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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 mpto1 1533) 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 x. 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 φ, ψ, and χ 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/ 1533. See Principia Mathematica page 22 and *10 for more information (especially *10.3 and *10.26). Expressions of the form "no φ is ψ " are consistently translated as ∀x(φ → ¬ ψ). These can also be expressed as ¬ ∃x(φ ∧ ψ), per alinexa 1578. We translate "all φ is ψ " to ∀x(φ → ψ), "some φ is ψ " to ∃x(φ ∧ ψ), and "some φ is not ψ " to ∃x(φ ∧ ¬ ψ). 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 x = A, x ∈ A, or x ⊆ A. 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 x ∈ A 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 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/ 1578. 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 2305, celaront 2306, cesaro 2311, camestros 2312, felapton 2317, darapti 2318, calemos 2322, fesapo 2323, and bamalip 2324. 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 2301 | "Barbara", one of the fundamental syllogisms of Aristotelian logic. All φ is ψ, and all χ is φ, therefore all χ is ψ. (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 ∀x(x ∈ H → x ∈ M) (all men are mortal) and ∀x(x = S → x ∈ H) (Socrates is a man) therefore ∀x(x = S → x ∈ M) (Socrates is mortal). Russell and Whitehead note that the "syllogism in Barbara is derived..." from syl 15. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1615. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm 1615, http://plato.stanford.edu/entries/aristotle-logic/ 1615, and https://en.wikipedia.org/wiki/Syllogism 1615. (Contributed by David A. Wheeler, 24-Aug-2016.) |
⊢ ∀x(φ → ψ) & ⊢ ∀x(χ → φ) ⇒ ⊢ ∀x(χ → ψ) | ||
Theorem | celarent 2302 | "Celarent", one of the syllogisms of Aristotelian logic. No φ is ψ, and all χ is φ, therefore no χ is ψ. (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.) |
⊢ ∀x(φ → ¬ ψ) & ⊢ ∀x(χ → φ) ⇒ ⊢ ∀x(χ → ¬ ψ) | ||
Theorem | darii 2303 | "Darii", one of the syllogisms of Aristotelian logic. All φ is ψ, and some χ is φ, therefore some χ is ψ. (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.) |
⊢ ∀x(φ → ψ) & ⊢ ∃x(χ ∧ φ) ⇒ ⊢ ∃x(χ ∧ ψ) | ||
Theorem | ferio 2304 | "Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No φ is ψ, and some χ is φ, therefore some χ is not ψ. (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.) |
⊢ ∀x(φ → ¬ ψ) & ⊢ ∃x(χ ∧ φ) ⇒ ⊢ ∃x(χ ∧ ¬ ψ) | ||
Theorem | barbari 2305 | "Barbari", one of the syllogisms of Aristotelian logic. All φ is ψ, all χ is φ, and some χ exist, therefore some χ is ψ. (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.) |
⊢ ∀x(φ → ψ) & ⊢ ∀x(χ → φ) & ⊢ ∃xχ ⇒ ⊢ ∃x(χ ∧ ψ) | ||
Theorem | celaront 2306 | "Celaront", one of the syllogisms of Aristotelian logic. No φ is ψ, all χ is φ, and some χ exist, therefore some χ is not ψ. (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.) |
⊢ ∀x(φ → ¬ ψ) & ⊢ ∀x(χ → φ) & ⊢ ∃xχ ⇒ ⊢ ∃x(χ ∧ ¬ ψ) | ||
Theorem | cesare 2307 | "Cesare", one of the syllogisms of Aristotelian logic. No φ is ψ, and all χ is ψ, therefore no χ is φ. (In Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to celarent 2302. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.) |
