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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | truan 1301 | True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.) |
Theorem | dfnot 1302 | Given falsum, we can define the negation of a wff as the statement that a contradiction follows from assuming . (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.) |
Theorem | inegd 1303 | Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Theorem | pm2.21fal 1304 | If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Theorem | pclem6 1305 | Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.) |
Syntax | wxo 1306 | Extend wff definition to include exclusive disjunction ('xor'). |
Definition | df-xor 1307 | Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. Contrast with (wa 102), (wo 661), and (wi 4) . (Contributed by FL, 22-Nov-2010.) (Modified by Jim Kingdon, 1-Mar-2018.) |
Theorem | xoranor 1308 | One way of defining exclusive or. Equivalent to df-xor 1307. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.) |
Theorem | excxor 1309 | This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.) (Proof rewritten by Jim Kingdon, 5-May-2018.) |
Theorem | xoror 1310 | XOR implies OR. (Contributed by BJ, 19-Apr-2019.) |
Theorem | xorbi2d 1311 | Deduction joining an equivalence and a left operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
Theorem | xorbi1d 1312 | Deduction joining an equivalence and a right operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
Theorem | xorbi12d 1313 | Deduction joining two equivalences to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
Theorem | xorbi12i 1314 | Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.) |
Theorem | xorbin 1315 | A consequence of exclusive or. In classical logic the converse also holds. (Contributed by Jim Kingdon, 8-Mar-2018.) |
Theorem | pm5.18im 1316 | One direction of pm5.18dc 810, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.) |
Theorem | xornbi 1317 | A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1322. (Contributed by Jim Kingdon, 10-Mar-2018.) |
Theorem | xor3dc 1318 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.) |
DECID DECID | ||
Theorem | xorcom 1319 | is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.) |
Theorem | pm5.15dc 1320 | 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 DECID | ||
Theorem | xor2dc 1321 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.) |
DECID DECID | ||
Theorem | xornbidc 1322 | Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.) |
DECID DECID | ||
Theorem | xordc 1323 | 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 DECID | ||
Theorem | xordc1 1324 | Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.) |
DECID | ||
Theorem | nbbndc 1325 | Move negation outside of biconditional, for decidable propositions. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.) |
DECID DECID | ||
Theorem | biassdc 1326 |
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 DECID DECID | ||
Theorem | bilukdc 1327 | 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 DECID DECID | ||
Theorem | dfbi3dc 1328 | 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 DECID | ||
Theorem | pm5.24dc 1329 | Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.) |
DECID DECID | ||
Theorem | xordidc 1330 | 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 DECID DECID | ||
Theorem | anxordi 1331 | Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro and Jim Kingdon, 7-Oct-2018.) |
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 102, (disjunction aka logical inclusive 'or') wo 661, (implies) wi 4, (not) wn 3, (logical equivalence) df-bi 115, and (exclusive or) df-xor 1307. | ||
Theorem | truantru 1332 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | truanfal 1333 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | falantru 1334 | A identity. (Contributed by David A. Wheeler, 23-Feb-2018.) |
Theorem | falanfal 1335 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | truortru 1336 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | truorfal 1337 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | falortru 1338 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | falorfal 1339 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | truimtru 1340 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | truimfal 1341 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | falimtru 1342 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | falimfal 1343 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | nottru 1344 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | notfal 1345 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | trubitru 1346 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | trubifal 1347 | A identity. (Contributed by David A. Wheeler, 23-Feb-2018.) |
Theorem | falbitru 1348 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | falbifal 1349 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | truxortru 1350 | A identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
Theorem | truxorfal 1351 | A identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
Theorem | falxortru 1352 | A identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
Theorem | falxorfal 1353 | 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 7, modus tollendo tollens (modus tollens) mto 620, modus ponendo tollens I mptnan 1354, modus ponendo tollens II mptxor 1355, and modus tollendo ponens (exclusive-or version) mtpxor 1357. 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 1357 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 1356. This set of indemonstrables is not the entire system of Stoic logic. | ||
Theorem | mptnan 1354 | 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 1355) 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.) |
Theorem | mptxor 1355 | 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.) |
Theorem | mtpor 1356 | Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1357, 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 is not true, and or (or both) are true, then 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.) |
Theorem | mtpxor 1357 | Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1356, one of the five "indemonstrables" in Stoic logic. The rule says, "if is not true, and either or (exclusively) are true, then must be true." Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtpor 1356. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1355, that is, it is exclusive-or df-xor 1307), 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 1355), 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.) |
Theorem | stoic2a 1358 |
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 , ; in Metamath we will represent that construct as . This version a is without the phrase "or both"; see stoic2b 1359 for the version with the phrase "or both". We already have this rule as syldan 276, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic2b 1359 |
Stoic logic Thema 2 version b. See stoic2a 1358.
