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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Pre-logic | ||
| Dummy link theorem for assisting proof development | ||
| Theorem | dummylink 1 |
(Note: This theorem will never appear in a completed proof and can be
ignored if you are using this database to learn logic - please start
with the next statement, wn 2.)
This is a technical theorem 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 theorem 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 theorem 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 */n/b/e 3syl,we?,wsb' to clean up (discard) all dummylink references automatically. This theorem was originally designed to assist importing partially completed Proof Worksheets from Mel O'Cat's mmj2 Proof Assistant GUI, but it can also be useful on its own. Interestingly, this "theorem" - or more precisely, inference - requires no axioms for its proof. |
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| Propositional calculus | ||
| Recursively define primitive wffs for propositional calculus | ||
| Syntax | wn 2 |
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 992 and wel 994).
|
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| Syntax | wi 3 |
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.
|
| | ||
| The axioms of propositional calculus | ||
| Axiom | ax-1 4 |
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 4 through ax-3 6 and rule ax-mp 7) 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 7) the wffs ax-1 4, ax-2 5, pm2.04 30, con3 94, notnot2 84, and notnot1 86. 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 30) and replacing the last three with our ax-3 6. (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 3 and wn 2) 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, and the much shorter proofs that we show here were found manually. Truth tables grow exponentially with the number of variables, but it is unknown if the same is true of proofs - an answer to this would resolve the P=NP conjecture in complexity theory. |
| | ||
| Axiom | ax-2 5 | 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 pm5.41 167. |
|
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| Axiom | ax-3 6 | 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 7 |
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. |
| | ||
| Logical implication | ||
| Theorem | a1i 8 | Inference derived from axiom ax-1 4. See a1d 12 for an explanation of our informal use of the terms "inference" and "deduction." See also the comment in syld 27. |
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| Theorem | a2i 9 | Inference derived from axiom ax-2 5. |
| | ||
| Theorem | syl 10 |
An inference version of the transitive laws for implication imim2 14
and
imim1 15, 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."
(A bit of trivia: this is the most commonly referenced assertion in our database. In second place is ax-mp 7, followed by visset 1858, bitri 171, imp 348, and ex 371. The Metamath program command 'show usage' shows the number of references.) |
| | ||
| Theorem | com12 11 | Inference that swaps (commutes) antecedents in an implication. |
|
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| Theorem | a1d 12 |
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 |
| | ||
| Theorem | a2d 13 | Deduction distributing an embedded antecedent. |
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| Theorem | imim2 14 | A closed form of syllogism (see syl 10). Theorem *2.05 of [WhiteheadRussell] p. 100. |
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| Theorem | imim1 15 | A closed form of syllogism (see syl 10). Theorem *2.06 of [WhiteheadRussell] p. 100. |
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| Theorem | imim1i 16 | Inference adding common consequents in an implication, thereby interchanging the original antecedent and consequent. |
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| Theorem | imim2i 17 | Inference adding common antecedents in an implication. |
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| Theorem | imim12i 18 | Inference joining two implications. |
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| Theorem | imim3i 19 | Inference adding three nested antecedents. |
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| Theorem | 3syl 20 | Inference chaining two syllogisms. |
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| Theorem | syl5 21 | A syllogism rule of inference. The second premise is used to replace the second antecedent of the first premise. |
|
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| Theorem | syl6 22 | A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. |
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| Theorem | syl7 23 | A syllogism rule of inference. The second premise is used to replace the third antecedent of the first premise. |
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| Theorem | syl8 24 | A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. |
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| Theorem | imim2d 25 | Deduction adding nested antecedents. |
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| Theorem | mpd 26 | A modus ponens deduction. |
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| Theorem | syld 27 |
Syllogism deduction. (The proof was shortened by O'Cat, 19-Feb-2008.)
