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Type | Label | Description |
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Statement | ||
Theorem | moanim 2301 | Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.) |
Theorem | euan 2302 | Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 24-Dec-2018.) |
Theorem | moanimv 2303* | Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.) |
Theorem | moanmo 2304 | Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.) |
Theorem | moaneu 2305 | Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) |
Theorem | euanv 2306* | Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.) |
Theorem | mopick 2307 | "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 17-Sep-2019.) |
Theorem | mopickOLD 2308 | Obsolete proof of mopick 2307 as of 15-Sep-2019. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 29-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | eupick 2309 | Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing such that is true, and there is also an (actually the same one) such that and are both true, then implies regardless of . This theorem can be useful for eliminating existential quantifiers in a hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by NM, 10-Jul-1994.) |
Theorem | eupicka 2310 | Version of eupick 2309 with closed formulas. (Contributed by NM, 6-Sep-2008.) |
Theorem | eupickb 2311 | Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) |
Theorem | eupickbi 2312 | Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) |
Theorem | mopick2 2313 | "At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1700. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | moexex 2314 | "At most one" double quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.) |
Theorem | moexexv 2315* | "At most one" double quantification. (Contributed by NM, 26-Jan-1997.) |
Theorem | 2moex 2316 | Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.) |
Theorem | 2euex 2317 | Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | 2eumo 2318 | Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.) |
Theorem | 2eu2ex 2319 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |
Theorem | 2moswap 2320 | A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.) |
Theorem | 2euswap 2321 | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.) |
Theorem | 2exeu 2322 | Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) |
Theorem | 2mo2 2323* | This theorem extends the idea of "at most one" to expressions in two set variables ("at most one pair and ". Note: this is not expressed by ). 2eu4 2331 relates this extension to double existential uniqueness, if at least one pair exists. (Contributed by Wolf Lammen, 26-Oct-2019.) |
Theorem | 2mo 2324* | Two equivalent expressions for double "at most one." (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Nov-2019.) |
Theorem | 2moOLD 2325* | Obsolete proof of 2mo 2324 as of 2-Nov-2019. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 25-Oct-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | 2mos 2326* | Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.) |
Theorem | 2eu1 2327 | Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 11-Nov-2019.) |
Theorem | 2eu1OLD 2328 | Obsolete proof of 2eu1 2327 as of 11-Nov-2019. (Contributed by NM, 3-Dec-2001.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | 2eu2 2329 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |
Theorem | 2eu3 2330 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |
Theorem | 2eu4 2331* | This theorem provides us with a definition of double existential uniqueness ("exactly one and exactly one "). Naively one might think (incorrectly) that it could be defined by . See 2eu1 2327 for a condition under which the naive definition holds and 2exeu 2322 for a one-way implication. See 2eu5 2333 and 2eu8 2337 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.) |
Theorem | 2eu4OLD 2332* | Obsolete proof of 2eu4 2331 as of 14-Sep-2019. (Contributed by NM, 3-Dec-2001.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | 2eu5 2333* | An alternate definition of double existential uniqueness (see 2eu4 2331). A mistake sometimes made in the literature is to use to mean "exactly one and exactly one ." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining as an additional condition. The correct definition apparently has never been published. ( means "exists at most one.") (Contributed by NM, 26-Oct-2003.) |
Theorem | 2eu6 2334* | Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Oct-2019.) |
Theorem | 2eu6OLD 2335* | Obsolete proof of 2eu6 2334 as of 21-Sep-2019. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | 2eu7 2336 | Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.) |
Theorem | 2eu8 2337 | Two equivalent expressions for double existential uniqueness. Curiously, we can put on either of the internal conjuncts but not both. We can also commute using 2eu7 2336. (Contributed by NM, 20-Feb-2005.) |
