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TypeLabelDescription
Statement

Theoremsb8mo 2301 Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremcbveu 2302 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremeu1 2303* An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremmo 2304* Equivalent definitions of "there exists at most one." (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremeuex 2305 Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremeumo0 2306* Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)

Theoremeu2 2307* An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.)

Theoremeu3 2308* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)

Theoremeuor 2309 Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.)

Theoremeuorv 2310* Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremmo2 2311* Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.)

Theoremsbmo 2312* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremmo3 2313* 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.)

Theoremmo4f 2314* "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)

Theoremmo4 2315* "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Theoremmobid 2316 Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)

Theoremmobidv 2317* Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremmobii 2318 Formula-building rule for "at most one" quantifier (inference rule). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)

Theoremcbvmo 2319 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)

Theoremeu5 2320 Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.)

Theoremeu4 2321* Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Theoremeumo 2322 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.)

Theoremeumoi 2323 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)

Theoremexmoeu 2324 Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.)

Theoremexmoeu2 2325 Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)

Theoremmoabs 2326 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)

Theoremexmo 2327 Something exists or at most one exists. (Contributed by NM, 8-Mar-1995.)

Theoremmoim 2328 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)

Theoremmoimi 2329 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)

Theoremmorimv 2330* Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)

Theoremeuimmo 2331 Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)

Theoremeuim 2332 Add existential uniqueness 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.)

Theoremmoan 2333 "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)

Theoremmoani 2334 "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)

Theoremmoor 2335 "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)

Theoremmooran1 2336 "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmooran2 2337 "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoanim 2338 Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.)

Theoremeuan 2339 Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoanimv 2340* Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremmoaneu 2341 Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.)

Theoremmoanmo 2342 Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.)

Theoremeuanv 2343* Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremmopick 2344 "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)

Theoremeupick 2345 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.)

Theoremeupicka 2346 Version of eupick 2345 with closed formulas. (Contributed by NM, 6-Sep-2008.)

Theoremeupickb 2347 Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)

Theoremeupickbi 2348 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremmopick2 2349 "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 1620. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremeuor2 2350 Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoexex 2351 "At most one" double quantification. (Contributed by NM, 3-Dec-2001.)

Theoremmoexexv 2352* "At most one" double quantification. (Contributed by NM, 26-Jan-1997.)

Theorem2moex 2353 Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)

Theorem2euex 2354 Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theorem2eumo 2355 Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)

Theorem2eu2ex 2356 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)

Theorem2moswap 2357 A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.)

Theorem2euswap 2358 A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.)

Theorem2exeu 2359 Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)

Theorem2mo 2360* Two equivalent expressions for double "at most one." (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.)

Theorem2mos 2361* Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.)

Theorem2eu1 2362 Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.)

Theorem2eu2 2363 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)

Theorem2eu3 2364 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)

Theorem2eu4 2365* This theorem provides us with a definition of double existential uniqueness ("exactly one and exactly one "). Naively one might think (incorrectly) that it could be defined by . See 2eu1 2362 for a condition under which the naive definition holds and 2exeu 2359 for a one-way implication. See 2eu5 2366 and 2eu8 2369 for alternate definitions. (Contributed by NM, 3-Dec-2001.)

Theorem2eu5 2366* An alternate definition of double existential uniqueness (see 2eu4 2365). A mistake sometimes made in the literature is to use to mean "exactly one and exactly one ." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining as an additional condition. The correct definition apparently has never been published. ( means "exists at most one.") (Contributed by NM, 26-Oct-2003.)

Theorem2eu6 2367* Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.)

Theorem2eu7 2368 Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)

Theorem2eu8 2369 Two equivalent expressions for double existential uniqueness. Curiously, we can put on either of the internal conjuncts but not both. We can also commute using 2eu7 2368. (Contributed by NM, 20-Feb-2005.)

Theoremeuequ1 2370* Equality has existential uniqueness. Special case of eueq1 3108 proved using only predicate calculus. (Contributed by Stefan Allan, 4-Dec-2008.)

Theoremexists1 2371* Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru 4391. (Contributed by NM, 5-Apr-2004.)

Theoremexists2 2372 A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

1.8  Other axiomatizations related to classical predicate calculus

1.8.1  Predicate calculus with all distinct variables

Axiomax-7d 2373* Distinct variable version of ax-7 1750. (Contributed by Mario Carneiro, 14-Aug-2015.)

Axiomax-8d 2374* Distinct variable version of ax-8 1688. (Contributed by Mario Carneiro, 14-Aug-2015.)

Axiomax-9d1 2375 Distinct variable version of ax-9 1667, equal variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)

Axiomax-9d2 2376* Distinct variable version of ax-9 1667, distinct variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)

Axiomax-10d 2377* Distinct variable version of ax10 2026. (Contributed by Mario Carneiro, 14-Aug-2015.)

Axiomax-11d 2378* Distinct variable version of ax-11 1762. (Contributed by Mario Carneiro, 14-Aug-2015.)

1.8.2  Aristotelian logic: Assertic syllogisms

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 1543) 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/. 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 1589. 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 non-existent entities in his logic. This is a thoughtful choice, not an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is "a phrase signifying a thing's essence." (Topics, I.5.102a37, Pickard-Cambridge.)... Because non-existent entities cannot be anything, they do not, in Aristotle's mind, possess an essence... This is why he leaves no place for fictional entities like goat-stags (or unicorns)." Source: Louis F. Groarke, "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy (A Peer-Reviewed Academic Resource), http://www.iep.utm.edu/aris-log/. Thus, some syllogisms have "extra" existence hypotheses that do not directly appear in Aristotle's original materials (since they were always assumed); they are added where they are needed. This affects barbari 2383, celaront 2384, cesaro 2389, camestros 2390, felapton 2395, darapti 2396, calemos 2400, fesapo 2401, and bamalip 2402.

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.

Theorembarbara 2379 "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 16. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1626. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm, http://plato.stanford.edu/entries/aristotle-logic/, and https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)

Theoremcelarent 2380 "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.)

Theoremdarii 2381 "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.)

Theoremferio 2382 "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.)

Theorembarbari 2383 "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.)

Theoremcelaront 2384 "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.)

Theoremcesare 2385 "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 2380. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.)

Theoremcamestres 2386 "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.)

Theoremfestino 2387 "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.)

Theorembaroco 2388 "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.)

Theoremcesaro 2389 "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.)

Theoremcamestros 2390 "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.)

Theoremdatisi 2391 "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.)

Theoremdisamis 2392 "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.)

Theoremferison 2393 "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.)

Theorembocardo 2394 "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 2392; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.)

Theoremfelapton 2395 "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.)

Theoremdarapti 2396 "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.)

Theoremcalemes 2397 "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.)

Theoremdimatis 2398 "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 2381 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)

Theoremfresison 2399 "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.)

Theoremcalemos 2400 "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.)

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