Notice that syld 27 can be obtained from syl 10 by
replacing each
hypothesis and conclusion |
| | ||
| Theorem | imim1d 28 | Deduction adding nested consequents. |
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| Theorem | imim12d 29 | Deduction combining antecedents and consequents. |
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| Theorem | pm2.04 30 | Swap antecedents. Theorem *2.04 of [WhiteheadRussell] p. 100. |
|
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| Theorem | pm2.83 31 | Theorem *2.83 of [WhiteheadRussell] p. 108. |
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| Theorem | com23 32 | Commutation of antecedents. Swap 2nd and 3rd. |
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| Theorem | com13 33 | Commutation of antecedents. Swap 1st and 3rd. |
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| Theorem | com3l 34 | Commutation of antecedents. Rotate left. |
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| Theorem | com3r 35 | Commutation of antecedents. Rotate right. |
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| Theorem | com34 36 | Commutation of antecedents. Swap 3rd and 4th. |
|
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| Theorem | com24 37 | Commutation of antecedents. Swap 2nd and 4th. |
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| Theorem | com14 38 | Commutation of antecedents. Swap 1st and 4th. |
|
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| Theorem | com4l 39 | Commutation of antecedents. Rotate left. (The proof was shortened by O'Cat, 15-Aug-2004.) |
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| Theorem | com4t 40 | Commutation of antecedents. Rotate twice. |
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| Theorem | com4r 41 | Commutation of antecedents. Rotate right. |
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| Theorem | a1dd 42 | Deduction introducing a nested embedded antecedent. (The proof was shortened by O'Cat, 15-Jan-2008.) |
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| Theorem | mp2 43 | A double modus ponens inference. |
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| Theorem | mpi 44 | A nested modus ponens inference. (The proof was shortened by Stefan Allan, 20-Mar-2006.) |
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| Theorem | mpii 45 | A doubly nested modus ponens inference. |
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| Theorem | mpdd 46 | A nested modus ponens deduction. |
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| Theorem | mpid 47 | A nested modus ponens deduction. |
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| Theorem | mpdi 48 | A nested modus ponens deduction. (The proof was shortened by O'Cat, 15-Jan-2008.) |
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| Theorem | mpcom 49 | Modus ponens inference with commutation of antecedents. |
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| Theorem | syldd 50 | Nested syllogism deduction. |
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| Theorem | sylcom 51 | Syllogism inference with commutation of antecedents. (The proof was shortened by O'Cat, 2-Feb-2006 and shortened further by Stefan Allan, 23-Feb-2006.) |
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| Theorem | syl5com 52 | Syllogism inference with commuted antecedents. |
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| Theorem | syl6com 53 | Syllogism inference with commuted antecedents. |
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| Theorem | syli 54 | Syllogism inference with common nested antecedent. |
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| Theorem | syl5d 55 | A nested syllogism deduction. (The proof was shortened by Josh Purinton, 29-Dec-2000 and shortened further by O'Cat, 2-Feb-2006.) |
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| Theorem | syl6d 56 | A nested syllogism deduction. (The proof was shortened by Josh Purinton, 29-Dec-2000 and shortened further by O'Cat, 2-Feb-2006.) |
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| Theorem | syl9 57 | A nested syllogism inference with different antecedents. (The proof was shortened by Josh Purinton, 29-Dec-2000.) |
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| Theorem | syl9r 58 | A nested syllogism inference with different antecedents. |
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| Theorem | id 59 | Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. For another version of the proof directly from axioms, see id1 60. (The proof was shortened by Stefan Allan, 20-Mar-2006.) |
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| Theorem | id1 60 | 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. 16 (PDF p. 22) 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 59. |
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| Theorem | idd 61 | Principle of identity with antecedent. |
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| Theorem | pm2.27 62 | This theorem, called "Assertion," can be thought of as closed form of modus ponens ax-mp 7. Theorem *2.27 of [WhiteheadRussell] p. 104. |
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| Theorem | pm2.43 63 | 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.) |
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| Theorem | pm2.43i 64 | Inference absorbing redundant antecedent. (The proof was shortened by O'Cat, 28-Nov-2008.) |
