HomeHome Intuitionistic Logic Explorer
Theorem List (p. 21 of 132)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 2001-2100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcbveu 2001 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E! x ph  <->  E! y ps )
 
Theoremeu1 2002* An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( E! x ph  <->  E. x ( ph  /\  A. y ( [ y  /  x ] ph  ->  x  =  y ) ) )
 
Theoremeuor 2003 Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 21-Oct-2005.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( ( -.  ph  /\ 
 E! x ps )  ->  E! x ( ph  \/  ps ) )
 
Theoremeuorv 2004* Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( ( -.  ph  /\ 
 E! x ps )  ->  E! x ( ph  \/  ps ) )
 
Theoremmo2n 2005* There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.)
 |- 
 F/ y ph   =>    |-  ( -.  E. x ph 
 ->  E. y A. x ( ph  ->  x  =  y ) )
 
Theoremmon 2006 There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.)
 |-  ( -.  E. x ph 
 ->  E* x ph )
 
Theoremeuex 2007 Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( E! x ph  ->  E. x ph )
 
Theoremeumo0 2008* Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( E! x ph  ->  E. y A. x ( ph  ->  x  =  y ) )
 
Theoremeumo 2009 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)
 |-  ( E! x ph  ->  E* x ph )
 
Theoremeumoi 2010 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
 |- 
 E! x ph   =>    |- 
 E* x ph
 
Theoremmobidh 2011 Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E* x ps  <->  E* x ch )
 )
 
Theoremmobid 2012 Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E* x ps  <->  E* x ch )
 )
 
Theoremmobidv 2013* Formula-building rule for "at most one" quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E* x ps  <->  E* x ch )
 )
 
Theoremmobii 2014 Formula-building rule for "at most one" quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)
 |-  ( ps  <->  ch )   =>    |-  ( E* x ps  <->  E* x ch )
 
Theoremhbmo1 2015 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.)
 |-  ( E* x ph  ->  A. x E* x ph )
 
Theoremhbmo 2016 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( E* y ph  ->  A. x E* y ph )
 
Theoremcbvmo 2017 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E* x ph  <->  E* y ps )
 
Theoremmo23 2018* An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.)
 |- 
 F/ y ph   =>    |-  ( E. y A. x ( ph  ->  x  =  y )  ->  A. x A. y ( ( ph  /\  [
 y  /  x ] ph )  ->  x  =  y ) )
 
Theoremmor 2019* Converse of mo23 2018 with an additional  E. x ph condition. (Contributed by Jim Kingdon, 25-Jun-2018.)
 |- 
 F/ y ph   =>    |-  ( E. x ph  ->  ( A. x A. y ( ( ph  /\ 
 [ y  /  x ] ph )  ->  x  =  y )  ->  E. y A. x ( ph  ->  x  =  y ) ) )
 
Theoremmodc 2020* Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.)
 |- 
 F/ y ph   =>    |-  (DECID 
 E. x ph  ->  ( E. y A. x ( ph  ->  x  =  y )  <->  A. x A. y
 ( ( ph  /\  [
 y  /  x ] ph )  ->  x  =  y ) ) )
 
Theoremeu2 2021* An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( (
 ph  /\  [ y  /  x ] ph )  ->  x  =  y ) ) )
 
Theoremeu3h 2022* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
 
Theoremeu3 2023* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
 
Theoremeu5 2024 Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)
 |-  ( E! x ph  <->  ( E. x ph  /\  E* x ph ) )
 
Theoremexmoeu2 2025 Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)
 |-  ( E. x ph  ->  ( E* x ph  <->  E! x ph ) )
 
Theoremmoabs 2026 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)
 |-  ( E* x ph  <->  ( E. x ph  ->  E* x ph ) )
 
Theoremexmodc 2027 If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)
 |-  (DECID 
 E. x ph  ->  ( E. x ph  \/  E* x ph ) )
 
Theoremexmonim 2028 There is at most one of something which does not exist. Unlike exmodc 2027 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.)
 |-  ( -.  E. x ph 
 ->  E* x ph )
 