Theorem | euequ1OLD 2338* | Obsolete proof of euequ1 2244 as of 8-Sep-2019. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | exists1 2339* | Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru 4584. (Contributed by NM, 5-Apr-2004.) |
Theorem | exists2 2340 | A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Model the Aristotelian assertic syllogisms using modern notation. This section shows that the Aristotelian assertic syllogisms can be proven with our axioms of logic, and also provides generally useful theorems. In antiquity Aristotelian logic and Stoic logic (see mptnan 1621) were the leading logical systems. Aristotelian logic became the leading system in medieval Europe; this section models this system (including later refinements to it). Aristotle defined syllogisms very generally ("a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so") Aristotle, Prior Analytics 24b18-20. However, in Prior Analytics he limits himself to categorical syllogisms that consist of three categorical propositions with specific structures. The syllogisms are the valid subset of the possible combinations of these structures. The medieval schools used vowels to identify the types of terms (a=all, e=none, i=some, and o=some are not), and named the different syllogisms with Latin words that had the vowels in the intended order. "There is a surprising amount of scholarly debate about how best to formalize Aristotle's syllogisms..." according to Aristotle's Modal Proofs: Prior Analytics A8-22 in Predicate Logic, Adriane Rini, Springer, 2011, ISBN 978-94-007-0049-9, page 28. For example, Lukasiewicz believes it is important to note that "Aristotle does not introduce singular terms or premisses into his system". Lukasiewicz also believes that Aristotelian syllogisms are predicates (having a true/false value), not inference rules: "The characteristic sign of an inference is the word 'therefore'... no syllogism is formulated by Aristotle primarily as an inference, but they are all implications." Jan Lukasiewicz, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, Second edition, Oxford, 1957, page 1-2. Lukasiewicz devised a specialized prefix notation for representing Aristotelian syllogisms instead of using standard predicate logic notation. We instead translate each Aristotelian syllogism into an inference rule, and each rule is defined using standard predicate logic notation and predicates. The predicates are represented by wff variables that may depend on the quantified variable . Our translation is essentially identical to the one use in Rini page 18, Table 2 "Non-Modal Syllogisms in Lower Predicate Calculus (LPC)", which uses standard predicate logic with predicates. Rini states, "the crucial point is that we capture the meaning Aristotle intends, and the method by which we represent that meaning is less important." There are two differences: we make the existence criteria explicit, and we use , , and in the order they appear (a common Metamath convention). Patzig also uses standard predicate logic notation and predicates (though he interprets them as conditional propositions, not as inference rules); see Gunther Patzig, Aristotle's Theory of the Syllogism second edition, 1963, English translation by Jonathan Barnes, 1968, page 38. Terms such as "all" and "some" are translated into predicate logic using the approach devised by Frege and Russell. "Frege (and Russell) devised an ingenious procedure for regimenting binary quantifiers like "every" and "some" in terms of unary quantifiers like "everything" and "something": they formalized sentences of the form "Some A is B" and "Every A is B" as exists x (Ax and Bx) and all x (Ax implies Bx), respectively." "Quantifiers and Quantification", Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/quantification/. See Principia Mathematica page 22 and *10 for more information (especially *10.3 and *10.26). Expressions of the form "no is " are consistently translated as . These can also be expressed as , per alinexa 1684. We translate "all is " to , "some is " to , and "some is not " to . It is traditional to use the singular verb "is", not the plural verb "are", in the generic expressions. By convention the major premise is listed first. In traditional Aristotelian syllogisms the predicates have a restricted form ("x is a ..."); those predicates could be modeled in modern notation by more specific constructs such as , , or . Here we use wff variables instead of specialized restricted forms. This generalization makes the syllogisms more useful in more circumstances. In addition, these expressions make it clearer that the syllogisms of Aristotelian logic are the forerunners of predicate calculus. If we used restricted forms like instead, we would not only unnecessarily limit their use, but we would also need to use set and class axioms, making their relationship to predicate calculus less clear. Using