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| Theorem | pm2.43d 65 | Deduction absorbing redundant antecedent. (The proof was shortened by O'Cat, 28-Nov-2008.) |
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| Theorem | pm2.43a 66 | Inference absorbing redundant antecedent. (The proof was shortened by O'Cat, 28-Nov-2008.) |
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| Theorem | pm2.43b 67 | Inference absorbing redundant antecedent. |
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| Theorem | sylc 68 | A syllogism inference combined with contraction. |
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| Theorem | pm2.86 69 | Converse of axiom ax-2 5. Theorem *2.86 of [WhiteheadRussell] p. 108. |
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| Theorem | pm2.86i 70 | Inference based on pm2.86 69. |
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| Theorem | pm2.86d 71 | Deduction based on pm2.86 69. |
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| Theorem | loolin 72 | The Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. (Contributed by O'Cat, 12-Aug-2004.) |
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| Theorem | loowoz 73 | 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.) |
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| Logical negation | ||
| Theorem | con4i 74 | Inference rule derived from axiom ax-3 6. |
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| Theorem | con4d 75 | Deduction derived from axiom ax-3 6. |
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| Theorem | pm2.21 76 | From a wff and its negation, anything is true. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. |
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| Theorem | pm2.21i 77 | A contradiction implies anything. Inference from pm2.21 76. |
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| Theorem | pm2.21d 78 | A contradiction implies anything. Deduction from pm2.21 76. |
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| Theorem | pm2.24 79 | Theorem *2.24 of [WhiteheadRussell] p. 104. |
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| Theorem | pm2.24ii 80 | A contradiction implies anything. Inference from pm2.24 79. |
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| Theorem | pm2.18 81 | Proof by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. Also called the Law of Clavius. |
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| Theorem | peirce 82 | Peirce's axiom. This odd-looking theorem is the "difference" between an intuitionistic system of propositional calculus and a classical system and is not accepted by intuitionists. When Peirce's axiom is added to an intuitionistic system, the system becomes equivalent to our classical system ax-1 4 through ax-3 6. A curious fact about this theorem is that it requires ax-3 6 for its proof even though the result has no negation connectives in it. |
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| Theorem | looinv 83 | The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. Using dfor2 227, we can see that this essentially expresses "disjunction commutes." Theorem *2.69 of [WhiteheadRussell] p. 108. |
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| Theorem | notnot2 84 | Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102. (The proof was shortened by David Harvey, 5-Sep-1999. An even shorter proof found by Josh Purinton, 29-Dec-2000.) |
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| Theorem | notnotri 85 | Inference from double negation. |
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| Theorem | notnot1 86 | Converse of double negation. Theorem *2.12 of [WhiteheadRussell] p. 101. |
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| Theorem | notnoti 87 | Infer double negation. |
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| Theorem | pm2.01 88 | Reductio ad absurdum. Theorem *2.01 of [WhiteheadRussell] p. 100. (The proof was shortened by O'Cat, 21-Nov-2008. |
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| Theorem | pm2.01d 89 | Deduction based on reductio ad absurdum. |
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| Theorem | con2 90 | Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. |
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| Theorem | con2d 91 | A contraposition deduction. |
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| Theorem | con1 92 | Contraposition. Theorem *2.15 of [WhiteheadRussell] p. 102. |
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| Theorem | con1d 93 | A contraposition deduction. |
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| Theorem | con3 94 | Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. |
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| Theorem | con3d 95 | A contraposition deduction. |
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| Theorem | con1i 96 | A contraposition inference. (The proof was shortened by O'Cat, 28-Nov-2008.) |
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| Theorem | con2i 97 | A contraposition inference. (The proof was shortened by O'Cat, 28-Nov-2008.) |
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| Theorem | con3i 98 | A contraposition inference. |
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| Theorem | pm2.5 99 | Theorem *2.5 of [WhiteheadRussell] p. 107. |
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| Theorem | pm2.51 100 | Theorem *2.51 of [WhiteheadRussell] p. 107. |
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