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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | weu 2301 | Extend wff definition to include existential uniqueness ("there exists a unique such that "). |
Syntax | wmo 2302 | Extend wff definition to include uniqueness ("there exists at most one such that "). |
Theorem | eujust 2303* | A soundness justification theorem for df-eu 2305, showing that the definition is equivalent to itself with its dummy variable renamed. Note that and needn't be distinct variables. See eujustALT 2304 for a proof that provides an example of how it can be achieved through the use of dvelim 2173. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | eujustALT 2304* | Alternate proof of eujust 2303 illustrating the use of dvelim 2173. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.) |
Definition | df-eu 2305* | Define existential uniqueness, i.e. "there exists exactly one such that ." Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2341, eu2 2340, eu3v 2329, and eu5 2327 (which in some cases we show with a hypothesis in place of a distinct variable condition on and ). Double uniqueness is tricky: does not mean "exactly one and one " (see 2eu4 2387). (Contributed by NM, 12-Aug-1993.) |
Definition | df-mo 2306 | Define "there exists at most one such that ." Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 2338. For other possible definitions see mo2 2310 and mo4 2348. (Contributed by NM, 8-Mar-1995.) |
Theorem | euequ1 2307* | Equality has existential uniqueness. Special case of eueq1 3213 proved using only predicate calculus. The proof needs be free of . This is ensured by having and be distinct. Alternatively, a distinctor could have been used instead. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof shortened by Wolf Lammen, 8-Sep-2019.) |
Theorem | mo2v 2308* | Alternate definition of "at most one." Unlike mo2 2310, which is slightly more general, it does not depend on ax-11 1922 and ax-13 2093, whence it is preferable within predicate logic. Elsewhere, most theorems depend on these axioms anyway, so this advantage is no longer important. (Contributed by Wolf Lammen, 27-May-2019.) (Proof shortened by Wolf Lammen, 10-Nov-2019.) |
Theorem | euf 2309* | A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2018.) |
Theorem | mo2 2310* | Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.) Restrict dummy variable z. (Revised by Wolf Lammen, 28-May-2019.) |
Theorem | nfeu1 2311 | Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Theorem | nfmo1 2312 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Theorem | nfeud2 2313 | Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.) |
Theorem | nfmod2 2314 | Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.) |
Theorem | nfeud 2315 | Deduction version of nfeu 2317. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Theorem | nfmod 2316 | Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.) |
Theorem | nfeu 2317 | Bound-variable hypothesis builder for uniqueness. Note that and needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Theorem | nfmo 2318 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.) |
Theorem | eubid 2319 | Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.) |
Theorem | mobid 2320 | Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.) |
Theorem | eubidv 2321* | Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.) |
Theorem | mobidv 2322* | Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.) |
Theorem | eubii 2323 | Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | mobii 2324 | Formula-building rule for "at most one" quantifier (inference rule). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.) |
Theorem | euex 2325 | Existential uniqueness implies existence. For a shorter proof using more axioms, see euexALT 2342. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2018.) |
Theorem | exmo 2326 | Something exists or at most one exists. (Contributed by NM, 8-Mar-1995.) |
Theorem | eu5 2327 | Uniqueness in terms of "at most one." Revised to reduce dependencies on axioms. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 25-May-2019.) |
Theorem | exmoeu2 2328 | Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.) |
Theorem | eu3v 2329* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) Add a distinct variable condition on . (Revised by Wolf Lammen, 29-May-2019.) |
Theorem | eumo 2330 | Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) |
Theorem | eumoi 2331 | "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
Theorem | moabs 2332 | Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.) |
Theorem | exmoeu 2333 | Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.) |
Theorem | sb8eu 2334 | Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.) |
Theorem | sb8mo 2335 | Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | cbveu 2336 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Theorem | cbvmo 2337 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) |
Theorem | mo3 2338* | 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.) (Proof shortened by Wolf Lammen, 18-Aug-2019.) |
Theorem | mo 2339* | Equivalent definitions of "there exists at most one." (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Dec-2018.) |
Theorem | eu2 2340* | An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) (Proof shortened by Wolf Lammen, 2-Dec-2018.) |
Theorem | eu1 2341* | 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.) (Proof shortened by Wolf Lammen, 29-Oct-2018.) |
Theorem | euexALT 2342 | Alternate proof of euex 2325. Shorter but uses more axioms. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | euor 2343 | Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) |
Theorem | euorv 2344* | Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.) |
Theorem | euor2 2345 | Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) |
Theorem | sbmo 2346* | Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | mo4f 2347* | "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.) |
Theorem | mo4 2348* | "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
Theorem | eu4 2349* | Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
Theorem | moim 2350 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.) |