such specific constructs would also be anti-historical; Aristotle and others who directly followed his work focused on relating wholes to their parts, an approach now called part-whole theory. The work of Cantor and Peano (over 2,000 years later) led to a sharper distinction between inclusion () and membership (); this distinction was not directly made in Aristotle's work. There are some widespread misconceptions about the existential assumptions made by Aristotle (aka "existential import"). Aristotle was not trying to develop something exactly corresponding to modern logic. Aristotle devised "a companion-logic for science. He relegates fictions like fairy godmothers and mermaids and unicorns to the realms of poetry and literature. In his mind, they exist outside the ambit of science. This is why he leaves no room for such non-existent entities in his logic. This is a thoughtful choice, not an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is "a phrase signifying a thing's essence." (Topics, I.5.102a37, Pickard-Cambridge.)... Because non-existent entities cannot be anything, they do not, in Aristotle's mind, possess an essence... This is why he leaves no place for fictional entities like goat-stags (or unicorns)." Source: Louis F. Groarke, "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy (A Peer-Reviewed Academic Resource), http://www.iep.utm.edu/aris-log/. Thus, some syllogisms have "extra" existence hypotheses that do not directly appear in Aristotle's original materials (since they were always assumed); they are added where they are needed. This affects barbari 2345, celaront 2346, cesaro 2351, camestros 2352, felapton 2357, darapti 2358, calemos 2362, fesapo 2363, and bamalip 2364. These are only the assertic syllogisms. Aristotle also defined modal syllogisms that deal with modal qualifiers such as "necessarily" and "possibly". Historically Aristotelian modal syllogisms were not as widely used. For more about modal syllogisms in a modern context, see Rini as well as Aristotle's Modal Syllogistic by Marko Malink, Harvard University Press, November 2013. We do not treat them further here. Aristotelian logic is essentially the forerunner of predicate calculus (as well as set theory since it discusses membership in groups), while Stoic logic is essentially the forerunner of propositional calculus. | ||
Theorem | barbara 2341 | "Barbara", one of the fundamental syllogisms of Aristotelian logic. All is , and all is , therefore all is . (In Aristotelian notation, AAA-1: MaP and SaM therefore SaP.) For example, given "All men are mortal" and "Socrates is a man", we can prove "Socrates is mortal". If H is the set of men, M is the set of mortal beings, and S is Socrates, these word phrases can be represented as (all men are mortal) and (Socrates is a man) therefore (Socrates is mortal). Russell and Whitehead note that the "syllogism in Barbara is derived..." from syl 17. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1724. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm, http://plato.stanford.edu/entries/aristotle-logic/, and https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) |
Theorem | celarent 2342 | "Celarent", one of the syllogisms of Aristotelian logic. No is , and all is , therefore no is . (In Aristotelian notation, EAE-1: MeP and SaM therefore SeP.) For example, given the "No reptiles have fur" and "All snakes are reptiles", therefore "No snakes have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | darii 2343 | "Darii", one of the syllogisms of Aristotelian logic. All is , and some is , therefore some is . (In Aristotelian notation, AII-1: MaP and SiM therefore SiP.) For example, given "All rabbits have fur" and "Some pets are rabbits", therefore "Some pets have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) |
Theorem | ferio 2344 | "Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No is , and some is , therefore some is not . (In Aristotelian notation, EIO-1: MeP and SiM therefore SoP.) For example, given "No homework is fun" and "Some reading is homework", therefore "Some reading is not fun". This is essentially a logical axiom in Aristotelian logic. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | barbari 2345 | "Barbari", one of the syllogisms of Aristotelian logic. All is , all is , and some exist, therefore some is . (In Aristotelian notation, AAI-1: MaP and SaM therefore SiP.) For example, given "All men are mortal", "All Greeks are men", and "Greeks exist", therefore "Some Greeks are mortal". Note the existence hypothesis (to prove the "some" in the conclusion). Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 30-Aug-2016.) |