Theoremmo2r 2029* A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.)
 |- 
 F/ y ph   =>    |-  ( E. y A. x ( ph  ->  x  =  y )  ->  E* x ph )
 
Theoremmo3h 2030* Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that  y not occur in  ph in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (New usage is discouraged.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( E* x ph  <->  A. x A. y ( (
 ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
 
Theoremmo3 2031* Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that  y not occur in  ph in place of our hypothesis. (Contributed by NM, 8-Mar-1995.)
 |- 
 F/ y ph   =>    |-  ( E* x ph  <->  A. x A. y ( (
 ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
 
Theoremmo2dc 2032* Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.)
 |- 
 F/ y ph   =>    |-  (DECID 
 E. x ph  ->  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) ) )
 
Theoremeuan 2033 Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( E! x (
 ph  /\  ps )  <->  (
 ph  /\  E! x ps ) )
 
Theoremeuanv 2034* Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( E! x (
 ph  /\  ps )  <->  (
 ph  /\  E! x ps ) )
 
Theoremeuor2 2035 Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( -.  E. x ph 
 ->  ( E! x (
 ph  \/  ps )  <->  E! x ps ) )
 
Theoremsbmo 2036* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( [ y  /  x ] E* z ph  <->  E* z [ y  /  x ] ph )
 
Theoremmo4f 2037* "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E* x ph  <->  A. x A. y ( ( ph  /\  ps )  ->  x  =  y ) )
 
Theoremmo4 2038* "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E* x ph  <->  A. x A. y ( ( ph  /\  ps )  ->  x  =  y ) )
 
Theoremeu4 2039* Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( (
 ph  /\  ps )  ->  x  =  y ) ) )
 
Theoremexmoeudc 2040 Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.)
 |-  (DECID 
 E. x ph  ->  ( E. x ph  <->  ( E* x ph 
 ->  E! x ph )
 ) )
 
Theoremmoim 2041 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E* x ps  ->  E* x ph ) )
 
Theoremmoimi 2042 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)
 |-  ( ph  ->  ps )   =>    |-  ( E* x ps  ->  E* x ph )
 
Theoremmoimv 2043* Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)
 |-  ( E* x (
 ph  ->  ps )  ->  ( ph  ->  E* x ps )
 )
 
Theoremeuimmo 2044 Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E! x ps  ->  E* x ph ) )
 
Theoremeuim 2045 Add existential unique existential 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.)
 |-  ( ( E. x ph 
 /\  A. x ( ph  ->  ps ) )  ->  ( E! x ps  ->  E! x ph ) )
 
Theoremmoan 2046 "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)
 |-  ( E* x ph  ->  E* x ( ps 
 /\  ph ) )
 
Theoremmoani 2047 "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)
 |- 
 E* x ph   =>    |- 
 E* x ( ps 
 /\  ph )
 
Theoremmoor 2048 "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)
 |-  ( E* x (
 ph  \/  ps )  ->  E* x ph )
 
Theoremmooran1 2049 "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( ( E* x ph 
 \/  E* x ps )  ->  E* x ( ph  /\ 
 ps ) )
 
Theoremmooran2 2050 "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( E* x (
 ph  \/  ps )  ->  ( E* x ph  /\ 
 E* x ps )
 )
 
Theoremmoanim 2051 Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 3-Dec-2001.)
 |- 
 F/ x ph   =>    |-  ( E* x (
 ph  /\  ps )  <->  (
 ph  ->  E* x ps )
 )
 
Theoremmoanimv 2052* Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( E* x (
 ph  /\  ps )  <->  (
 ph  ->  E* x ps )
 )
 
Theoremmoaneu 2053 Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.)
 |- 
 E* x ( ph  /\ 
 E! x ph )
 
Theoremmoanmo 2054 Nested at-most-one quantifiers. (Contributed by NM, 25-Jan-2006.)
 |- 
 E* x ( ph  /\ 
 E* x ph )
 
