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Type | Label | Description |
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Statement | ||
Theorem | mo2n 2001* | There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.) |
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Theorem | mon 2002 | There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.) |
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Theorem | euex 2003 | Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Theorem | eumo0 2004* | Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.) |
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Theorem | eumo 2005 | Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) |
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Theorem | eumoi 2006 | "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
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Theorem | mobidh 2007 | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) |
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Theorem | mobid 2008 | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) |
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Theorem | mobidv 2009* | Formula-building rule for "at most one" quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.) |
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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.) |
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Theorem | hbmo1 2011 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) |
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Theorem | hbmo 2012 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.) |
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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.) |
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Theorem | mo23 2014* | An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.) |
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Theorem | mor 2015* |
Converse of mo23 2014 with an additional ![]() ![]() ![]() |
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Theorem | modc 2016* | Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.) |
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Theorem | eu2 2017* | An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) |
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Theorem | eu3h 2018* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.) |
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Theorem | eu3 2019* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) |
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Theorem | eu5 2020 | Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) |
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Theorem | exmoeu2 2021 | Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.) |
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Theorem | moabs 2022 | Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.) |
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Theorem | exmodc 2023 | If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.) |
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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.) |
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Theorem | mo2r 2025* | A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.) |
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Theorem | mo3h 2026* |
Alternate definition of "at most one." Definition of [BellMachover]
p. 460, except that definition has the side condition that ![]() ![]() |
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Theorem | mo3 2027* |
Alternate definition of "at most one." Definition of [BellMachover]
p. 460, except that definition has the side condition that ![]() ![]() |
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Theorem | mo2dc 2028* | Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.) |
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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.) |
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Theorem | euanv 2030* | Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.) |
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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.) |
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Theorem | sbmo 2032* | Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | mo4f 2033* | "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.) |
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Theorem | mo4 2034* | "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
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Theorem | eu4 2035* | Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
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Theorem | exmoeudc 2036 | Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.) |
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Theorem | moim 2037 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.) |
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Theorem | moimi 2038 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.) |
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Theorem | moimv 2039* | Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.) |
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Theorem | euimmo 2040 | Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.) |
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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.) |
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Theorem | moan 2042 | "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.) |
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Theorem | moani 2043 | "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.) |
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Theorem | moor 2044 | "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.) |
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Theorem | mooran1 2045 | "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Theorem | mooran2 2046 | "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Theorem | moanim 2047 | Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 3-Dec-2001.) |
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Theorem | moanimv 2048* | Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 23-Mar-1995.) |
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Theorem | moaneu 2049 | Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.) |
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Theorem | moanmo 2050 | Nested at-most-one quantifiers. (Contributed by NM, 25-Jan-2006.) |
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Theorem | mopick 2051 | "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) |
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Theorem | eupick 2052 |
Existential uniqueness "picks" a variable value for which another wff
is
true. If there is only one thing ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | eupicka 2053 | Version of eupick 2052 with closed formulas. (Contributed by NM, 6-Sep-2008.) |
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Theorem | eupickb 2054 | Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) |
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Theorem | eupickbi 2055 | Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) |
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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.) |
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Theorem | moexexdc 2057 | "At most one" double quantification. (Contributed by Jim Kingdon, 5-Jul-2018.) |
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Theorem | euexex 2058 | Existential uniqueness and "at most one" double quantification. (Contributed by Jim Kingdon, 28-Dec-2018.) |
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Theorem | 2moex 2059 | Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.) |
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Theorem | 2euex 2060 | Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Theorem | 2eumo 2061 | Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.) |
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Theorem | 2eu2ex 2062 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |
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Theorem | 2moswapdc 2063 | A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.) |
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Theorem | 2euswapdc 2064 | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.) |
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Theorem | 2exeu 2065 | Double existential uniqueness implies double unique existential quantification. (Contributed by NM, 3-Dec-2001.) |
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Theorem | 2eu4 2066* |
This theorem provides us with a definition of double existential
uniqueness ("exactly one ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | 2eu7 2067 | Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.) |
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Theorem | euequ1 2068* | Equality has existential uniqueness. (Contributed by Stefan Allan, 4-Dec-2008.) |
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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.) |
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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.) |
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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
Expressions of the form "no
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
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
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Theorem | celarent 2072 |
"Celarent", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | darii 2073 |
"Darii", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ferio 2074 |
"Ferio" ("Ferioque"), one of the syllogisms of Aristotelian
logic. No
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Theorem | barbari 2075 |
"Barbari", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | celaront 2076 |
"Celaront", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | cesare 2077 |
"Cesare", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | camestres 2078 |
"Camestres", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | festino 2079 |
"Festino", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | baroco 2080 |
"Baroco", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | cesaro 2081 |
"Cesaro", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | camestros 2082 |
"Camestros", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | datisi 2083 |
"Datisi", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | disamis 2084 |
"Disamis", one of the syllogisms of Aristotelian logic. Some ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ferison 2085 |
"Ferison", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | bocardo 2086 |
"Bocardo", one of the syllogisms of Aristotelian logic. Some ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | felapton 2087 |
"Felapton", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | darapti 2088 |
"Darapti", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | calemes 2089 |
"Calemes", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dimatis 2090 |
"Dimatis", one of the syllogisms of Aristotelian logic. Some ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | fresison 2091 |
"Fresison", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | calemos 2092 |
"Calemos", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | fesapo 2093 |
"Fesapo", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | bamalip 2094 |
"Bamalip", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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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
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
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 |
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Theorem | axext3 2096* |
A generalization of the Axiom of Extensionality in which ![]() ![]() |
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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.) |
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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.) |
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Syntax | cab 2099 |
Introduce the class builder or class abstraction notation ("the class of
sets ![]() ![]() ![]() ![]() |
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Definition | df-clab 2100 |
Define class abstraction notation (so-called by Quine), also called a
"class builder" in the literature. ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
This is our first use of the 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 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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