⊢ ∀x(φ → ¬ ψ) & ⊢ ∀x(χ → ψ) ⇒ ⊢ ∀x(χ → ¬ φ) | ||
Theorem | camestres 2308 | "Camestres", one of the syllogisms of Aristotelian logic. All φ is ψ, and no χ is ψ, therefore no χ is φ. (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.) |
⊢ ∀x(φ → ψ) & ⊢ ∀x(χ → ¬ ψ) ⇒ ⊢ ∀x(χ → ¬ φ) | ||
Theorem | festino 2309 | "Festino", one of the syllogisms of Aristotelian logic. No φ is ψ, and some χ is ψ, therefore some χ is not φ. (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.) |
⊢ ∀x(φ → ¬ ψ) & ⊢ ∃x(χ ∧ ψ) ⇒ ⊢ ∃x(χ ∧ ¬ φ) | ||
Theorem | baroco 2310 | "Baroco", one of the syllogisms of Aristotelian logic. All φ is ψ, and some χ is not ψ, therefore some χ is not φ. (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.) |
⊢ ∀x(φ → ψ) & ⊢ ∃x(χ ∧ ¬ ψ) ⇒ ⊢ ∃x(χ ∧ ¬ φ) | ||
Theorem | cesaro 2311 | "Cesaro", one of the syllogisms of Aristotelian logic. No φ is ψ, all χ is ψ, and χ exist, therefore some χ is not φ. (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.) |
⊢ ∀x(φ → ¬ ψ) & ⊢ ∀x(χ → ψ) & ⊢ ∃xχ ⇒ ⊢ ∃x(χ ∧ ¬ φ) | ||
Theorem | camestros 2312 | "Camestros", one of the syllogisms of Aristotelian logic. All φ is ψ, no χ is ψ, and χ exist, therefore some χ is not φ. (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.) |
⊢ ∀x(φ → ψ) & ⊢ ∀x(χ → ¬ ψ) & ⊢ ∃xχ ⇒ ⊢ ∃x(χ ∧ ¬ φ) | ||
Theorem | datisi 2313 | "Datisi", one of the syllogisms of Aristotelian logic. All φ is ψ, and some φ is χ, therefore some χ is ψ. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.) |
⊢ ∀x(φ → ψ) & ⊢ ∃x(φ ∧ χ) ⇒ ⊢ ∃x(χ ∧ ψ) | ||
Theorem | disamis 2314 | "Disamis", one of the syllogisms of Aristotelian logic. Some φ is ψ, and all φ is χ, therefore some χ is ψ. (In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.) |
⊢ ∃x(φ ∧ ψ) & ⊢ ∀x(φ → χ) ⇒ ⊢ ∃x(χ ∧ ψ) | ||
Theorem | ferison 2315 | "Ferison", one of the syllogisms of Aristotelian logic. No φ is ψ, and some φ is χ, therefore some χ is not ψ. (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.) |
⊢ ∀x(φ → ¬ ψ) & ⊢ ∃x(φ ∧ χ) ⇒ ⊢ ∃x(χ ∧ ¬ ψ) | ||
Theorem | bocardo 2316 | "Bocardo", one of the syllogisms of Aristotelian logic. Some φ is not ψ, and all φ is χ, therefore some χ is not ψ. (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 2314; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.) |
⊢ ∃x(φ ∧ ¬ ψ) & ⊢ ∀x(φ → χ) ⇒ ⊢ ∃x(χ ∧ ¬ ψ) | ||
Theorem | felapton 2317 | "Felapton", one of the syllogisms of Aristotelian logic. No φ is ψ, all φ is χ, and some φ exist, therefore some χ is not ψ. (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.) |
⊢ ∀x(φ → ¬ ψ) & ⊢ ∀x(φ → χ) & ⊢ ∃xφ ⇒ ⊢ ∃x(χ ∧ ¬ ψ) | ||
Theorem | darapti 2318 | "Darapti", one of the syllogisms of Aristotelian logic. All φ is ψ, all φ is χ, and some φ exist, therefore some χ is ψ. (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.) |
⊢ ∀x(φ → ψ) & ⊢ ∀x(φ → χ) & ⊢ ∃xφ ⇒ ⊢ ∃x(χ ∧ ψ) | ||
Theorem | calemes 2319 | "Calemes", one of the syllogisms of Aristotelian logic. All φ is ψ, and no ψ is χ, therefore no χ is φ. (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.) |
⊢ ∀x(φ → ψ) & ⊢ ∀x(ψ → ¬ χ) ⇒ ⊢ ∀x(χ → ¬ φ) | ||
