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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | mp3and 1301 | A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) & ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)) ⇒ ⊢ (𝜑 → 𝜏) | ||
Theorem | mp3an12i 1302 | mp3an 1298 with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ (𝜒 → 𝜃) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) ⇒ ⊢ (𝜒 → 𝜏) | ||
Theorem | mp3an2i 1303 | mp3an 1298 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.) |
⊢ 𝜑 & ⊢ (𝜓 → 𝜒) & ⊢ (𝜓 → 𝜃) & ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ (𝜓 → 𝜏) | ||
Theorem | mp3an3an 1304 | mp3an 1298 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.) |
⊢ 𝜑 & ⊢ (𝜓 → 𝜒) & ⊢ (𝜃 → 𝜏) & ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂) ⇒ ⊢ ((𝜓 ∧ 𝜃) → 𝜂) | ||
Theorem | mp3an2ani 1305 | An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) |
⊢ 𝜑 & ⊢ (𝜓 → 𝜒) & ⊢ ((𝜓 ∧ 𝜃) → 𝜏) & ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂) ⇒ ⊢ ((𝜓 ∧ 𝜃) → 𝜂) | ||
Theorem | biimp3a 1306 | Infer implication from a logical equivalence. Similar to biimpa 292. (Contributed by NM, 4-Sep-2005.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | biimp3ar 1307 | Infer implication from a logical equivalence. Similar to biimpar 293. (Contributed by NM, 2-Jan-2009.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒) | ||
Theorem | 3anandis 1308 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.) |
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃)) → 𝜏) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏) | ||
Theorem | 3anandirs 1309 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.) (Revised by NM, 18-Apr-2007.) |
⊢ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜃) ∧ (𝜒 ∧ 𝜃)) → 𝜏) ⇒ ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) | ||
Theorem | ecased 1310 | Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.) |
⊢ (𝜑 → ¬ 𝜒) & ⊢ (𝜑 → (𝜓 ∨ 𝜒)) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | ecase23d 1311 | Variation of ecased 1310 with three disjuncts instead of two. (Contributed by NM, 22-Apr-1994.) (Revised by Jim Kingdon, 9-Dec-2017.) |
⊢ (𝜑 → ¬ 𝜒) & ⊢ (𝜑 → ¬ 𝜃) & ⊢ (𝜑 → (𝜓 ∨ 𝜒 ∨ 𝜃)) ⇒ ⊢ (𝜑 → 𝜓) | ||
Even though it isn't ordinarily part of propositional calculus, the universal quantifier ∀ is introduced here so that the soundness of definition df-tru 1317 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in axiom ax-5 1406 in the predicate calculus section below. For those who want propositional calculus to be self-contained i.e. to use wff variables only, the alternate definition dftru2 1322 may be adopted and this subsection moved down to the start of the subsection with wex 1451 below. However, the use of dftru2 1322 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid. | ||
Syntax | wal 1312 | Extend wff definition to include the universal quantifier ('for all'). ∀𝑥𝜑 is read "𝜑 (phi) is true for all 𝑥." Typically, in its final application 𝜑 would be replaced with a wff containing a (free) occurrence of the variable 𝑥, for example 𝑥 = 𝑦. In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of 𝑥. When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same. |
wff ∀𝑥𝜑 | ||
Even though it isn't ordinarily part of propositional calculus, the equality predicate = is introduced here so that the soundness of definition df-tru 1317 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in axiom ax-8 1465 in the predicate calculus section below. For those who want propositional calculus to be self-contained i.e. to use wff variables only, the alternate definition dftru2 1322 may be adopted and this subsection moved down to just above weq 1462 below. However, the use of dftru2 1322 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid. | ||
Syntax | cv 1313 |
This syntax construction states that a variable 𝑥, which has been
declared to be a setvar variable by $f statement vx, is also a class
expression. This can be justified informally as follows. We know that
the class builder {𝑦 ∣ 𝑦 ∈ 𝑥} is a class by cab 2101.