Version b is with the phrase "or both". We already have this rule as mpd3an3 1269, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic3 1360 |
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.) |
Theorem | stoic4a 1361 |
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 to represent the "external" assertibles. This is version a, which is without the phrase "or both"; see stoic4b 1362 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic4b 1362 |
Stoic logic Thema 4 version b.
This is version b, which is with the phrase "or both". See stoic4a 1361 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | syl6an 1363 | A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.) |
Theorem | syl10 1364 | A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.) |
Theorem | exbir 1365 | Exportation implication also converting head from biconditional to conditional. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | 3impexp 1366 | impexp 259 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | 3impexpbicom 1367 | 3impexp 1366 with biconditional consequent of antecedent that is commuted in consequent. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | 3impexpbicomi 1368 | Deduction form of 3impexpbicom 1367. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | ancomsimp 1369 | Closed form of ancoms 264. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | expcomd 1370 | Deduction form of expcom 114. (Contributed by Alan Sare, 22-Jul-2012.) |
Theorem | expdcom 1371 | Commuted form of expd 254. (Contributed by Alan Sare, 18-Mar-2012.) |
Theorem | simplbi2comg 1372 | Implication form of simplbi2com 1373. (Contributed by Alan Sare, 22-Jul-2012.) |
Theorem | simplbi2com 1373 | A deduction eliminating a conjunct, similar to simplbi2 377. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.) |
Theorem | syl6ci 1374 | A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.) |
Theorem | mpisyl 1375 | A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.) |
The universal quantifier was introduced above in wal 1282 for use by df-tru 1287. See the comments in that section. In this section, we continue with the first "real" use of it. | ||
Axiom | ax-5 1376 | Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-7 1377 | Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. One of the predicate logic axioms which do not involve equality. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-gen 1378 | Rule of Generalization. The postulated inference rule of predicate calculus. See e.g. Rule 2 of [Hamilton] p. 74. This rule says that if something is unconditionally true, then it is true for all values of a variable. For example, if we have proved , we can conclude or even . Theorem spi 1469 shows we can go the other way also: in other words we can add or remove universal quantifiers from the beginning of any theorem as required. (Contributed by NM, 5-Aug-1993.) |
Theorem | gen2 1379 | Generalization applied twice. (Contributed by NM, 30-Apr-1998.) |
Theorem | mpg 1380 | Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.) |
Theorem | mpgbi 1381 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
Theorem | mpgbir 1382 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
Theorem | a7s 1383 | Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.) |
Theorem | alimi 1384 | Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.) |
Theorem | 2alimi 1385 | Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.) |
Theorem | alim 1386 | Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 30-Mar-2008.) |
Theorem | al2imi 1387 | Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.) |
Theorem | alanimi 1388 | Variant of al2imi 1387 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.) |
Syntax | wnf 1389 | Extend wff definition to include the not-free predicate. |
Definition | df-nf 1390 |
Define the not-free predicate for wffs. This is read " is not free
in ".
Not-free means that the value of cannot affect the
value of ,
e.g., any occurrence of in
is effectively
bound by a "for all" or something that expands to one (such as
"there
exists"). In particular, substitution for a variable not free in a
wff
does not affect its value (sbf 1700). An example of where this is used is
stdpc5 1516. See nf2 1598 for an alternate definition which
does not involve
nested quantifiers on the same variable.
Not-free is a commonly used constraint, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the not-free notion within the logic itself rather than as a metalogical side condition. To be precise, our definition really means "effectively not free," because it is slightly less restrictive than the usual textbook definition for not-free (which only considers syntactic freedom). For example, is effectively not free in the bare expression , even though would be considered free in the usual textbook definition, because the value of in the expression cannot affect the truth of the expression (and thus substitution will not change the result). (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfi 1391 | Deduce that is not free in from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbth 1392 |
No variable is (effectively) free in a theorem.
This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form from smaller formulas of this form. These are useful for constructing hypotheses that state " is (effectively) not free in ." (Contributed by NM, 5-Aug-1993.) |
Theorem | nfth 1393 | No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfnth 1394 | No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) |
Theorem | nftru 1395 | The true constant has no free variables. (This can also be proven in one step with nfv 1461, but this proof does not use ax-17 1459.) (Contributed by Mario Carneiro, 6-Oct-2016.) |
Theorem | alimdh 1396 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.) |
Theorem | albi 1397 | Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrimih 1398 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | albii 1399 | Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.) |
Theorem | 2albii 1400 | Inference adding 2 universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.) |
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