Theorem | dimatis 2320 | "Dimatis", one of the syllogisms of Aristotelian logic. Some φ is ψ, and all ψ is χ, therefore some χ is φ. (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 2303 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) |
⊢ ∃x(φ ∧ ψ) & ⊢ ∀x(ψ → χ) ⇒ ⊢ ∃x(χ ∧ φ) | ||
Theorem | fresison 2321 | "Fresison", one of the syllogisms of Aristotelian logic. No φ is ψ (PeM), and some ψ is χ (MiS), therefore some χ is not φ (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.) |
⊢ ∀x(φ → ¬ ψ) & ⊢ ∃x(ψ ∧ χ) ⇒ ⊢ ∃x(χ ∧ ¬ φ) | ||
Theorem | calemos 2322 | "Calemos", one of the syllogisms of Aristotelian logic. All φ is ψ (PaM), no ψ is χ (MeS), and χ exist, therefore some χ is not φ (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.) |
⊢ ∀x(φ → ψ) & ⊢ ∀x(ψ → ¬ χ) & ⊢ ∃xχ ⇒ ⊢ ∃x(χ ∧ ¬ φ) | ||
Theorem | fesapo 2323 | "Fesapo", one of the syllogisms of Aristotelian logic. No φ is ψ, all ψ is χ, and ψ exist, therefore some χ is not φ. (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.) |
⊢ ∀x(φ → ¬ ψ) & ⊢ ∀x(ψ → χ) & ⊢ ∃xψ ⇒ ⊢ ∃x(χ ∧ ¬ φ) | ||
Theorem | bamalip 2324 | "Bamalip", one of the syllogisms of Aristotelian logic. All φ is ψ, all ψ is χ, and φ exist, therefore some χ is φ. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2305. (Contributed by David A. Wheeler, 28-Aug-2016.) |
⊢ ∀x(φ → ψ) & ⊢ ∀x(ψ → χ) & ⊢ ∃xφ ⇒ ⊢ ∃x(χ ∧ φ) | ||
Intuitionistic (constructive) logic is similar to classical logic with the notable omission of ax-3 7 and theorems such as exmid 404 or peirce 172. 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 359 and df-ex 1542 which are not valid in intitionistic 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. | ||
Theorem | axi4 2325 | Specialization (intuitionistic logic axiom ax-4). This is just sp 1747 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) |
⊢ (∀xφ → φ) | ||
Theorem | axi5r 2326 | Converse of ax-5o (intuitionistic logic axiom ax-i5r). (Contributed by Jim Kingdon, 31-Dec-2017.) |
⊢ ((∀xφ → ∀xψ) → ∀x(∀xφ → ψ)) | ||
Theorem | axial 2327 | x is not free in ∀xφ (intuitionistic logic axiom ax-ial). (Contributed by Jim Kingdon, 31-Dec-2017.) |
⊢ (∀xφ → ∀x∀xφ) | ||
Theorem | axie1 2328 | x is bound in ∃xφ (intuitionistic logic axiom ax-ie1). (Contributed by Jim Kingdon, 31-Dec-2017.) |
⊢ (∃xφ → ∀x∃xφ) | ||
Theorem | axie2 2329 | A key property of existential quantification (intuitionistic logic axiom ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017.) |
⊢ (∀x(ψ → ∀xψ) → (∀x(φ → ψ) ↔ (∃xφ → ψ))) | ||
Theorem | axi9 2330 | Axiom of existence (intuitionistic logic axiom ax-i9). In classical logic, this is equivalent to ax-9 1654 but in intuitionistic logic it needs to be stated using the existential quantifier. (Contributed by Jim Kingdon, 31-Dec-2017.) |
⊢ ∃x x = y | ||
Theorem | axi10 2331 | Axiom of Quantifier Substitution (intuitionistic logic axiom ax-10). This is just ax10 1944 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) |
⊢ (∀x x = y → ∀y y = x) | ||
Theorem | axi11e 2332 | Axiom of Variable Substitution for Existence (intuitionistic logic axiom ax-i11e). This can be derived from ax-11 1746 in a classical context but a separate axiom is needed for intuitionistic predicate calculus. (Contributed by Jim Kingdon, 31-Dec-2017.) |
⊢ (x = y → (∃x(x = y ∧ φ) → ∃yφ)) | ||
Theorem | axi12 2333 |
Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12).