Theoremmopick 2055 "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)
 |-  ( ( E* x ph 
 /\  E. x ( ph  /\ 
 ps ) )  ->  ( ph  ->  ps )
 )
 
Theoremeupick 2056 Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing  x such that 
ph is true, and there is also an  x (actually the same one) such that  ph and  ps are both true, then  ph implies  ps regardless of  x. 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.)
 |-  ( ( E! x ph 
 /\  E. x ( ph  /\ 
 ps ) )  ->  ( ph  ->  ps )
 )
 
Theoremeupicka 2057 Version of eupick 2056 with closed formulas. (Contributed by NM, 6-Sep-2008.)
 |-  ( ( E! x ph 
 /\  E. x ( ph  /\ 
 ps ) )  ->  A. x ( ph  ->  ps ) )
 
Theoremeupickb 2058 Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)
 |-  ( ( E! x ph 
 /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph  <->  ps ) )
 
Theoremeupickbi 2059 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( E. x (
 ph  /\  ps )  <->  A. x ( ph  ->  ps ) ) )
 
Theoremmopick2 2060 "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 1595. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( ( E* x ph 
 /\  E. x ( ph  /\ 
 ps )  /\  E. x ( ph  /\  ch ) )  ->  E. x ( ph  /\  ps  /\  ch ) )
 
Theoremmoexexdc 2061 "At most one" double quantification. (Contributed by Jim Kingdon, 5-Jul-2018.)
 |- 
 F/ y ph   =>    |-  (DECID 
 E. x ph  ->  ( ( E* x ph  /\ 
 A. x E* y ps )  ->  E* y E. x ( ph  /\  ps ) ) )
 
Theoremeuexex 2062 Existential uniqueness and "at most one" double quantification. (Contributed by Jim Kingdon, 28-Dec-2018.)
 |- 
 F/ y ph   =>    |-  ( ( E! x ph 
 /\  A. x E* y ps )  ->  E* y E. x ( ph  /\  ps ) )
 
Theorem2moex 2063 Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)
 |-  ( E* x E. y ph  ->  A. y E* x ph )
 
Theorem2euex 2064 Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( E! x E. y ph  ->  E. y E! x ph )
 
Theorem2eumo 2065 Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)
 |-  ( E! x E* y ph  ->  E* x E! y ph )
 
Theorem2eu2ex 2066 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
 |-  ( E! x E! y ph  ->  E. x E. y ph )
 
Theorem2moswapdc 2067 A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.)
 |-  (DECID 
 E. x E. y ph  ->  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ph ) ) )
 
Theorem2euswapdc 2068 A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.)
 |-  (DECID 
 E. x E. y ph  ->  ( A. x E* y ph  ->  ( E! x E. y ph  ->  E! y E. x ph ) ) )
 
Theorem2exeu 2069 Double existential uniqueness implies double unique existential quantification. (Contributed by NM, 3-Dec-2001.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E! x E! y ph )
 
Theorem2eu4 2070* This theorem provides us with a definition of double existential uniqueness ("exactly one 
x and exactly one  y"). Naively one might think (incorrectly) that it could be defined by  E! x E! y ph. See 2exeu 2069 for a one-way implication. (Contributed by NM, 3-Dec-2001.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y
 ( ph  ->  ( x  =  z  /\  y  =  w ) ) ) )
 
Theorem2eu7 2071 Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  E! x E! y ( E. x ph  /\  E. y ph ) )
 
Theoremeuequ1 2072* Equality has existential uniqueness. (Contributed by Stefan Allan, 4-Dec-2008.)
 |- 
 E! x  x  =  y
 
Theoremexists1 2073* 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.)
 |-  ( E! x  x  =  x  <->  A. x  x  =  y )
 
Theoremexists2 2074 A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( ( E. x ph 
 /\  E. x  -.  ph )  ->  -.  E! x  x  =  x )
 
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 1386) 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  x. 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  ph,  ps, and  ch 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/ 1386. See Principia Mathematica page 22 and *10 for more information (especially *10.3 and *10.26).