Theorem | celaront 2346 | "Celaront", one of the syllogisms of Aristotelian logic. No is , all is , and some exist, therefore some is not . (In Aristotelian notation, EAO-1: MeP and SaM therefore SoP.) For example, given "No reptiles have fur", "All snakes are reptiles.", and "Snakes exist.", prove "Some snakes have no fur". Note the existence hypothesis. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | cesare 2347 | "Cesare", one of the syllogisms of Aristotelian logic. No is , and all is , therefore no is . (In Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to celarent 2342. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.) |
Theorem | camestres 2348 | "Camestres", one of the syllogisms of Aristotelian logic. All is , and no is , therefore no is . (In Aristotelian notation, AEE-2: PaM and SeM therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | festino 2349 | "Festino", one of the syllogisms of Aristotelian logic. No is , and some is , therefore some is not . (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.) |
Theorem | baroco 2350 | "Baroco", one of the syllogisms of Aristotelian logic. All is , and some is not , therefore some is not . (In Aristotelian notation, AOO-2: PaM and SoM therefore SoP.) For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative." (Contributed by David A. Wheeler, 28-Aug-2016.) |
Theorem | cesaro 2351 | "Cesaro", one of the syllogisms of Aristotelian logic. No is , all is , and exist, therefore some is not . (In Aristotelian notation, EAO-2: PeM and SaM therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | camestros 2352 | "Camestros", one of the syllogisms of Aristotelian logic. All is , no is , and exist, therefore some is not . (In Aristotelian notation, AEO-2: PaM and SeM therefore SoP.) For example, "All horses have hooves", "No humans have hooves", and humans exist, therefore "Some humans are not horses". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | datisi 2353 | "Datisi", one of the syllogisms of Aristotelian logic. All is , and some is , therefore some is . (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.) |
Theorem | disamis 2354 | "Disamis", one of the syllogisms of Aristotelian logic. Some is , and all is , therefore some is . (In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.) |
Theorem | ferison 2355 | "Ferison", one of the syllogisms of Aristotelian logic. No is , and some is , therefore some is not . (In Aristotelian notation, EIO-3: MeP and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | bocardo 2356 | "Bocardo", one of the syllogisms of Aristotelian logic. Some is not , and all is , therefore some is not . (In Aristotelian notation, OAO-3: MoP and MaS therefore SoP.) For example, "Some cats have no tails", "All cats are mammals", therefore "Some mammals have no tails". A reorder of disamis 2354; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.) |
Theorem | felapton 2357 | "Felapton", one of the syllogisms of Aristotelian logic. No is , all is , and some exist, therefore some is not . (In Aristotelian notation, EAO-3: MeP and MaS therefore SoP.) For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | darapti 2358 | "Darapti", one of the syllogisms of Aristotelian logic. All is , all is , and some exist, therefore some is . (In Aristotelian notation, AAI-3: MaP and MaS therefore SiP.) For example, "All squares are rectangles" and "All squares are rhombuses", therefore "Some rhombuses are rectangles". (Contributed by David A. Wheeler, 28-Aug-2016.) |
Theorem | calemes 2359 | "Calemes", one of the syllogisms of Aristotelian logic. All is , and no is , therefore no is . (In Aristotelian notation, AEE-4: PaM and MeS therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | dimatis 2360 | "Dimatis", one of the syllogisms of Aristotelian logic. Some is , and all is , therefore some is . (In Aristotelian notation, IAI-4: PiM and MaS therefore SiP.) For example, "Some pets are rabbits.", "All rabbits have fur", therefore "Some fur bearing animals are pets". Like darii 2343 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) |
Theorem | fresison 2361 | "Fresison", one of the syllogisms of Aristotelian logic. No is (PeM), and some is (MiS), therefore some is not (SoP). (In Aristotelian notation, EIO-4: PeM and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | calemos 2362 | "Calemos", one of the syllogisms of Aristotelian logic. All is (PaM), no is (MeS), and exist, therefore some is not (SoP). (In Aristotelian notation, AEO-4: PaM and MeS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | fesapo 2363 | "Fesapo", one of the syllogisms of Aristotelian logic. No is , all is , and exist, therefore some is not . (In Aristotelian notation, EAO-4: PeM and MaS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | bamalip 2364 | "Bamalip", one of the syllogisms of Aristotelian logic. All is , all is , and exist, therefore some is . (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2345. (Contributed by David A. Wheeler, 28-Aug-2016.) |