In classical logic, this is mostly a restatement of ax12o 1934 (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.) |
⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y))) | ||
Here we introduce New Foundations set theory. We first introduce the axiom of extensionality in ax-ext 2334. We later add set construction axioms from Hailperin, such as ax-nin 4078, that are designed to implement the Stratification Axiom from Quine. We then introduce ordered pairs, relationships, and functions. Note that the definition of an ordered pair (in df-op 4566) is different than the Kuratowski ordered pair definition (in df-opk 4058) typically used in ZFC, because the Kuratowski definition is not type-level. We conclude with orderings. | ||
Axiom | ax-ext 2334* |
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 (∀w(w ∈ x ↔ w ∈ y) → (x ∈ z → y ∈ z)), and equality x = y is defined as ∀w(w ∈ x ↔ w ∈ y). 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 1675 through ax-16 2144 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 x in ax-ext 2334 can represent any actual variable v1, v2, v3,... . Distinct variable restrictions ($d) prevent us from substituting say v1 for both x and z. This is in contrast to typical textbook presentations that present actual axioms (except for ZFC Replacement ax-rep in set.mm, 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 2334 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y) | ||
Theorem | axext2 2335* | The Axiom of Extensionality (ax-ext 2334) restated so that it postulates the existence of a set z given two arbitrary sets x and y. 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.) |
⊢ ∃z((z ∈ x ↔ z ∈ y) → x = y) | ||
Theorem | axext3 2336* | A generalization of the Axiom of Extensionality in which x and y need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y) | ||
Theorem | axext4 2337* | 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 2334 and df-cleq 2346. (Contributed by NM, 14-Nov-2008.) |
⊢ (x = y ↔ ∀z(z ∈ x ↔ z ∈ y)) | ||
Theorem | bm1.1 2338* | 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.) |
⊢ Ⅎxφ ⇒ ⊢ (∃x∀y(y ∈ x ↔ φ) → ∃!x∀y(y ∈ x ↔ φ)) | ||
Syntax | cab 2339 | Introduce the class builder or class abstraction notation ("the class of sets x such that φ is true"). Our class variables A, B, etc. range over class builders (implicitly in the case of defined class terms such as df-nul 3551). Note that a setvar variable can be expressed as a class builder per theorem cvjust 2348, justifying the assignment of setvar variables to class variables via the use of cv 1641. |
class {x ∣ φ} | ||
Definition | df-clab 2340 |
Define class abstraction notation (so-called by Quine), also called a
"class builder" in the literature. x and y need not be distinct.
