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Theorem List for Intuitionistic Logic Explorer - 1901-2000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxwmo 1901 Extend wff definition to include uniqueness ("there exists at most one such that ").

Theoremeujust 1902* A soundness justification theorem for df-eu 1903, showing that the definition is equivalent to itself with its dummy variable renamed. Note that and needn't be distinct variables. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Definitiondf-eu 1903* 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 1925, eu2 1944, eu3 1946, and eu5 1947 (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 1993). (Contributed by NM, 5-Aug-1993.)

Definitiondf-mo 1904 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 1954. For another possible definition see mo4 1961. (Contributed by NM, 5-Aug-1993.)

Theoremeuf 1905* A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)

Theoremeubidh 1906 Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Theoremeubid 1907 Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Theoremeubidv 1908* Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Theoremeubii 1909 Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremhbeu1 1910 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.)

Theoremnfeu1 1911 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmo1 1912 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremsb8eu 1913 Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremsb8mo 1914 Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremnfeudv 1915* Deduction version of nfeu 1919. Similar to nfeud 1916 but has the additional constraint that and must be distinct. (Contributed by Jim Kingdon, 25-May-2018.)

Theoremnfeud 1916 Deduction version of nfeu 1919. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 25-May-2018.)

Theoremnfmod 1917 Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfeuv 1918* Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 1919 but has the additional constraint that and must be distinct. (Contributed by Jim Kingdon, 23-May-2018.)

Theoremnfeu 1919 Bound-variable hypothesis builder for existential uniqueness. Note that and needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 23-May-2018.)

Theoremnfmo 1920 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)

Theoremhbeu 1921 Bound-variable hypothesis builder for uniqueness. Note that and needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Proof rewritten by Jim Kingdon, 24-May-2018.)

Theoremhbeud 1922 Deduction version of hbeu 1921. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 25-May-2018.)

Theoremsb8euh 1923 Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.)

Theoremcbveu 1924 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremeu1 1925* An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.)

Theoremeuor 1926 Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.)

Theoremeuorv 1927* Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremmo2n 1928* There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.)

Theoremmon 1929 There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.)

Theoremeuex 1930 Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremeumo0 1931* Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)

Theoremeumo 1932 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)

Theoremeumoi 1933 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)

Theoremmobidh 1934 Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)

Theoremmobid 1935 Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)

Theoremmobidv 1936* Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremmobii 1937 Formula-building rule for "at most one" quantifier (inference rule). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)

Theoremhbmo1 1938 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.)

Theoremhbmo 1939 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)

Theoremcbvmo 1940 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)

Theoremmo23 1941* An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.)

Theoremmor 1942* Converse of mo23 1941 with an additional condition. (Contributed by Jim Kingdon, 25-Jun-2018.)

Theoremmodc 1943* Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.)
DECID

Theoremeu2 1944* An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.)

Theoremeu3h 1945* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.)

Theoremeu3 1946* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)

Theoremeu5 1947 Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)

Theoremexmoeu2 1948 Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)

Theoremmoabs 1949 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)

Theoremexmodc 1950 If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)
DECID

Theoremexmonim 1951 There is at most one of something which does not exist. Unlike exmodc 1950 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.)

Theoremmo2r 1952* A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.)

Theoremmo3h 1953* 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.) (New usage is discouraged.)

Theoremmo3 1954* 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.)

Theoremmo2dc 1955* Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.)
DECID

Theoremeuan 1956 Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremeuanv 1957* Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremeuor2 1958 Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremsbmo 1959* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremmo4f 1960* "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)

Theoremmo4 1961* "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Theoremeu4 1962* Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Theoremexmoeudc 1963 Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.)
DECID

Theoremmoim 1964 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)

Theoremmoimi 1965 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)

Theoremmoimv 1966* Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)

Theoremeuimmo 1967 Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)

Theoremeuim 1968 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 1969 "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)

Theoremmoani 1970 "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)

Theoremmoor 1971 "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)

Theoremmooran1 1972 "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmooran2 1973 "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoanim 1974 Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.)

Theoremmoanimv 1975* Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremmoaneu 1976 Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.)

Theoremmoanmo 1977 Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.)

Theoremmopick 1978 "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)

Theoremeupick 1979 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 1980 Version of eupick 1979 with closed formulas. (Contributed by NM, 6-Sep-2008.)

Theoremeupickb 1981 Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)

Theoremeupickbi 1982 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremmopick2 1983 "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 1522. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoexexdc 1984 "At most one" double quantification. (Contributed by Jim Kingdon, 5-Jul-2018.)
DECID

Theoremeuexex 1985 Existential uniqueness and "at most one" double quantification. (Contributed by Jim Kingdon, 28-Dec-2018.)

Theorem2moex 1986 Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)

Theorem2euex 1987 Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theorem2eumo 1988 Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)

Theorem2eu2ex 1989 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)

Theorem2moswapdc 1990 A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.)
DECID

Theorem2euswapdc 1991 A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.)
DECID

Theorem2exeu 1992 Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.)

Theorem2eu4 1993* 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 2exeu 1992 for a one-way implication. (Contributed by NM, 3-Dec-2001.)

Theorem2eu7 1994 Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)

Theoremeuequ1 1995* Equality has existential uniqueness. (Contributed by Stefan Allan, 4-Dec-2008.)

Theoremexists1 1996* 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. (Contributed by NM, 5-Apr-2004.)

Theoremexists2 1997 A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

1.4.7  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 mptnan 1314) 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 1494. 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 2002, celaront 2003, cesaro 2008, camestros 2009, felapton 2014, darapti 2015, calemos 2019, fesapo 2020, and bamalip 2021.

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 1998 "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 14. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1526. 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 1999 "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 2000 "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.)

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