Since (when
𝑦 is distinct from 𝑥) we
have 𝑥 =
{𝑦 ∣ 𝑦 ∈ 𝑥} by
cvjust 2110, we can argue that the syntax "class 𝑥 " can be viewed as
an abbreviation for "class {𝑦 ∣ 𝑦 ∈ 𝑥}". See the discussion
under the definition of class in [Jech] p.
4 showing that "Every set can
be considered to be a class."
While it is tempting and perhaps occasionally useful to view cv 1313 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1313 is intrinsically no different from any other class-building syntax such as cab 2101, cun 3037, or c0 3331. For a general discussion of the theory of classes and the role of cv 1313, see https://us.metamath.org/mpeuni/mmset.html#class 1313. (The description above applies to set theory, not predicate calculus. The purpose of introducing class 𝑥 here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1462 of predicate calculus from the wceq 1314 of set theory, so that we don't overload the = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.) |
class 𝑥 | ||
Syntax | wceq 1314 |
Extend wff definition to include class equality.
For a general discussion of the theory of classes, see https://us.metamath.org/mpeuni/mmset.html#class. (The purpose of introducing wff 𝐴 = 𝐵 here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1462 of predicate calculus in terms of the wceq 1314 of set theory, so that we don't "overload" the = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. For example, some parsers - although not the Metamath program - stumble on the fact that the = in 𝑥 = 𝑦 could be the = of either weq 1462 or wceq 1314, although mathematically it makes no difference. The class variables 𝐴 and 𝐵 are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2108 for more information on the set theory usage of wceq 1314.) |
wff 𝐴 = 𝐵 | ||
Syntax | wtru 1315 | ⊤ is a wff. |
wff ⊤ | ||
Theorem | trujust 1316 | Soundness justification theorem for df-tru 1317. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.) |
⊢ ((∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) ↔ (∀𝑦 𝑦 = 𝑦 → ∀𝑦 𝑦 = 𝑦)) | ||
Definition | df-tru 1317 | Definition of the truth value "true", or "verum", denoted by ⊤. This is a tautology, as proved by tru 1318. In this definition, an instance of id 19 is used as the definiens, although any tautology, such as an axiom, can be used in its place. This particular id 19 instance was chosen so this definition can be checked by the same algorithm that is used for predicate calculus. This definition should be referenced directly only by tru 1318, and other proofs should depend on tru 1318 (directly or indirectly) instead of this definition, since there are many alternate ways to define ⊤. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by NM, 11-Jul-2019.) (New usage is discouraged.) |
⊢ (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)) | ||
Theorem | tru 1318 | The truth value ⊤ is provable. (Contributed by Anthony Hart, 13-Oct-2010.) |
⊢ ⊤ | ||
Syntax | wfal 1319 | ⊥ is a wff. |
wff ⊥ | ||
Definition | df-fal 1320 | Definition of the truth value "false", or "falsum", denoted by ⊥. See also df-tru 1317. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (⊥ ↔ ¬ ⊤) | ||
Theorem | fal 1321 | The truth value ⊥ is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.) |
⊢ ¬ ⊥ | ||
Theorem | dftru2 1322 | An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.) |
⊢ (⊤ ↔ (𝜑 → 𝜑)) | ||
Theorem | mptru 1323 | Eliminate ⊤ as an antecedent. A proposition implied by ⊤ is true. (Contributed by Mario Carneiro, 13-Mar-2014.) |
⊢ (⊤ → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | tbtru 1324 | A proposition is equivalent to itself being equivalent to ⊤. (Contributed by Anthony Hart, 14-Aug-2011.) |
⊢ (𝜑 ↔ (𝜑 ↔ ⊤)) | ||
Theorem | nbfal 1325 | The negation of a proposition is equivalent to itself being equivalent to ⊥. (Contributed by Anthony Hart, 14-Aug-2011.) |
⊢ (¬ 𝜑 ↔ (𝜑 ↔ ⊥)) | ||
Theorem | bitru 1326 | A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.) |
⊢ 𝜑 ⇒ ⊢ (𝜑 ↔ ⊤) | ||
Theorem | bifal 1327 | A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.) |
⊢ ¬ 𝜑 ⇒ ⊢ (𝜑 ↔ ⊥) | ||
Theorem | falim 1328 | The truth value ⊥ implies anything. Also called the principle of explosion, or "ex falso quodlibet". (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.) |