Definition 2.1 of [Quine] p. 16. Typically,
φ will have y as a
free variable, and "{y
∣ φ} " is read "the class of
all sets y
such that φ(y) is true." We do not define {y ∣
φ} 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 1710, which extends or "overloads" the wel 1711 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2346 and df-clel 2349, we introduce a new kind of variable (class variable) that can substituted with expressions such as {y ∣ φ}. In the present definition, the x on the left-hand side is a setvar variable. Syntax definition cv 1641 allows us to substitute a setvar variable x for a class variable: all sets are classes by cvjust 2348 (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 2458 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 2914 which is used, for example, to convert elirrv in set.mm to elirr in set.mm. 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 {y ∣ φ} a "class term". For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class 2914. (Contributed by NM, 5-Aug-1993.) |
⊢ (x ∈ {y ∣ φ} ↔ [x / y]φ) | ||
Theorem | abid 2341 | Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 5-Aug-1993.) |
⊢ (x ∈ {x ∣ φ} ↔ φ) | ||
Theorem | hbab1 2342* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.) |
⊢ (y ∈ {x ∣ φ} → ∀x y ∈ {x ∣ φ}) | ||
Theorem | nfsab1 2343* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎx y ∈ {x ∣ φ} | ||
Theorem | hbab 2344* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (z ∈ {y ∣ φ} → ∀x z ∈ {y ∣ φ}) | ||
Theorem | nfsab 2345* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎxφ ⇒ ⊢ Ⅎx z ∈ {y ∣ φ} | ||
Definition | df-cleq 2346* |
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 y = z ↔ ∀x(x ∈ y ↔ x ∈ z), which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see theorem axext4 2337). 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 x =_{2} y ↔ x = y 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 2340, df-clel 2349, and abeq2 2458. In the form of dfcleq 2347, 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. For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class 2347. (Contributed by NM, 15-Sep-1993.) |
⊢ (∀x(x ∈ y ↔ x ∈ z) → y = z) ⇒ ⊢ (A = B ↔ ∀x(x ∈ A ↔ x ∈ B)) | ||
Theorem | dfcleq 2347* | The same as df-cleq 2346 with the hypothesis removed using the Axiom of Extensionality ax-ext 2334. (Contributed by NM, 15-Sep-1993.) |
⊢ (A = B ↔ ∀x(x ∈ A ↔ x ∈ B)) | ||
Theorem | cvjust 2348* | Every setvar 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 1641, which allows us to substitute a setvar variable for a class variable. See also cab 2339 and df-clab 2340. Note that this is not a rigorous justification, because cv 1641 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.) |
⊢ x = {y ∣ y ∈ x} | ||
Definition | df-clel 2349* |
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 2346 it extends or "overloads" the
use of the existing membership symbol, but unlike df-cleq 2346 it does not
strengthen the set of valid wffs of logic when the class variables are
replaced with setvar variables (see cleljust 2014), so we don't include
any set theory axiom as a hypothesis. See also comments about the
syntax under df-clab 2340. Alternate definitions of A ∈ B (but that
require either A or
B to be a set) are shown by
clel2 2975,
clel3 2977, and clel4 2978.
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. For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class 2978. (Contributed by NM, 5-Aug-1993.) |
⊢ (A ∈ B ↔ ∃x(x = A ∧ x ∈ B)) | ||
Theorem | eqriv 2350* | Infer equality of classes from equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
⊢ (x ∈ A ↔ x ∈ B) ⇒ ⊢ A = B | ||
Theorem | eqrdv 2351* | Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.) |
⊢ (φ → (x ∈ A ↔ x ∈ B)) ⇒ ⊢ (φ → A = B) | ||
Theorem | eqrdav 2352* | Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) |
⊢ ((φ ∧ x ∈ A) → x ∈ C) & ⊢ ((φ ∧ x ∈ B) → x ∈ C) & ⊢ ((φ ∧ x ∈ C) → (x ∈ A ↔ x ∈ B)) ⇒ ⊢ (φ → A = B) | ||
Theorem | eqid 2353 |
Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine]
p. 41.
