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Type | Label | Description |
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Statement | ||
Theorem | a1tru 1201 | Anything implies . (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.) |
Theorem | truan 1202 | True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) |
Theorem | dfnot 1203 | One definition of negation in logics that take as axiomatic is via "implies contradiction", i.e. . (Contributed by Mario Carneiro, 2-Feb-2015.) |
Theorem | inegd 1204 | Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Theorem | pclem6 1205 | Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.) |
Syntax | wxo 1206 | Extend wff definition to include exclusive disjunction ('xor'). |
Definition | df-xor 1207 | Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. Contrast with (wa 95), (wo 608), and (wi 4) . (Contributed by FL, 22-Nov-2010.) (Modified by Jim Kingdon, 1-Mar-2018.) |
Theorem | xoranor 1208 | One way of defining exclusive or. Equivalent to df-xor 1207. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.) |
Theorem | excxor 1209 | This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.) (Proof rewritten by Jim Kingdon, 5-May-2018.) |
Theorem | xorbin 1210 | A consequence of exclusive or. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 8-Mar-2018.) |
Theorem | pm5.18im 1211 | One direction of pm5.18dc 750, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.) |
Theorem | xornbi 1212 | A consequence of exclusive or. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 10-Mar-2018.) |
Theorem | xor3dc 1213 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.) |
DECID DECID | ||
Theorem | pm5.15dc 1214 | 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 1215 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.) |
DECID DECID | ||
Theorem | xornbidc 1216 | Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.) |
DECID DECID | ||
Theorem | xordc 1217 | 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 1218 | Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.) |
DECID | ||
Theorem | nbbndc 1219 | 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 1220 |
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 1221 | 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 1222 | An alternate definition of the biconditional for decicable propositions. Theorem *5.23 of [WhiteheadRussell] p. 124, but with decidability conditions. (Contributed by Jim Kingdon, 5-May-2018.) |
DECID DECID | ||
Theorem | pm5.24dc 1223 | Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.) |
DECID DECID | ||
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 95, (disjunction aka logical inclusive 'or') wo 608, (implies) wi 4, (not) wn 3, (logical equivalence) df-bi 108. | ||
Theorem | truantru 1224 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | truanfal 1225 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | falantru 1226 | A identity. (Contributed by David A. Wheeler, 23-Feb-2018.) |
Theorem | falanfal 1227 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | truortru 1228 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | truorfal 1229 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | falortru 1230 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | falorfal 1231 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | truimtru 1232 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | truimfal 1233 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | falimtru 1234 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | falimfal 1235 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | nottru 1236 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | notfal 1237 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | trubitru 1238 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | trubifal 1239 | A identity. (Contributed by David A. Wheeler, 23-Feb-2018.) |
Theorem | falbitru 1240 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | falbifal 1241 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | truxortru 1242 | A identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
Theorem | truxorfal 1243 | A identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
Theorem | falxortru 1244 | A identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
Theorem | falxorfal 1245 | 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 569, modus ponendo tollens I mpto1 1246, modus ponendo tollens II mpto2 1247, and modus tollendo ponens (exclusive-or version) mtp-xor 1248. 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 mtp-xor 1248 originally used exclusive-or, but over time the name modus tollendo ponens has increasingly referred to an inclusive-or variation, which is proved in mtp-or 1249. This set of indemonstrables is not the entire system of Stoic logic. | ||
Theorem | mpto1 1246 | 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 mpto2 1247) 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, 2-Mar-2018.) |
Theorem | mpto2 1247 | 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 | mtp-xor 1248 | Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, 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 mtp-or 1249. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mpto2 1247, that is, it is exclusive-or df-xor 1207), 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 mpto2 1247), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 2-Mar-2018.) |
Theorem | mtp-or 1249 | Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtp-xor 1248, 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 | ee22 1250 | Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 2-May-2011.) |
Theorem | ee12an 1251 | Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 28-Oct-2011.) |
Theorem | ee23 1252 | Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 17-Jul-2011.) |
Theorem | exbir 1253 | Exportation implication also converting head from biconditional to conditional. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | 3impexp 1254 | impexp 248 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | 3impexpbicom 1255 | 3impexp 1254 with biconditional consequent of antecedent that is commuted in consequent. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | 3impexpbicomi 1256 | Deduction form of 3impexpbicom 1255. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | ancomsimp 1257 | Closed form of ancoms 253. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | exp3acom3r 1258 | Export and commute antecedents. (Contributed by Alan Sare, 18-Mar-2012.) |
Theorem | exp3acom23g 1259 | Implication form of exp3acom23 1260. (Contributed by Alan Sare, 22-Jul-2012.) |
Theorem | exp3acom23 1260 | The exportation deduction exp3a 243 with commutation of the conjoined wwfs. (Contributed by Alan Sare, 22-Jul-2012.) |
Theorem | simplbi2comg 1261 | Implication form of simplbi2com 1262. (Contributed by Alan Sare, 22-Jul-2012.) |
Theorem | simplbi2com 1262 | A deduction eliminating a conjunct, similar to simplbi2 364. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.) |
Theorem | ee21 1263 | Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 18-Mar-2012.) |
Theorem | ee10 1264 | Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by NM, 25-Jul-2011.) |
Theorem | ee02 1265 | Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by NM, 22-Jul-2012.) |
Syntax | wal 1266 | Extend wff definition to include the universal quantifier ('for all'). is read " (phi) is true for all ." Typically, in its final application would be replaced with a wff containing a (free) occurrence of the variable , for example . 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 . 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. |
Axiom | ax-5 1267 | Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-7 1268 | 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 1269 | 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 a4i 1361 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 1270 | Generalization applied twice. (Contributed by NM, 30-Apr-1998.) |
Theorem | mpg 1271 | Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.) |
Theorem | mpgbi 1272 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
Theorem | mpgbir 1273 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
Theorem | a7s 1274 | Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.) |
Theorem | alimi 1275 | Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.) |
Theorem | 2alimi 1276 | Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.) |
Theorem | alim 1277 | Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 30-Mar-2008.) |
Theorem | al2imi 1278 | Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.) |
Theorem | alanimi 1279 | Variant of al2imi 1278 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.) |
Syntax | wnf 1280 | Extend wff definition to include the not-free predicate. |
Definition | df-nf 1281 |
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 1547). An example of where this is used is
stdpc5 1382. See nf2 1462 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 1282 | Deduce that is not free in from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbth 1283 |
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 1284 | No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfnth 1285 | No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) |
Theorem | nftru 1286 | The true constant has no free variables. (This can also be proven in one step with nfv 1352, but this proof does not use ax-17 1350.) (Contributed by Mario Carneiro, 6-Oct-2016.) |
Theorem | alimd 1287 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.) |
Theorem | albi 1288 | Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrimih 1289 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | albii 1290 | Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.) |
Theorem | 2albii 1291 | Inference adding 2 universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.) |
Theorem | hbxfrbi 1292 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Theorem | nfbii 1293 | Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfxfr 1294 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfxfrd 1295 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | hbal 1296 | If is not free in , it is not free in . (Contributed by NM, 5-Aug-1993.) |
Theorem | alcom 1297 | Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrimd 1298 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | albid 1299 | Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.26 1300 | 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.) |
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