Intuitionistic (constructive) logic is similar to classical logic with the notable omission of ax-3 8 and theorems such as exmid 413 or peirce 181. We mostly treat intuitionistic logic in a separate file, iset.mm, which is known as the Intuitionistic Logic Explorer on the web site. However, iset.mm has a number of additional axioms (mainly to replace definitions like df-or 368 and df-ex 1634 which are not valid in intuitionistic logic) and we want to prove those axioms here to demonstrate that adding those axioms in iset.mm does not make iset.mm any less consistent than set.mm. The following axioms are unchanged between set.mm and iset.mm: ax-1 6, ax-2 7, ax-mp 5, ax-4 1652, ax-11 1866, ax-gen 1639, ax-7 1814, ax-12 1878, ax-8 1844, ax-9 1846, and ax-5 1725. In this list of axioms, the ones that repeat earlier theorems are marked "(New usage is discouraged.)" so that the earlier theorems will be used consistently in other proofs. | ||
Theorem | axia1 2365 | Left 'and' elimination (intuitionistic logic axiom ax-ia1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.) |
Theorem | axia2 2366 | Right 'and' elimination (intuitionistic logic axiom ax-ia2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.) |
Theorem | axia3 2367 | 'And' introduction (intuitionistic logic axiom ax-ia3). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.) |
Theorem | axin1 2368 | 'Not' introduction (intuitionistic logic axiom ax-in1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.) |
Theorem | axin2 2369 | 'Not' elimination (intuitionistic logic axiom ax-in2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.) |
Theorem | axio 2370 | Definition of 'or' (intuitionistic logic axiom ax-io). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.) |
Theorem | axi4 2371 | Specialization (intuitionistic logic axiom ax-4). This is just sp 1883 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.) |
Theorem | axi5r 2372 | Converse of ax-c4 (intuitionistic logic axiom ax-i5r). (Contributed by Jim Kingdon, 31-Dec-2017.) |
Theorem | axial 2373 | is not free in (intuitionistic logic axiom ax-ial). (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.) |
Theorem | axie1 2374 | is not free in (intuitionistic logic axiom ax-ie1). (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.) |
Theorem | axie2 2375 | A key property of existential quantification (intuitionistic logic axiom ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017.) |
Theorem | axi9 2376 | Axiom of existence (intuitionistic logic axiom ax-i9). In classical logic, this is equivalent to ax-6 1771 but in intuitionistic logic it needs to be stated using the existential quantifier. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.) |
Theorem | axi10 2377 | Axiom of Quantifier Substitution (intuitionistic logic axiom ax-10). This is just axc11n 2075 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.) |
Theorem | axi12 2378 | Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12). In classical logic, this is mostly a restatement of axc9 2072 (with one additional quantifier). But in intuitionistic logic, changing the negations and implications to disjunctions makes it stronger. (Contributed by Jim Kingdon, 31-Dec-2017.) |
Theorem | axbnd 2379 | Axiom of Bundling (intuitionistic logic axiom ax-bnd). In classical logic, this and axi12 2378 are fairly straightforward consequences of axc9 2072. But in intuitionistic logic, it is not easy to add the extra to axi12 2378 and so we treat the two as separate axioms. (Contributed by Jim Kingdon, 22-Mar-2018.) |
Set theory uses the formalism of propositional and predicate calculus to assert properties of arbitrary mathematical objects called "sets." A set can be an element of another set, and this relationship is indicated by the symbol. Starting with the simplest mathematical object, called the empty set, set theory builds up more and more complex structures whose existence follows from the axioms, eventually resulting in extremely complicated sets that we identify with the real numbers and other familiar mathematical objects. A simplistic concept of sets, sometimes called "naive set theory", is vulnerable to a paradox called "Russell's Paradox" (ru 3275), a discovery that revolutionized the foundations of mathematics and logic. Russell's Paradox spawned the development of set theories that countered the paradox, including the ZF set theory that is most widely used and is defined here. Except for Extensionality, the axioms basically say, "given an arbitrary set x (and, in the cases of Replacement and Regularity, provided that an antecedent is satisfied), there exists another set y based on or constructed from it, with the stated properties." (The axiom of Extensionality can also be restated this way as shown by axext2 2381.) The individual axiom links provide more detailed descriptions. We derive the redundant ZF axioms of Separation, Null Set, and Pairing from the others as theorems. | ||
Axiom | ax-ext 2380* |
Axiom of Extensionality. An axiom of Zermelo-Fraenkel set theory. It
states that two sets are identical if they contain the same elements.