This law is thought to have originated with Aristotle (Metaphysics, Zeta, 17, 1041 a, 10-20: "Therefore, inquiring why a thing is itself, it's inquiring nothing; ... saying that the thing is itself constitutes the sole reasoning and the sole cause, in every case, to the question of why the man is man or the musician musician."). (Thanks to Stefan Allan and Benoît Jubin for this information.) (Contributed by NM, 5-Aug-1993.) (Revised by Benoît Jubin, 14-Oct-2017.) |
⊢ A = A | ||
Theorem | eqidd 2354 | Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.) |
⊢ (φ → A = A) | ||
Theorem | eqcom 2355 | Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 5-Aug-1993.) |
⊢ (A = B ↔ B = A) | ||
Theorem | eqcoms 2356 | Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 5-Aug-1993.) |
⊢ (A = B → φ) ⇒ ⊢ (B = A → φ) | ||
Theorem | eqcomi 2357 | Inference from commutative law for class equality. (Contributed by NM, 5-Aug-1993.) |
⊢ A = B ⇒ ⊢ B = A | ||
Theorem | eqcomd 2358 | Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → B = A) | ||
Theorem | eqeq1 2359 | Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
⊢ (A = B → (A = C ↔ B = C)) | ||
Theorem | eqeq1i 2360 | Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
⊢ A = B ⇒ ⊢ (A = C ↔ B = C) | ||
Theorem | eqeq1d 2361 | Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → (A = C ↔ B = C)) | ||
Theorem | eqeq2 2362 | Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
⊢ (A = B → (C = A ↔ C = B)) | ||
Theorem | eqeq2i 2363 | Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
⊢ A = B ⇒ ⊢ (C = A ↔ C = B) | ||
Theorem | eqeq2d 2364 | Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → (C = A ↔ C = B)) | ||
Theorem | eqeq12 2365 | Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.) |
⊢ ((A = B ∧ C = D) → (A = C ↔ B = D)) | ||
Theorem | eqeq12i 2366 | A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ A = B & ⊢ C = D ⇒ ⊢ (A = C ↔ B = D) | ||
Theorem | eqeq12d 2367 | A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (φ → A = B) & ⊢ (φ → C = D) ⇒ ⊢ (φ → (A = C ↔ B = D)) | ||
Theorem | eqeqan12d 2368 | A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (φ → A = B) & ⊢ (ψ → C = D) ⇒ ⊢ ((φ ∧ ψ) → (A = C ↔ B = D)) | ||
Theorem | eqeqan12rd 2369 | A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) |
⊢ (φ → A = B) & ⊢ (ψ → C = D) ⇒ ⊢ ((ψ ∧ φ) → (A = C ↔ B = D)) | ||
Theorem | eqtr 2370 | Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.) |
⊢ ((A = B ∧ B = C) → A = C) | ||
Theorem | eqtr2 2371 | A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ ((A = B ∧ A = C) → B = C) | ||
Theorem | eqtr3 2372 | A transitive law for class equality. (Contributed by NM, 20-May-2005.) |
⊢ ((A = C ∧ B = C) → A = B) | ||
Theorem | eqtri 2373 | An equality transitivity inference. (Contributed by NM, 5-Aug-1993.) |
⊢ A = B & ⊢ B = C ⇒ ⊢ A = C | ||
Theorem | eqtr2i 2374 | An equality transitivity inference. (Contributed by NM, 21-Feb-1995.) |
⊢ A = B & ⊢ B = C ⇒ ⊢ C = A | ||
Theorem | eqtr3i 2375 | An equality transitivity inference. (Contributed by NM, 6-May-1994.) |
⊢ A = B & ⊢ A = C ⇒ ⊢ B = C | ||
Theorem | eqtr4i 2376 | An equality transitivity inference. (Contributed by NM, 5-Aug-1993.) |
⊢ A = B & ⊢ C = B ⇒ ⊢ A = C | ||
Theorem | 3eqtri 2377 | An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.) |
