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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nbbndc 1301 | 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 1302 |
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 1303 | 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 1304 | 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 1305 | Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.) |
DECID DECID | ||
Theorem | xordidc 1306 | 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 1307 | 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 101, (disjunction aka logical inclusive 'or') wo 639, (implies) wi 4, (not) wn 3, (logical equivalence) df-bi 114, and (exclusive or) df-xor 1283. | ||
Theorem | truantru 1308 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | truanfal 1309 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | falantru 1310 | A identity. (Contributed by David A. Wheeler, 23-Feb-2018.) |
Theorem | falanfal 1311 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | truortru 1312 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | truorfal 1313 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | falortru 1314 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | falorfal 1315 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | truimtru 1316 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | truimfal 1317 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | falimtru 1318 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | falimfal 1319 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | nottru 1320 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | notfal 1321 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | trubitru 1322 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | trubifal 1323 | A identity. (Contributed by David A. Wheeler, 23-Feb-2018.) |
Theorem | falbitru 1324 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | falbifal 1325 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | truxortru 1326 | A identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
Theorem | truxorfal 1327 | A identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
Theorem | falxortru 1328 | A identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
Theorem | falxorfal 1329 | 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 598, modus ponendo tollens I mptnan 1330, modus ponendo tollens II mptxor 1331, and modus tollendo ponens (exclusive-or version) mtpxor 1333. 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 1333 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 1332. This set of indemonstrables is not the entire system of Stoic logic. | ||
Theorem | mptnan 1330 | 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 1331) 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 1331 | 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 1332 | Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1333, 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 alternative 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 1333 | Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1332, 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 1332. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1331, that is, it is exclusive-or df-xor 1283), 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 1331), 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 1334 |
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 1335 for the version with the phrase "or both". We already have this rule as syldan 270, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic2b 1335 |
Stoic logic Thema 2 version b. See stoic2a 1334.
Version b is with the phrase "or both". We already have this rule as mpd3an3 1244, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic3 1336 |
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 1337 |
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 1338 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic4b 1338 |
Stoic logic Thema 4 version b.
This is version b, which is with the phrase "or both". See stoic4a 1337 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | syl6an 1339 | A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.) |
Theorem | syl10 1340 | A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.) |
Theorem | exbir 1341 | Exportation implication also converting head from biconditional to conditional. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | 3impexp 1342 | impexp 254 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | 3impexpbicom 1343 | 3impexp 1342 with biconditional consequent of antecedent that is commuted in consequent. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | 3impexpbicomi 1344 | Deduction form of 3impexpbicom 1343. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | ancomsimp 1345 | Closed form of ancoms 259. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | expcomd 1346 | Deduction form of expcom 113. (Contributed by Alan Sare, 22-Jul-2012.) |
Theorem | expdcom 1347 | Commuted form of expd 249. (Contributed by Alan Sare, 18-Mar-2012.) |
Theorem | simplbi2comg 1348 | Implication form of simplbi2com 1349. (Contributed by Alan Sare, 22-Jul-2012.) |
Theorem | simplbi2com 1349 | A deduction eliminating a conjunct, similar to simplbi2 371. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.) |
Theorem | syl6ci 1350 | A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.) |
Theorem | mpisyl 1351 | A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.) |
The universal quantifier was introduced above in wal 1257 for use by df-tru 1262. See the comments in that section. In this section, we continue with the first "real" use of it. | ||
Axiom | ax-5 1352 | Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-7 1353 | 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 1354 | 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 1445 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 1355 | Generalization applied twice. (Contributed by NM, 30-Apr-1998.) |
Theorem | mpg 1356 | Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.) |
Theorem | mpgbi 1357 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
Theorem | mpgbir 1358 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
Theorem | a7s 1359 | Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.) |
Theorem | alimi 1360 | Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.) |
Theorem | 2alimi 1361 | Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.) |
Theorem | alim 1362 | Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 30-Mar-2008.) |
Theorem | al2imi 1363 | Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.) |
Theorem | alanimi 1364 | Variant of al2imi 1363 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.) |
Syntax | wnf 1365 | Extend wff definition to include the not-free predicate. |
Definition | df-nf 1366 |
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 1676). An example of where this is used is
stdpc5 1492. See nf2 1574 for an alternative 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 1367 | Deduce that is not free in from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbth 1368 |
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 1369 | No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfnth 1370 | No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) |
Theorem | nftru 1371 | The true constant has no free variables. (This can also be proven in one step with nfv 1437, but this proof does not use ax-17 1435.) (Contributed by Mario Carneiro, 6-Oct-2016.) |
Theorem | alimdh 1372 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.) |
Theorem | albi 1373 | Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrimih 1374 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | albii 1375 | Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.) |
Theorem | 2albii 1376 | Inference adding 2 universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.) |
Theorem | hbxfrbi 1377 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Theorem | nfbii 1378 | Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfxfr 1379 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfxfrd 1380 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | alcoms 1381 | Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.) |
Theorem | hbal 1382 | If is not free in , it is not free in . (Contributed by NM, 5-Aug-1993.) |
Theorem | alcom 1383 | Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrimdh 1384 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | albidh 1385 | Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.26 1386 | Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) |
Theorem | 19.26-2 1387 | Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.) |
Theorem | 19.26-3an 1388 | Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.) |
Theorem | 19.33 1389 | Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrot3 1390 | Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
Theorem | alrot4 1391 | Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 28-Jun-2014.) |
Theorem | albiim 1392 | Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.) |
Theorem | 2albiim 1393 | Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.) |
Theorem | hband 1394 | Deduction form of bound-variable hypothesis builder hban 1455. (Contributed by NM, 2-Jan-2002.) |
Theorem | hb3and 1395 | Deduction form of bound-variable hypothesis builder hb3an 1458. (Contributed by NM, 17-Feb-2013.) |
Theorem | hbald 1396 | Deduction form of bound-variable hypothesis builder hbal 1382. (Contributed by NM, 2-Jan-2002.) |
Syntax | wex 1397 | Extend wff definition to include the existential quantifier ("there exists"). |
Axiom | ax-ie1 1398 | is bound in . One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Axiom | ax-ie2 1399 | Define existential quantification. means "there exists at least one set such that is true." One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Theorem | hbe1 1400 | is not free in . (Contributed by NM, 5-Aug-1993.) |
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