Axiom Ext of [BellMachover] p.
461.
Set theory can also be formulated with a single primitive predicate on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes , and equality is defined as . All of the usual axioms of equality then become theorems of set theory. See, for example, Axiom 1 of [TakeutiZaring] p. 8. To use the above "equality-free" version of Extensionality with Metamath's logical axioms, we would rewrite ax-7 1814 through ax-9 1846 with equality expanded according to the above definition. Some of those axioms could be proved from set theory and would be redundant. Not all of them are redundant, since our axioms of predicate calculus make essential use of equality for the proper substitution that is a primitive notion in traditional predicate calculus. A study of such an axiomatization would be an interesting project for someone exploring the foundations of logic. General remarks: Our set theory axioms are presented using defined connectives (, , etc.) for convenience. However, it is implicitly understood that the actual axioms use only the primitive connectives , , , , and . It is straightforward to establish the equivalence between the actual axioms and the ones we display, and we will not do so. It is important to understand that strictly speaking, all of our set theory axioms are really schemes that represent an infinite number of actual axioms. This is inherent in the design of Metamath ("metavariable math"), which manipulates only metavariables. For example, the metavariable in ax-ext 2380 can represent any actual variable v1, v2, v3,... . Distinct variable restrictions ($d) prevent us from substituting say v1 for both and . This is in contrast to typical textbook presentations that present actual axioms (except for Replacement ax-rep 4506, which involves a wff metavariable). In practice, though, the theorems and proofs are essentially the same. The $d restrictions make each of the infinite axioms generated by the ax-ext 2380 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 21-May-1993.) |
Theorem | axext2 2381* | The Axiom of Extensionality (ax-ext 2380) restated so that it postulates the existence of a set given two arbitrary sets and . This way to express it follows the general idea of the other ZFC axioms, which is to postulate the existence of sets given other sets. (Contributed by NM, 28-Sep-2003.) |
Theorem | axext3 2382* | A generalization of the Axiom of Extensionality in which and need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) Remove dependencies on ax-10 1861, ax-12 1878, ax-13 2026. (Revised by Wolf Lammen, 9-Dec-2019.) |
Theorem | axext3OLD 2383* | Obsolete proof of axext3 2382 as of 9-Dec-2019. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | axext4 2384* | A bidirectional version of Extensionality. Although this theorem "looks" like it is just a definition of equality, it requires the Axiom of Extensionality for its proof under our axiomatization. See the comments for ax-ext 2380 and df-cleq 2394. (Contributed by NM, 14-Nov-2008.) |
Theorem | bm1.1 2385* | Any set defined by a property is the only set defined by that property. Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Wolf Lammen, 13-Nov-2019.) |
Theorem | bm1.1OLD 2386* | Obsolete proof of bm1.1 2385 as of 12-Nov-2019. (Contributed by NM, 30-Jun-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
Syntax | cab 2387 | Introduce the class builder or class abstraction notation ("the class of sets such that is true"). Our class variables , , etc. range over class builders (implicitly in the case of defined class terms such as df-nul 3738). Note that a setvar variable can be expressed as a class builder per theorem cvjust 2396, justifying the assignment of setvar variables to class variables via the use of cv 1404. |
Definition | df-clab 2388 |
Define class abstraction notation (so-called by Quine), also called a
"class builder" in the literature. and need not be distinct.
Definition 2.1 of [Quine] p. 16.
Typically, will
have as a
free variable, and " " is read "the class of all sets
such that is true." We do not define in
isolation but only as part of an expression that extends or
"overloads"
the
relationship.