⊢ (⊥ → 𝜑) | ||
Theorem | falimd 1329 | The truth value ⊥ implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ ((𝜑 ∧ ⊥) → 𝜓) | ||
Theorem | a1tru 1330 | Anything implies ⊤. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.) |
⊢ (𝜑 → ⊤) | ||
Theorem | truan 1331 | True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.) |
⊢ ((⊤ ∧ 𝜑) ↔ 𝜑) | ||
Theorem | dfnot 1332 | Given falsum, we can define the negation of a wff 𝜑 as the statement that a contradiction follows from assuming 𝜑. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.) |
⊢ (¬ 𝜑 ↔ (𝜑 → ⊥)) | ||
Theorem | inegd 1333 | Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ ((𝜑 ∧ 𝜓) → ⊥) ⇒ ⊢ (𝜑 → ¬ 𝜓) | ||
Theorem | pm2.21fal 1334 | If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → ¬ 𝜓) ⇒ ⊢ (𝜑 → ⊥) | ||
Theorem | pclem6 1335 | Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.) |
⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓) | ||
Syntax | wxo 1336 | Extend wff definition to include exclusive disjunction ('xor'). |
wff (𝜑 ⊻ 𝜓) | ||
Definition | df-xor 1337 | Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. Contrast with ∧ (wa 103), ∨ (wo 680), and → (wi 4) . (Contributed by FL, 22-Nov-2010.) (Modified by Jim Kingdon, 1-Mar-2018.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) | ||
Theorem | xoranor 1338 | One way of defining exclusive or. Equivalent to df-xor 1337. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓))) | ||
Theorem | excxor 1339 | This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.) (Proof rewritten by Jim Kingdon, 5-May-2018.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓))) | ||
Theorem | xoror 1340 | XOR implies OR. (Contributed by BJ, 19-Apr-2019.) |
⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ∨ 𝜓)) | ||
Theorem | xorbi2d 1341 | Deduction joining an equivalence and a left operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ((𝜃 ⊻ 𝜓) ↔ (𝜃 ⊻ 𝜒))) | ||
Theorem | xorbi1d 1342 | Deduction joining an equivalence and a right operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ((𝜓 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜃))) | ||
Theorem | xorbi12d 1343 | Deduction joining two equivalences to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → (𝜃 ↔ 𝜏)) ⇒ ⊢ (𝜑 → ((𝜓 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜏))) | ||
Theorem | xorbi12i 1344 | Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ↔ 𝜃) ⇒ ⊢ ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃)) | ||
Theorem | xorbin 1345 | A consequence of exclusive or. In classical logic the converse also holds. (Contributed by Jim Kingdon, 8-Mar-2018.) |
⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ↔ ¬ 𝜓)) | ||
Theorem | pm5.18im 1346 | One direction of pm5.18dc 851, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.) |
⊢ ((𝜑 ↔ 𝜓) → ¬ (𝜑 ↔ ¬ 𝜓)) | ||
Theorem | xornbi 1347 | A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1352. (Contributed by Jim Kingdon, 10-Mar-2018.) |
⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ↔ 𝜓)) | ||
Theorem | xor3dc 1348 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓)))) | ||
Theorem | xorcom 1349 | ⊻ is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑)) | ||
Theorem | pm5.15dc 1350 | A decidable proposition is equivalent to a decidable proposition or its negation. Based on theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ∨ (𝜑 ↔ ¬ 𝜓)))) | ||
Theorem | xor2dc 1351 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))))) | ||
Theorem | xornbidc 1352 | Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)))) | ||
Theorem | xordc 1353 | Two ways to express "exclusive or" between decidable propositions. Theorem *5.22 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))))) | ||
Theorem | xordc1 1354 | Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.) |
⊢ ((𝜑 ⊻ 𝜓) → DECID 𝜑) | ||
Theorem | nbbndc 1355 | Move negation outside of biconditional, for decidable propositions. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)))) | ||
Theorem | biassdc 1356 |
Associative law for the biconditional, for decidable propositions.
