Home | Metamath
Proof ExplorerTheorem List
(p. 24 of 328)
| < Previous Next > |

Browser slow? Try the
Unicode version. |

Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs

Color key: | Metamath Proof Explorer
(1-22422) |
Hilbert Space Explorer
(22423-23945) |
Users' Mathboxes
(23946-32763) |

Type | Label | Description |
---|---|---|

Statement | ||

Theorem | sb8mo 2301 | Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

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

Theorem | eu1 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.) |

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

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

Theorem | eumo0 2306* | Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.) |

Theorem | eu2 2307* | An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) |

Theorem | eu3 2308* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) |

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

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

Theorem | mo2 2311* | Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.) |

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

Theorem | mo3 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.) |

Theorem | mo4f 2314* | "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.) |

Theorem | mo4 2315* | "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.) |

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

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

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

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

Theorem | eu5 2320 | Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.) |

Theorem | eu4 2321* | Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.) |

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

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

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

Theorem | exmoeu2 2325 | Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.) |

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

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

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

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

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

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

Theorem | euim 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.) |

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

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

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

Theorem | mooran1 2336 | "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |

Theorem | mooran2 2337 | "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |

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

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

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

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

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

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

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

Theorem | eupick 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.) |

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

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

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

Theorem | mopick2 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.) |

Theorem | euor2 2350 | Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |

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

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

Theorem | 2moex 2353 | Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.) |

Theorem | 2euex 2354 | Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |

Theorem | 2eumo 2355 | Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.) |

Theorem | 2eu2ex 2356 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |

Theorem | 2moswap 2357 | A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.) |

Theorem | 2euswap 2358 | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.) |

Theorem | 2exeu 2359 | Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) |

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

Theorem | 2mos 2361* | Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.) |

Theorem | 2eu1 2362 | Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.) |

Theorem | 2eu2 2363 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |

Theorem | 2eu3 2364 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |

Theorem | 2eu4 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.) |

Theorem | 2eu5 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.) |

Theorem | 2eu6 2367* | Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) |

Theorem | 2eu7 2368 | Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.) |

Theorem | 2eu8 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.) |

Theorem | euequ1 2370* | Equality has existential uniqueness. Special case of eueq1 3108 proved using only predicate calculus. (Contributed by Stefan Allan, 4-Dec-2008.) |

Theorem | exists1 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.) |

Theorem | exists2 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 | ||

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

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

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

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

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

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

1.8.2 Aristotelian logic: Assertic
syllogismsModel 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,
"There is a surprising amount of scholarly debate
about how best to formalize Aristotle's syllogisms..." according to
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, 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),
These are only the 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 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.) |

Theorem | celarent 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.) |

Theorem | darii 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.) |

Theorem | ferio 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.) |

Theorem | barbari 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.) |

Theorem | celaront 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.) |

Theorem | cesare 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.) |

Theorem | camestres 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.) |

Theorem | festino 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.) |

Theorem | baroco 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.) |

Theorem | cesaro 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.) |

Theorem | camestros 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.) |

Theorem | datisi 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.) |

Theorem | disamis 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.) |

Theorem | ferison 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.) |

Theorem | bocardo 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.) |

Theorem | felapton 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.) |

Theorem | darapti 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.) |

Theorem | calemes 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.) |

Theorem | dimatis 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.) |

Theorem | fresison 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.) |

Theorem | calemos 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.) |

< Previous Next > |

Copyright terms: Public domain | < Previous Next > |