Theorem | moimi 2351 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.) |
Theorem | morimOLD 2352 | TODO-NM: AV and BJ propose to keep this theorem as morim . Obsolete theorem as of 22-Dec-2018. Use moimi 2351 applied to ax-1 6, as demonstrated in the proof. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Wolf Lammen, 22-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | euimmo 2353 | Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.) |
Theorem | euim 2354 | 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 2355 | "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.) |
Theorem | moani 2356 | "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.) |
Theorem | moor 2357 | "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.) |
Theorem | mooran1 2358 | "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | mooran2 2359 | "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | moanim 2360 | Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.) |
Theorem | euan 2361 | Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 24-Dec-2018.) |
Theorem | moanimv 2362* | Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.) |
Theorem | moanmo 2363 | Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.) |
Theorem | moaneu 2364 | Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) |
Theorem | euanv 2365* | Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.) |
Theorem | mopick 2366 | "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 17-Sep-2019.) |
Theorem | eupick 2367 | 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 2368 | Version of eupick 2367 with closed formulas. (Contributed by NM, 6-Sep-2008.) |
Theorem | eupickb 2369 | Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) |
Theorem | eupickbi 2370 | Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) |
Theorem | mopick2 2371 | "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 1733. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | moexex 2372 | "At most one" double quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.) |
Theorem | moexexv 2373* | "At most one" double quantification. (Contributed by NM, 26-Jan-1997.) |
Theorem | 2moex 2374 | Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.) |
Theorem | 2euex 2375 | Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | 2eumo 2376 | Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.) |
Theorem | 2eu2ex 2377 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |
Theorem | 2moswap 2378 | A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.) |
Theorem | 2euswap 2379 | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.) |
Theorem | 2exeu 2380 | Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) |
Theorem | 2mo2 2381* | This theorem extends the idea of "at most one" to expressions in two set variables ("at most one pair and ". Note: this is not expressed by ). 2eu4 2387 relates this extension to double existential uniqueness, if at least one pair exists. (Contributed by Wolf Lammen, 26-Oct-2019.) |
Theorem | 2mo 2382* | Two equivalent expressions for double "at most one." (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Nov-2019.) |
Theorem | 2mos 2383* | Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.) |
Theorem | 2eu1 2384 | Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 11-Nov-2019.) |
Theorem | 2eu2 2385 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |
Theorem | 2eu3 2386 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |
Theorem | 2eu4 2387* | 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 2384 for a condition under which the naive definition holds and 2exeu 2380 for a one-way implication. See 2eu5 2388 and 2eu8 2391 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.) |
Theorem | 2eu5 2388* | An alternate definition of double existential uniqueness (see 2eu4 2387). 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 2389* | Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Oct-2019.) |
Theorem | 2eu7 2390 | Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.) |
Theorem | 2eu8 2391 | 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 2390. (Contributed by NM, 20-Feb-2005.) |
Theorem | exists1 2392* | 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 4597. (Contributed by NM, 5-Apr-2004.) |
Theorem | exists2 2393 | A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Model the Aristotelian assertic syllogisms using modern notation. This section shows that the Aristotelian assertic syllogisms can be proven with our axioms of logic, and also provides generally useful theorems. In antiquity Aristotelian logic and Stoic logic (see mptnan 1653) 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 approach 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 1715. 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 more specific 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 Aristotelian 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. Using such specific constructs would also be anti-historical; Aristotle and others who directly followed his work focused on relating wholes to their parts, an approach now called part-whole theory. The work of Cantor and Peano (over 2,000 years later) led to a sharper distinction between inclusion () and membership (); this distinction was not directly made in Aristotle's work. 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 2398, celaront 2399, cesaro 2404, camestros 2405, felapton 2410, darapti 2411, calemos 2415, fesapo 2416, and bamalip 2417. 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. Aristotelian 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 2394 | "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 17. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1759. 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 2395 | "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 2396 | "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 2397 | "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 2398 | "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 2399 | "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 2400 | "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 2395. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.) |
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