The classical version (without the decidability conditions) is an axiom of system DS in Vladimir Lifschitz, "On calculational proofs", Annals of Pure and Applied Logic, 113:207-224, 2002, http://www.cs.utexas.edu/users/ai-lab/pub-view.php?PubID=26805, and, interestingly, was not included in Principia Mathematica but was apparently first noted by Jan Lukasiewicz circa 1923. (Contributed by Jim Kingdon, 4-May-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → (DECID 𝜒 → (((𝜑 ↔ 𝜓) ↔ 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒)))))) | ||
Theorem | bilukdc 1357 | Lukasiewicz's shortest axiom for equivalential calculus (but modified to require decidable propositions). Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), p. 96. (Contributed by Jim Kingdon, 5-May-2018.) |
⊢ (((DECID 𝜑 ∧ DECID 𝜓) ∧ DECID 𝜒) → ((𝜑 ↔ 𝜓) ↔ ((𝜒 ↔ 𝜓) ↔ (𝜑 ↔ 𝜒)))) | ||
Theorem | dfbi3dc 1358 | An alternate definition of the biconditional for decidable propositions. Theorem *5.23 of [WhiteheadRussell] p. 124, but with decidability conditions. (Contributed by Jim Kingdon, 5-May-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))))) | ||
Theorem | pm5.24dc 1359 | Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))))) | ||
Theorem | xordidc 1360 | Conjunction distributes over exclusive-or, for decidable propositions. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. (Contributed by Jim Kingdon, 14-Jul-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → (DECID 𝜒 → ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒)))))) | ||
Theorem | anxordi 1361 | Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro and Jim Kingdon, 7-Oct-2018.) |
⊢ ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒))) | ||
For classical logic, truth tables can be used to define propositional logic operations, by showing the results of those operations for all possible combinations of true (⊤) and false (⊥). Although the intuitionistic logic connectives are not as simply defined, ⊤ and ⊥ do play similar roles as in classical logic and most theorems from classical logic continue to hold. Here we show that our definitions and axioms produce equivalent results for ⊤ and ⊥ as we would get from truth tables for ∧ (conjunction aka logical 'and') wa 103, ∨ (disjunction aka logical inclusive 'or') wo 680, → (implies) wi 4, ¬ (not) wn 3, ↔ (logical equivalence) df-bi 116, and ⊻ (exclusive or) df-xor 1337. | ||
Theorem | truantru 1362 | A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊤ ∧ ⊤) ↔ ⊤) | ||
Theorem | truanfal 1363 | A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊤ ∧ ⊥) ↔ ⊥) | ||
Theorem | falantru 1364 | A ∧ identity. (Contributed by David A. Wheeler, 23-Feb-2018.) |
⊢ ((⊥ ∧ ⊤) ↔ ⊥) | ||
Theorem | falanfal 1365 | A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊥ ∧ ⊥) ↔ ⊥) | ||
Theorem | truortru 1366 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊤ ∨ ⊤) ↔ ⊤) | ||
Theorem | truorfal 1367 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊤ ∨ ⊥) ↔ ⊤) | ||
Theorem | falortru 1368 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊥ ∨ ⊤) ↔ ⊤) | ||
Theorem | falorfal 1369 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊥ ∨ ⊥) ↔ ⊥) | ||
Theorem | truimtru 1370 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊤ → ⊤) ↔ ⊤) | ||
Theorem | truimfal 1371 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊤ → ⊥) ↔ ⊥) | ||
Theorem | falimtru 1372 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊥ → ⊤) ↔ ⊤) | ||
Theorem | falimfal 1373 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊥ → ⊥) ↔ ⊤) | ||
Theorem | nottru 1374 | A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (¬ ⊤ ↔ ⊥) | ||
Theorem | notfal 1375 | A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (¬ ⊥ ↔ ⊤) | ||
Theorem | trubitru 1376 | A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊤ ↔ ⊤) ↔ ⊤) | ||
Theorem | trubifal 1377 | A ↔ identity. (Contributed by David A. Wheeler, 23-Feb-2018.) |
⊢ ((⊤ ↔ ⊥) ↔ ⊥) | ||
Theorem | falbitru 1378 | A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊥ ↔ ⊤) ↔ ⊥) | ||
Theorem | falbifal 1379 | A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊥ ↔ ⊥) ↔ ⊤) | ||
Theorem | truxortru 1380 | A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
⊢ ((⊤ ⊻ ⊤) ↔ ⊥) | ||
Theorem | truxorfal 1381 | A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
