 3-Apr-2013.) |
Theorem | loolinALT 94 | The Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. This version of loolin 755 does not use ax-3 7, meaning that this theorem is intuitionistically valid. (Contributed by O'Cat, 12-Aug-2004.) |
Theorem | loowoz 95 | An alternate for the Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, due to Barbara Wozniakowska, Reports on Mathematical Logic 10, 129-137 (1978). (Contributed by O'Cat, 8-Aug-2004.) |
Syntax | wa 96 | Extend wff definition to include conjunction ('and'). |
Syntax | wb 97 | Extend our wff definition to include the biconditional connective. |
Axiom | ax-ia1 98 | Left 'and' elimination. Axiom 1 of 10 for intuitionistic logic. |
Axiom | ax-ia2 99 | Right 'and' elimination. Axiom 2 of 10 for intuitionistic logic. |
Axiom | ax-ia3 100 | 'And' introduction. Axiom 3 of 10 for intuitionistic logic. |
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