Expressions of the form "no  ph is  ps " are consistently translated as  A. x (
ph  ->  -.  ps ). These can also be expressed as  -.  E. x
( ph  /\  ps ), per alinexa 1567. We translate "all  ph is  ps " to  A. x (
ph  ->  ps ), "some  ph is  ps " to  E. x
( ph  /\  ps ), and "some  ph is not  ps " to  E. x
( ph  /\  -.  ps ). 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  x  =  A,  x  e.  A, or  x  C_  A. 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  x  e.  A 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 nonexistent 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 nonexistent 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/ 1567. 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 2079, celaront 2080, cesaro 2085, camestros 2086, felapton 2091, darapti 2092, calemos 2096, fesapo 2097, and bamalip 2098.

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 2075 "Barbara", one of the fundamental syllogisms of Aristotelian logic. All  ph is  ps, and all  ch is  ph, therefore all  ch is  ps. (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  A. x ( x  e.  H  ->  x  e.  M ) (all men are mortal) and  A. x ( x  =  S  ->  x  e.  H ) (Socrates is a man) therefore  A. x ( x  =  S  ->  x  e.  M ) (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 1599. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm 1599, http://plato.stanford.edu/entries/aristotle-logic/ 1599, and https://en.wikipedia.org/wiki/Syllogism 1599. (Contributed by David A. Wheeler, 24-Aug-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ch  ->  ph )   =>    |-  A. x ( ch  ->  ps )
 
Theoremcelarent 2076 "Celarent", one of the syllogisms of Aristotelian logic. No  ph is  ps, and all  ch is  ph, therefore no  ch is  ps. (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.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ch  ->  ph )   =>    |-  A. x ( ch  ->  -.  ps )
 
Theoremdarii 2077 "Darii", one of the syllogisms of Aristotelian logic. All  ph is  ps, and some  ch is  ph, therefore some  ch is  ps. (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.)
 |- 
 A. x ( ph  ->  ps )   &    |-  E. x ( ch  /\  ph )   =>    |-  E. x ( ch  /\  ps )
 
Theoremferio 2078 "Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No  ph is  ps, and some  ch is  ph, therefore some  ch is not  ps. (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.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  E. x ( ch  /\  ph )   =>    |-  E. x ( ch  /\  -.  ps )
 
Theorembarbari 2079 "Barbari", one of the syllogisms of Aristotelian logic. All  ph is  ps, all  ch is  ph, and some  ch exist, therefore some  ch is  ps. (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.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ch  ->  ph )   &    |-  E. x ch   =>    |- 
 E. x ( ch 
 /\  ps )
 
Theoremcelaront 2080 "Celaront", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ch is  ph, and some  ch exist, therefore some  ch is not  ps. (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.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ch  ->  ph )   &    |-  E. x ch   =>    |- 
 E. x ( ch 
 /\  -.  ps )
 
Theoremcesare 2081 "Cesare", one of the syllogisms of Aristotelian logic. No  ph is  ps, and all  ch is  ps, therefore no  ch is  ph. (In Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to celarent 2076. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ch  ->  ps )   =>    |-  A. x ( ch  ->  -.  ph )
 
Theoremcamestres 2082 "Camestres", one of the syllogisms of Aristotelian logic. All  ph is  ps, and no  ch is  ps, therefore no  ch is  ph. (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.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ch  ->  -.  ps )   =>    |-  A. x ( ch  ->  -.  ph )
 
Theoremfestino 2083 "Festino", one of the syllogisms of Aristotelian logic. No  ph is  ps, and some  ch is  ps, therefore some  ch is not  ph. (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  E. x ( ch  /\  ps )   =>    |-  E. x ( ch  /\  -.  ph )
 
Theorembaroco 2084 "Baroco", one of the syllogisms of Aristotelian logic. All  ph is  ps, and some  ch is not  ps, therefore some  ch is not  ph. (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.)
 |- 
 A. x ( ph  ->  ps )   &    |-  E. x ( ch  /\  -.  ps )   =>    |- 
 E. x ( ch 
 /\  -.  ph )
 
