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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | mo2n 2001* | There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.) |
Theorem | mon 2002 | There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.) |
Theorem | euex 2003 | Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | eumo0 2004* | Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.) |
Theorem | eumo 2005 | Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) |
Theorem | eumoi 2006 | "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
Theorem | mobidh 2007 | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) |
Theorem | mobid 2008 | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) |
Theorem | mobidv 2009* | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.) |
Theorem | mobii 2010 | Formula-building rule for "at most one" quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.) |
Theorem | hbmo1 2011 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) |
Theorem | hbmo 2012 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.) |
Theorem | cbvmo 2013 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) |
Theorem | mo23 2014* | An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.) |
Theorem | mor 2015* | Converse of mo23 2014 with an additional condition. (Contributed by Jim Kingdon, 25-Jun-2018.) |
Theorem | modc 2016* | Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.) |
DECID | ||
Theorem | eu2 2017* | An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) |
Theorem | eu3h 2018* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.) |
Theorem | eu3 2019* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) |
Theorem | eu5 2020 | Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) |
Theorem | exmoeu2 2021 | Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.) |
Theorem | moabs 2022 | Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.) |
Theorem | exmodc 2023 | If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.) |
DECID | ||
Theorem | exmonim 2024 | There is at most one of something which does not exist. Unlike exmodc 2023 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.) |
Theorem | mo2r 2025* | A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.) |
Theorem | mo3h 2026* | Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that not occur in in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (New usage is discouraged.) |
Theorem | mo3 2027* | Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that not occur in in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) |
Theorem | mo2dc 2028* | Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.) |
DECID | ||
Theorem | euan 2029 | Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | euanv 2030* | Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.) |
Theorem | euor2 2031 | Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | sbmo 2032* | Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | mo4f 2033* | "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.) |
Theorem | mo4 2034* | "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
Theorem | eu4 2035* | Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
Theorem | exmoeudc 2036 | Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.) |
DECID | ||
Theorem | moim 2037 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.) |
Theorem | moimi 2038 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.) |
Theorem | moimv 2039* | Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.) |
Theorem | euimmo 2040 | Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.) |
Theorem | euim 2041 | Add existential unique existential quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Theorem | moan 2042 | "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.) |
Theorem | moani 2043 | "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.) |
Theorem | moor 2044 | "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.) |
Theorem | mooran1 2045 | "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | mooran2 2046 | "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | moanim 2047 | Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 3-Dec-2001.) |
Theorem | moanimv 2048* | Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 23-Mar-1995.) |
Theorem | moaneu 2049 | Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.) |
Theorem | moanmo 2050 | Nested at-most-one quantifiers. (Contributed by NM, 25-Jan-2006.) |
Theorem | mopick 2051 | "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) |
Theorem | eupick 2052 | 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 2053 | Version of eupick 2052 with closed formulas. (Contributed by NM, 6-Sep-2008.) |
Theorem | eupickb 2054 | Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) |
Theorem | eupickbi 2055 | Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Theorem | mopick2 2056 | "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 1591. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | moexexdc 2057 | "At most one" double quantification. (Contributed by Jim Kingdon, 5-Jul-2018.) |
DECID | ||
Theorem | euexex 2058 | Existential uniqueness and "at most one" double quantification. (Contributed by Jim Kingdon, 28-Dec-2018.) |
Theorem | 2moex 2059 | Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.) |
Theorem | 2euex 2060 | Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | 2eumo 2061 | Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.) |
Theorem | 2eu2ex 2062 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |
Theorem | 2moswapdc 2063 | A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.) |
DECID | ||
Theorem | 2euswapdc 2064 | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.) |
DECID | ||
Theorem | 2exeu 2065 | Double existential uniqueness implies double unique existential quantification. (Contributed by NM, 3-Dec-2001.) |
Theorem | 2eu4 2066* | 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 2exeu 2065 for a one-way implication. (Contributed by NM, 3-Dec-2001.) |
Theorem | 2eu7 2067 | Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.) |
Theorem | euequ1 2068* | Equality has existential uniqueness. (Contributed by Stefan Allan, 4-Dec-2008.) |
Theorem | exists1 2069* | 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. (Contributed by NM, 5-Apr-2004.) |
Theorem | exists2 2070 | 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 1382) 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 aproach 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/ 1382. 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 1563. 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 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 Aristolean 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. 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 nonexistent 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 nonexistent 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/ 1563. 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 2075, celaront 2076, cesaro 2081, camestros 2082, felapton 2087, darapti 2088, calemos 2092, fesapo 2093, and bamalip 2094. 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. Aristotelean 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 2071 | "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 14. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1595. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm 1595, http://plato.stanford.edu/entries/aristotle-logic/ 1595, and https://en.wikipedia.org/wiki/Syllogism 1595. (Contributed by David A. Wheeler, 24-Aug-2016.) |
Theorem | celarent 2072 | "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 2073 | "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 2074 | "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 2075 | "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 2076 | "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 2077 | "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 2072. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.) |
Theorem | camestres 2078 | "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 2079 | "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 2080 | "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 2081 | "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 2082 | "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 2083 | "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 2084 | "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 2085 | "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 2086 | "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 2084; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.) |
Theorem | felapton 2087 | "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 2088 | "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 2089 | "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 2090 | "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 2073 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) |
Theorem | fresison 2091 | "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 2092 | "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 2093 | "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 2094 | "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 2075. (Contributed by David A. Wheeler, 28-Aug-2016.) |
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. Here we develop set theory based on the Intuitionistic Zermelo-Fraenkel (IZF) system, mostly following the IZF axioms as laid out in [Crosilla]. Constructive Zermelo-Fraenkel (CZF), also described in Crosilla, is not as easy to formalize in Metamath because the statement of some of its axioms uses the notion of "bounded formula". Since Metamath has, purposefully, a very weak metalogic, that notion must be developed in the logic itself. This is similar to our treatment of substitution (df-sb 1717) and our definition of the nonfreeness predicate (df-nf 1418), whereas substitution and bound and free variables are ordinarily defined in the metalogic. The development of CZF has begun in BJ's mathbox, see wbd 12693. | ||
Axiom | ax-ext 2095* |
Axiom of Extensionality. It states that two sets are identical if they
contain the same elements. Axiom 1 of [Crosilla] p. "Axioms of CZF and
IZF" (with unnecessary quantifiers removed).
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-8 1463 through ax-16 1766 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. 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 2095 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 axioms which involve wff metavariables). 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 2095 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 5-Aug-1993.) |
Theorem | axext3 2096* | 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.) |
Theorem | axext4 2097* | 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 2095. (Contributed by NM, 14-Nov-2008.) |
Theorem | bm1.1 2098* | 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.) |
Syntax | cab 2099 | Introduce the class builder or class abstraction notation ("the class of sets such that is true"). Our class variables , , etc. range over class builders (sometimes implicitly). Note that a setvar variable can be expressed as a class builder per theorem cvjust 2108, justifying the assignment of setvar variables to class variables via the use of cv 1311. |
Definition | df-clab 2100 |
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 1461, which extends or "overloads" the wel 1462 definition connecting setvar variables, requires that both sides of be a class. In df-cleq 2106 and df-clel 2109, we introduce a new kind of variable (class variable) that can substituted with expressions such as . In the present definition, the on the left-hand side is a setvar variable. Syntax definition cv 1311 allows us to substitute a setvar variable for a class variable: all sets are classes by cvjust 2108 (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 2221 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 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". For a general discussion of the theory of classes, see https://us.metamath.org/mpeuni/mmset.html#class 2221. (Contributed by NM, 5-Aug-1993.) |
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