⊢ ((⊤ ⊻ ⊥) ↔ ⊤) | ||
Theorem | falxortru 1382 | A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
⊢ ((⊥ ⊻ ⊤) ↔ ⊤) | ||
Theorem | falxorfal 1383 | A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
⊢ ((⊥ ⊻ ⊥) ↔ ⊥) | ||
The Greek Stoics developed a system of logic. The Stoic Chrysippus, in particular, was often considered one of the greatest logicians of antiquity. Stoic logic is different from Aristotle's system, since it focuses on propositional logic, though later thinkers did combine the systems of the Stoics with Aristotle. Jan Lukasiewicz reports, "For anybody familiar with mathematical logic it is self-evident that the Stoic dialectic is the ancient form of modern propositional logic" ( On the history of the logic of proposition by Jan Lukasiewicz (1934), translated in: Selected Works - Edited by Ludwik Borkowski - Amsterdam, North-Holland, 1970 pp. 197-217, referenced in "History of Logic" https://www.historyoflogic.com/logic-stoics.htm). For more about Aristotle's system, see barbara and related theorems. A key part of the Stoic logic system is a set of five "indemonstrables" assigned to Chrysippus of Soli by Diogenes Laertius, though in general it is difficult to assign specific ideas to specific thinkers. The indemonstrables are described in, for example, [Lopez-Astorga] p. 11 , [Sanford] p. 39, and [Hitchcock] p. 5. These indemonstrables are modus ponendo ponens (modus ponens) ax-mp 5, modus tollendo tollens (modus tollens) mto 634, modus ponendo tollens I mptnan 1384, modus ponendo tollens II mptxor 1385, and modus tollendo ponens (exclusive-or version) mtpxor 1387. The first is an axiom, the second is already proved; in this section we prove the other three. Since we assume or prove all of indemonstrables, the system of logic we use here is as at least as strong as the set of Stoic indemonstrables. Note that modus tollendo ponens mtpxor 1387 originally used exclusive-or, but over time the name modus tollendo ponens has increasingly referred to an inclusive-or variation, which is proved in mtpor 1386. This set of indemonstrables is not the entire system of Stoic logic. | ||
Theorem | mptnan 1384 | Modus ponendo tollens 1, one of the "indemonstrables" in Stoic logic. See rule 1 on [Lopez-Astorga] p. 12 , rule 1 on [Sanford] p. 40, and rule A3 in [Hitchcock] p. 5. Sanford describes this rule second (after mptxor 1385) as a "safer, and these days much more common" version of modus ponendo tollens because it avoids confusion between inclusive-or and exclusive-or. (Contributed by David A. Wheeler, 3-Jul-2016.) |
⊢ 𝜑 & ⊢ ¬ (𝜑 ∧ 𝜓) ⇒ ⊢ ¬ 𝜓 | ||
Theorem | mptxor 1385 | Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic. Note that this uses exclusive-or ⊻. See rule 2 on [Lopez-Astorga] p. 12 , rule 4 on [Sanford] p. 39 and rule A4 in [Hitchcock] p. 5 . (Contributed by David A. Wheeler, 2-Mar-2018.) |
⊢ 𝜑 & ⊢ (𝜑 ⊻ 𝜓) ⇒ ⊢ ¬ 𝜓 | ||
Theorem | mtpor 1386 | Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1387, one of the five original "indemonstrables" in Stoic logic. However, in Stoic logic this rule used exclusive-or, while the name modus tollendo ponens often refers to a variant of the rule that uses inclusive-or instead. The rule says, "if 𝜑 is not true, and 𝜑 or 𝜓 (or both) are true, then 𝜓 must be true." An alternate phrasing is, "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth." -- Sherlock Holmes (Sir Arthur Conan Doyle, 1890: The Sign of the Four, ch. 6). (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.) |
⊢ ¬ 𝜑 & ⊢ (𝜑 ∨ 𝜓) ⇒ ⊢ 𝜓 | ||
Theorem | mtpxor 1387 | Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1386, one of the five "indemonstrables" in Stoic logic. The rule says, "if 𝜑 is not true, and either 𝜑 or 𝜓 (exclusively) are true, then 𝜓 must be true." Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtpor 1386. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1385, that is, it is exclusive-or df-xor 1337), rule 3 of [Sanford] p. 39 (where it is not as clearly stated which kind of "or" is used but it appears to be in the same sense as mptxor 1385), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 4-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.) (Proof shortened by BJ, 19-Apr-2019.) |
⊢ ¬ 𝜑 & ⊢ (𝜑 ⊻ 𝜓) ⇒ ⊢ 𝜓 | ||
Theorem | stoic2a 1388 |
Stoic logic Thema 2 version a.