Theoremcesaro 2085 "Cesaro", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ch is  ps, and  ch exist, therefore some  ch is not  ph. (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.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ch  ->  ps )   &    |-  E. x ch   =>    |- 
 E. x ( ch 
 /\  -.  ph )
 
Theoremcamestros 2086 "Camestros", one of the syllogisms of Aristotelian logic. All  ph is  ps, no  ch is  ps, and  ch exist, therefore some  ch is not  ph. (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.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ch  ->  -.  ps )   &    |-  E. x ch   =>    |- 
 E. x ( ch 
 /\  -.  ph )
 
Theoremdatisi 2087 "Datisi", one of the syllogisms of Aristotelian logic. All  ph is  ps, and some  ph is  ch, therefore some  ch is  ps. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  E. x (
 ph  /\  ch )   =>    |-  E. x ( ch  /\  ps )
 
Theoremdisamis 2088 "Disamis", one of the syllogisms of Aristotelian logic. Some  ph is  ps, and all  ph is  ch, therefore some  ch is  ps. (In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 E. x ( ph  /\ 
 ps )   &    |-  A. x (
 ph  ->  ch )   =>    |- 
 E. x ( ch 
 /\  ps )
 
Theoremferison 2089 "Ferison", one of the syllogisms of Aristotelian logic. No  ph is  ps, and some  ph is  ch, therefore some  ch is not  ps. (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.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  E. x ( ph  /\  ch )   =>    |-  E. x ( ch  /\  -.  ps )
 
Theorembocardo 2090 "Bocardo", one of the syllogisms of Aristotelian logic. Some  ph is not  ps, and all  ph is  ch, therefore some  ch is not  ps. (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 2088; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.)
 |- 
 E. x ( ph  /\ 
 -.  ps )   &    |-  A. x (
 ph  ->  ch )   =>    |- 
 E. x ( ch 
 /\  -.  ps )
 
Theoremfelapton 2091 "Felapton", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ph is  ch, and some  ph exist, therefore some  ch is not  ps. (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.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ph  ->  ch )   &    |-  E. x ph   =>    |-  E. x ( ch  /\  -. 
 ps )
 
Theoremdarapti 2092 "Darapti", one of the syllogisms of Aristotelian logic. All  ph is  ps, all  ph is  ch, and some  ph exist, therefore some  ch is  ps. (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.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x (
 ph  ->  ch )   &    |-  E. x ph   =>    |-  E. x ( ch  /\  ps )
 
Theoremcalemes 2093 "Calemes", one of the syllogisms of Aristotelian logic. All  ph is  ps, and no  ps is  ch, therefore no  ch is  ph. (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.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ps  ->  -.  ch )   =>    |-  A. x ( ch  ->  -.  ph )
 
Theoremdimatis 2094 "Dimatis", one of the syllogisms of Aristotelian logic. Some  ph is  ps, and all  ps is  ch, therefore some  ch is  ph. (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 2077 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 E. x ( ph  /\ 
 ps )   &    |-  A. x ( ps  ->  ch )   =>    |-  E. x ( ch  /\  ph )
 
Theoremfresison 2095 "Fresison", one of the syllogisms of Aristotelian logic. No  ph is  ps (PeM), and some  ps is  ch (MiS), therefore some  ch is not  ph (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.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  E. x ( ps  /\  ch )   =>    |-  E. x ( ch  /\  -.  ph )
 
Theoremcalemos 2096 "Calemos", one of the syllogisms of Aristotelian logic. All  ph is  ps (PaM), no  ps is  ch (MeS), and  ch exist, therefore some  ch is not  ph (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.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ps  ->  -.  ch )   &    |-  E. x ch   =>    |- 
 E. x ( ch 
 /\  -.  ph )
 