⊢ A = B & ⊢ B = C & ⊢ C = D ⇒ ⊢ A = D | ||
Theorem | 3eqtrri 2378 | An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ A = B & ⊢ B = C & ⊢ C = D ⇒ ⊢ D = A | ||
Theorem | 3eqtr2i 2379 | An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) |
⊢ A = B & ⊢ C = B & ⊢ C = D ⇒ ⊢ A = D | ||
Theorem | 3eqtr2ri 2380 | An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ A = B & ⊢ C = B & ⊢ C = D ⇒ ⊢ D = A | ||
Theorem | 3eqtr3i 2381 | An inference from three chained equalities. (Contributed by NM, 6-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ A = B & ⊢ A = C & ⊢ B = D ⇒ ⊢ C = D | ||
Theorem | 3eqtr3ri 2382 | An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.) |
⊢ A = B & ⊢ A = C & ⊢ B = D ⇒ ⊢ D = C | ||
Theorem | 3eqtr4i 2383 | An inference from three chained equalities. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ A = B & ⊢ C = A & ⊢ D = B ⇒ ⊢ C = D | ||
Theorem | 3eqtr4ri 2384 | An inference from three chained equalities. (Contributed by NM, 2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ A = B & ⊢ C = A & ⊢ D = B ⇒ ⊢ D = C | ||
Theorem | eqtrd 2385 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → A = B) & ⊢ (φ → B = C) ⇒ ⊢ (φ → A = C) | ||
Theorem | eqtr2d 2386 | An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.) |
⊢ (φ → A = B) & ⊢ (φ → B = C) ⇒ ⊢ (φ → C = A) | ||
Theorem | eqtr3d 2387 | An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.) |
⊢ (φ → A = B) & ⊢ (φ → A = C) ⇒ ⊢ (φ → B = C) | ||
Theorem | eqtr4d 2388 | An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.) |
⊢ (φ → A = B) & ⊢ (φ → C = B) ⇒ ⊢ (φ → A = C) | ||
Theorem | 3eqtrd 2389 | A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.) |
⊢ (φ → A = B) & ⊢ (φ → B = C) & ⊢ (φ → C = D) ⇒ ⊢ (φ → A = D) | ||
Theorem | 3eqtrrd 2390 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (φ → A = B) & ⊢ (φ → B = C) & ⊢ (φ → C = D) ⇒ ⊢ (φ → D = A) | ||
Theorem | 3eqtr2d 2391 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) |
⊢ (φ → A = B) & ⊢ (φ → C = B) & ⊢ (φ → C = D) ⇒ ⊢ (φ → A = D) | ||
Theorem | 3eqtr2rd 2392 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) |
⊢ (φ → A = B) & ⊢ (φ → C = B) & ⊢ (φ → C = D) ⇒ ⊢ (φ → D = A) | ||
Theorem | 3eqtr3d 2393 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (φ → A = B) & ⊢ (φ → A = C) & ⊢ (φ → B = D) ⇒ ⊢ (φ → C = D) | ||
Theorem | 3eqtr3rd 2394 | A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.) |
⊢ (φ → A = B) & ⊢ (φ → A = C) & ⊢ (φ → B = D) ⇒ ⊢ (φ → D = C) | ||
Theorem | 3eqtr4d 2395 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (φ → A = B) & ⊢ (φ → C = A) & ⊢ (φ → D = B) ⇒ ⊢ (φ → C = D) | ||
Theorem | 3eqtr4rd 2396 | A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.) |
⊢ (φ → A = B) & ⊢ (φ → C = A) & ⊢ (φ → D = B) ⇒ ⊢ (φ → D = C) | ||
Theorem | syl5eq 2397 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
⊢ A = B & ⊢ (φ → B = C) ⇒ ⊢ (φ → A = C) | ||
Theorem | syl5req 2398 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ A = B & ⊢ (φ → B = C) ⇒ ⊢ (φ → C = A) | ||
Theorem | syl5eqr 2399 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
⊢ B = A & ⊢ (φ → B = C) ⇒ ⊢ (φ → A = C) | ||
Theorem | syl5reqr 2400 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ B = A & ⊢ (φ → B = C) ⇒ ⊢ (φ → C = A) |
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