Statement T2 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 2 as follows: "When from two assertibles a third follows, and from the third and one (or both) of the two another follows, then this other follows from the first two." Bobzien uses constructs such as 𝜑, 𝜓⊢ 𝜒; in Metamath we will represent that construct as 𝜑 ∧ 𝜓 → 𝜒. This version a is without the phrase "or both"; see stoic2b 1389 for the version with the phrase "or both". We already have this rule as syldan 278, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜃) | ||
Theorem | stoic2b 1389 |
Stoic logic Thema 2 version b. See stoic2a 1388.
Version b is with the phrase "or both". We already have this rule as mpd3an3 1299, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜃) | ||
Theorem | stoic3 1390 |
Stoic logic Thema 3.
Statement T3 of [Bobzien] p. 116-117 discusses Stoic logic thema 3. "When from two (assemblies) a third follows, and from the one that follows (i.e., the third) together with another, external external assumption, another follows, then other follows from the first two and the externally co-assumed one. (Simp. Cael. 237.2-4)" (Contributed by David A. Wheeler, 17-Feb-2019.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) | ||
Theorem | stoic4a 1391 |
Stoic logic Thema 4 version a.
Statement T4 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 4: "When from two assertibles a third follows, and from the third and one (or both) of the two and one (or more) external assertible(s) another follows, then this other follows from the first two and the external(s)." We use 𝜃 to represent the "external" assertibles. This is version a, which is without the phrase "or both"; see stoic4b 1392 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) | ||
Theorem | stoic4b 1392 |
Stoic logic Thema 4 version b.
This is version b, which is with the phrase "or both". See stoic4a 1391 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ (((𝜒 ∧ 𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) | ||
Theorem | syl6an 1393 | A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → (𝜒 → 𝜃)) & ⊢ ((𝜓 ∧ 𝜃) → 𝜏) ⇒ ⊢ (𝜑 → (𝜒 → 𝜏)) | ||
Theorem | syl10 1394 | A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏))) & ⊢ (𝜒 → (𝜏 → 𝜂)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂))) | ||
Theorem | exbir 1395 | Exportation implication also converting head from biconditional to conditional. (Contributed by Alan Sare, 31-Dec-2011.) |
⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒)))) | ||
Theorem | 3impexp 1396 | impexp 261 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.) |
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃)))) | ||
Theorem | 3impexpbicom 1397 | 3impexp 1396 with biconditional consequent of antecedent that is commuted in consequent. (Contributed by Alan Sare, 31-Dec-2011.) |
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))) | ||
Theorem | 3impexpbicomi 1398 | Deduction form of 3impexpbicom 1397. (Contributed by Alan Sare, 31-Dec-2011.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃)))) | ||
Theorem | ancomsimp 1399 | Closed form of ancoms 266. (Contributed by Alan Sare, 31-Dec-2011.) |
⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜓 ∧ 𝜑) → 𝜒)) | ||
Theorem | expcomd 1400 | Deduction form of expcom 115. (Contributed by Alan Sare, 22-Jul-2012.) |
⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ⇒ ⊢ (𝜑 → (𝜒 → (𝜓 → 𝜃))) |
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