Theoremfesapo 2097 "Fesapo", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ps is  ch, and  ps exist, therefore some  ch is not  ph. (In Aristotelian notation, EAO-4: PeM and MaS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
 |- 
 A. x ( ph  ->  -.  ps )   &    |-  A. x ( ps  ->  ch )   &    |-  E. x ps   =>    |- 
 E. x ( ch 
 /\  -.  ph )
 
Theorembamalip 2098 "Bamalip", one of the syllogisms of Aristotelian logic. All  ph is  ps, all  ps is  ch, and  ph exist, therefore some  ch is  ph. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2079. (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ps  ->  ch )   &    |-  E. x ph   =>    |-  E. x ( ch  /\  ph )
 
PART 2  SET THEORY

Set theory uses the formalism of propositional and predicate calculus to assert properties of arbitrary mathematical objects called "sets." A set can be an element of another set, and this relationship is indicated by the  e. symbol. Starting with the simplest mathematical object, called the empty set, set theory builds up more and more complex structures whose existence follows from the axioms, eventually resulting in extremely complicated sets that we identify with the real numbers and other familiar mathematical objects.

Here we develop set theory based on the Intuitionistic Zermelo-Fraenkel (IZF) system, mostly following the IZF axioms as laid out in [Crosilla]. Constructive Zermelo-Fraenkel (CZF), also described in Crosilla, is not as easy to formalize in Metamath because the statement of some of its axioms uses the notion of "bounded formula". Since Metamath has, purposefully, a very weak metalogic, that notion must be developed in the logic itself. This is similar to our treatment of substitution (df-sb 1721) and our definition of the nonfreeness predicate (df-nf 1422), whereas substitution and bound and free variables are ordinarily defined in the metalogic. The development of CZF has begun in BJ's mathbox, see wbd 12937.

 
2.1  IZF Set Theory - start with the Axiom of Extensionality
 
2.1.1  Introduce the Axiom of Extensionality
 
Axiomax-ext 2099* Axiom of Extensionality. It states that two sets are identical if they contain the same elements. Axiom 1 of [Crosilla] p. "Axioms of CZF and IZF" (with unnecessary quantifiers removed).

Set theory can also be formulated with a single primitive predicate  e. on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes  ( A. w
( w  e.  x  <->  w  e.  y )  -> 
( x  e.  z  ->  y  e.  z ) ), and equality  x  =  y is defined as  A. w ( w  e.  x  <->  w  e.  y
). All of the usual axioms of equality then become theorems of set theory. See, for example, Axiom 1 of [TakeutiZaring] p. 8.

To use the above "equality-free" version of Extensionality with Metamath's logical axioms, we would rewrite ax-8 1467 through ax-16 1770 with equality expanded according to the above definition. Some of those axioms could be proved from set theory and would be redundant. Not all of them are redundant, since our axioms of predicate calculus make essential use of equality for the proper substitution that is a primitive notion in traditional predicate calculus. A study of such an axiomatization would be an interesting project for someone exploring the foundations of logic.

It is important to understand that strictly speaking, all of our set theory axioms are really schemes that represent an infinite number of actual axioms. This is inherent in the design of Metamath ("metavariable math"), which manipulates only metavariables. For example, the metavariable  x in ax-ext 2099 can represent any actual variable v1, v2, v3,... . Distinct variable restrictions ($d) prevent us from substituting say v1 for both  x and  z. This is in contrast to typical textbook presentations that present actual axioms (except for axioms which involve wff metavariables). In practice, though, the theorems and proofs are essentially the same. The $d restrictions make each of the infinite axioms generated by the ax-ext 2099 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 5-Aug-1993.)

 |-  ( A. z ( z  e.  x  <->  z  e.  y
 )  ->  x  =  y )
 
Theoremaxext3 2100* A generalization of the Axiom of Extensionality in which  x and  y need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  ( A. z ( z  e.  x  <->  z  e.  y
 )  ->  x  =  y )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13177
  Copyright terms: Public domain < Previous  Next >