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Type | Label | Description |
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Statement | ||
Theorem | nfeu 2301 | 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 2302 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.) |
Theorem | eubid 2303 | Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.) |
Theorem | mobid 2304 | Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.) |
Theorem | eubidv 2305* | Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.) |
Theorem | mobidv 2306* | Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.) |
Theorem | eubii 2307 | Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | mobii 2308 | 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 2309 | Existential uniqueness implies existence. For a shorter proof using more axioms, see euexALT 2329. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2018.) |
Theorem | exmo 2310 | Something exists or at most one exists. (Contributed by NM, 8-Mar-1995.) |
Theorem | eu5 2311 | 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 2312 | Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.) |
Theorem | eu3v 2313* | 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 2314 | Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) |
Theorem | eumoi 2315 | "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
Theorem | moabs 2316 | Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.) |
Theorem | exmoeu 2317 | Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.) |
Theorem | eumo0OLD 2318* | Obsolete as of 6-Jun-2018. Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | sb8eu 2319 | 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 | sb8euOLD 2320 | Obsolete proof of sb8eu 2319 as of 24-Aug-2019. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 8-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | sb8mo 2321 | Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | cbveu 2322 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Theorem | cbvmo 2323 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) |
Theorem | mo3 2324* | 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 | mo3OLD 2325* | Obsolete proof of mo3 2324 as of 20-Jul-2019. (Contributed by NM, 8-Mar-1995.) (Revised by Wolf Lammen, 3-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mo 2326* | 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 2327* | 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 2328* | 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 2329 | Alternate proof of euex 2309. 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 | eu3OLD 2330* | Obsolete theorem as of 29-May-2018. Superseded by eu3v 2313 that better fits common usage pattern. (Contributed by NM, 8-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | eu5OLD 2331 | Obsolete proof of eu5 2311 as of 25-May-2019 (Contributed by NM, 23-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | euor 2332 | Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) |
Theorem | euorv 2333* | Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.) |
Theorem | euor2 2334 | 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 | mo2OLD 2335* | Obsolete proof of mo2 2294 as of 28-May-2019. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | sbmo 2336* | Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | mo4f 2337* | "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.) |
Theorem | mo4 2338* | "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
Theorem | eu4 2339* | Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
Theorem | moim 2340 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.) |
Theorem | moimi 2341 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.) |
Theorem | morimOLD 2342 | TODO-NM: AV and BJ propose to keep this theorem as morim . Obsolete theorem as of 22-Dec-2018. Use moimi 2341 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 2343 | Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.) |
Theorem | euim 2344 | 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 2345 | "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.) |
Theorem | moani 2346 | "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.) |
Theorem | moor 2347 | "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.) |
Theorem | mooran1 2348 | "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | mooran2 2349 | "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | moanim 2350 | 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 2351 | 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 2352* | Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.) |
Theorem | moanmo 2353 | Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.) |
Theorem | moaneu 2354 | Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) |
Theorem | euanv 2355* | Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.) |
Theorem | mopick 2356 | "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 | mopickOLD 2357 | Obsolete proof of mopick 2356 as of 15-Sep-2019. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 29-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | eupick 2358 | 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 2359 | Version of eupick 2358 with closed formulas. (Contributed by NM, 6-Sep-2008.) |
Theorem | eupickb 2360 | Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) |
Theorem | eupickbi 2361 | Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) |
Theorem | mopick2 2362 | "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 1680. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | moexex 2363 | "At most one" double quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.) |
Theorem | moexexv 2364* | "At most one" double quantification. (Contributed by NM, 26-Jan-1997.) |
Theorem | 2moex 2365 | Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.) |
Theorem | 2euex 2366 | Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | 2eumo 2367 | Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.) |
Theorem | 2eu2ex 2368 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |
Theorem | 2moswap 2369 | A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.) |
Theorem | 2euswap 2370 | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.) |
Theorem | 2exeu 2371 | Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) |
Theorem | 2mo2 2372* | 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 2380 relates this extension to double existential uniqueness, if at least one pair exists. (Contributed by Wolf Lammen, 26-Oct-2019.) |
Theorem | 2mo 2373* | 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 | 2moOLD 2374* | Obsolete proof of 2mo 2373 as of 2-Nov-2019. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 25-Oct-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | 2mos 2375* | Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.) |
Theorem | 2eu1 2376 | 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 | 2eu1OLD 2377 | Obsolete proof of 2eu1 2376 as of 11-Nov-2019. (Contributed by NM, 3-Dec-2001.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | 2eu2 2378 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |
Theorem | 2eu3 2379 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |
Theorem | 2eu4 2380* | 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 2376 for a condition under which the naive definition holds and 2exeu 2371 for a one-way implication. See 2eu5 2382 and 2eu8 2386 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.) |
Theorem | 2eu4OLD 2381* | Obsolete proof of 2eu4 2380 as of 14-Sep-2019. (Contributed by NM, 3-Dec-2001.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | 2eu5 2382* | An alternate definition of double existential uniqueness (see 2eu4 2380). 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 2383* | 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 | 2eu6OLD 2384* | Obsolete proof of 2eu6 2383 as of 21-Sep-2019. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | 2eu7 2385 | Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.) |
Theorem | 2eu8 2386 | 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 2385. (Contributed by NM, 20-Feb-2005.) |
Theorem | euequ1OLD 2387* | Obsolete proof of euequ1 2289 as of 8-Sep-2019. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | exists1 2388* | 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 4647. (Contributed by NM, 5-Apr-2004.) |
Theorem | exists2 2389 | A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Axiom | ax-7d 2390* | Distinct variable version of ax-11 1843. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Axiom | ax-8d 2391* | Distinct variable version of ax-7 1791. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Axiom | ax-9d1 2392 | Distinct variable version of ax-6 1748, equal variables case. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Axiom | ax-9d2 2393* | Distinct variable version of ax-6 1748, distinct variables case. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Axiom | ax-10d 2394* | Distinct variable version of axc11n 2050. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Axiom | ax-11d 2395* | Distinct variable version of ax-12 1855. (Contributed by Mario Carneiro, 14-Aug-2015.) |
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 1601) 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 1664. 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 2400, celaront 2401, cesaro 2406, camestros 2407, felapton 2412, darapti 2413, calemos 2417, fesapo 2418, and bamalip 2419. 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 2396 | "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 1704. 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 2397 | "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 2398 | "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 2399 | "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 2400 | "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.) |
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