This is our first use of the symbol to connect classes instead of sets. The syntax definition wcel 1842, which extends or "overloads" the wel 1843 definition connecting setvar variables, requires that both sides of be classes. In df-cleq 2394 and df-clel 2397, we introduce a new kind of variable (class variable) that can be substituted with expressions such as . In the present definition, the on the left-hand side is a setvar variable. Syntax definition cv 1404 allows us to substitute a setvar variable for a class variable: all sets are classes by cvjust 2396 (but not necessarily vice-versa). For a full description of how classes are introduced and how to recover the primitive language, see the discussion in Quine (and under abeq2 2526 for a quick overview). Because class variables can be substituted with compound expressions and setvar variables cannot, it is often useful to convert a theorem containing a free setvar variable to a more general version with a class variable. This is done with theorems such as vtoclg 3116 which is used, for example, to convert elirrv 8056 to elirr 8057. This is called the "axiom of class comprehension" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. He calls the construction a "class term". While the three class definitions df-clab 2388, df-cleq 2394, and df-clel 2397 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker. For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 26-May-1993.) |
Theorem | abid 2389 | Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 26-May-1993.) |
Theorem | hbab1 2390* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-1993.) |
Theorem | nfsab1 2391* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbab 2392* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) |
Theorem | nfsab 2393* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Definition | df-cleq 2394* |
Define the equality connective between classes. Definition 2.7 of
[Quine] p. 18. Also Definition 4.5 of
[TakeutiZaring] p. 13; Chapter 4
provides its justification and methods for eliminating it. Note that
its elimination will not necessarily result in a single wff in the
original language but possibly a "scheme" of wffs.
This is an example of a somewhat "risky" definition, meaning that it has a more complex than usual soundness justification (outside of Metamath), because it "overloads" or reuses the existing equality symbol rather than introducing a new symbol. This allows us to make statements that may not hold for the original symbol. For example, it permits us to deduce , which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see theorem axext4 2384). We therefore include this axiom as a hypothesis, so that the use of Extensionality is properly indicated. We could avoid this complication by introducing a new symbol, say =_{2}, in place of . This would also have the advantage of making elimination of the definition straightforward, so that we could eliminate Extensionality as a hypothesis. We would then also have the advantage of being able to identify in various proofs exactly where Extensionality truly comes into play rather than just being an artifact of a definition. One of our theorems would then be =_{2} by invoking Extensionality. However, to conform to literature usage, we retain this overloaded definition. This also makes some proofs shorter and probably easier to read, without the constant switching between two kinds of equality. See also comments under df-clab 2388, df-clel 2397, and abeq2 2526. In the form of dfcleq 2395, this is called the "axiom of extensionality" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. While the three class definitions df-clab 2388, df-cleq 2394, and df-clel 2397 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker. For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 15-Sep-1993.) |
Theorem | dfcleq 2395* | The same as df-cleq 2394 with the hypothesis removed using the Axiom of Extensionality ax-ext 2380. (Contributed by NM, 15-Sep-1993.) |
Theorem | cvjust 2396* | Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a setvar variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 1404, which allows us to substitute a setvar variable for a class variable. See also cab 2387 and df-clab 2388. Note that this is not a rigorous justification, because cv 1404 is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." (Contributed by NM, 7-Nov-2006.) |
Definition | df-clel 2397* |
Define the membership connective between classes. Theorem 6.3 of
[Quine] p. 41, or Proposition 4.6 of [TakeutiZaring] p. 13, which we
adopt as a definition. See these references for its metalogical
justification. Note that like df-cleq 2394 it extends or "overloads" the
use of the existing membership symbol, but unlike df-cleq 2394 it does not
strengthen the set of valid wffs of logic when the class variables are
replaced with setvar variables (see cleljust 2133), so we don't include
any set theory axiom as a hypothesis. See also comments about the
syntax under df-clab 2388. Alternate definitions of
(but that
require either
or to be a set) are
shown by clel2 3185,
clel3 3187, and clel4 3188.
This is called the "axiom of membership" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. While the three class definitions df-clab 2388, df-cleq 2394, and df-clel 2397 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker. For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 26-May-1993.) |
Theorem | eqriv 2398* | Infer equality of classes from equivalence of membership. (Contributed by NM, 21-Jun-1993.) |
Theorem | eqrdv 2399* | Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.) |
Theorem | eqrdav 2400* | Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Nov-2019.) |
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