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Type | Label | Description |
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Statement | ||
Definition | df-mo 1901 | Define "there exists at most one x such that φ." Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 1951. For another possible definition see mo4 1958. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃*xφ ↔ (∃xφ → ∃!xφ)) | ||
Theorem | euf 1902* | A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.) |
⊢ (φ → ∀yφ) ⇒ ⊢ (∃!xφ ↔ ∃y∀x(φ ↔ x = y)) | ||
Theorem | eubidh 1903 | Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.) |
⊢ (φ → ∀xφ) & ⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∃!xψ ↔ ∃!xχ)) | ||
Theorem | eubid 1904 | Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.) |
⊢ Ⅎxφ & ⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∃!xψ ↔ ∃!xχ)) | ||
Theorem | eubidv 1905* | Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∃!xψ ↔ ∃!xχ)) | ||
Theorem | eubii 1906 | Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
⊢ (φ ↔ ψ) ⇒ ⊢ (∃!xφ ↔ ∃!xψ) | ||
Theorem | hbeu1 1907 | Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) |
⊢ (∃!xφ → ∀x∃!xφ) | ||
Theorem | nfeu1 1908 | Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎx∃!xφ | ||
Theorem | nfmo1 1909 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎx∃*xφ | ||
Theorem | sb8eu 1910 | Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎyφ ⇒ ⊢ (∃!xφ ↔ ∃!y[y / x]φ) | ||
Theorem | sb8mo 1911 | Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ Ⅎyφ ⇒ ⊢ (∃*xφ ↔ ∃*y[y / x]φ) | ||
Theorem | nfeudv 1912* | Deduction version of nfeu 1916. Similar to nfeud 1913 but has the additional constraint that x and y must be distinct. (Contributed by Jim Kingdon, 25-May-2018.) |
⊢ Ⅎyφ & ⊢ (φ → Ⅎxψ) ⇒ ⊢ (φ → Ⅎx∃!yψ) | ||
Theorem | nfeud 1913 | Deduction version of nfeu 1916. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 25-May-2018.) |
⊢ Ⅎyφ & ⊢ (φ → Ⅎxψ) ⇒ ⊢ (φ → Ⅎx∃!yψ) | ||
Theorem | nfmod 1914 | Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Ⅎyφ & ⊢ (φ → Ⅎxψ) ⇒ ⊢ (φ → Ⅎx∃*yψ) | ||
Theorem | nfeuv 1915* | Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 1916 but has the additional constraint that x and y must be distinct. (Contributed by Jim Kingdon, 23-May-2018.) |
⊢ Ⅎxφ ⇒ ⊢ Ⅎx∃!yφ | ||
Theorem | nfeu 1916 | Bound-variable hypothesis builder for existential uniqueness. Note that x and y 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.) |
⊢ Ⅎxφ ⇒ ⊢ Ⅎx∃!yφ | ||
Theorem | nfmo 1917 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.) |
⊢ Ⅎxφ ⇒ ⊢ Ⅎx∃*yφ | ||
Theorem | hbeu 1918 | Bound-variable hypothesis builder for uniqueness. Note that x and y needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Proof rewritten by Jim Kingdon, 24-May-2018.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (∃!yφ → ∀x∃!yφ) | ||
Theorem | hbeud 1919 | Deduction version of hbeu 1918. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 25-May-2018.) |
⊢ (φ → ∀xφ) & ⊢ (φ → ∀yφ) & ⊢ (φ → (ψ → ∀xψ)) ⇒ ⊢ (φ → (∃!yψ → ∀x∃!yψ)) | ||
Theorem | sb8euh 1920 | Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.) |
⊢ (φ → ∀yφ) ⇒ ⊢ (∃!xφ ↔ ∃!y[y / x]φ) | ||
Theorem | cbveu 1921 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
⊢ Ⅎyφ & ⊢ Ⅎxψ & ⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃!xφ ↔ ∃!yψ) | ||
Theorem | eu1 1922* | An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) |
⊢ (φ → ∀yφ) ⇒ ⊢ (∃!xφ ↔ ∃x(φ ∧ ∀y([y / x]φ → x = y))) | ||
Theorem | euor 1923 | Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) |
⊢ (φ → ∀xφ) ⇒ ⊢ ((¬ φ ∧ ∃!xψ) → ∃!x(φ ∨ ψ)) | ||
Theorem | euorv 1924* | Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.) |
⊢ ((¬ φ ∧ ∃!xψ) → ∃!x(φ ∨ ψ)) | ||
Theorem | mo2n 1925* | There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.) |
⊢ Ⅎyφ ⇒ ⊢ (¬ ∃xφ → ∃y∀x(φ → x = y)) | ||
Theorem | mon 1926 | There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.) |
⊢ (¬ ∃xφ → ∃*xφ) | ||
Theorem | euex 1927 | Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (∃!xφ → ∃xφ) | ||
Theorem | eumo0 1928* | Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.) |
⊢ (φ → ∀yφ) ⇒ ⊢ (∃!xφ → ∃y∀x(φ → x = y)) | ||
Theorem | eumo 1929 | Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) |
⊢ (∃!xφ → ∃*xφ) | ||
Theorem | eumoi 1930 | "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
⊢ ∃!xφ ⇒ ⊢ ∃*xφ | ||
Theorem | mobidh 1931 | Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.) |
⊢ (φ → ∀xφ) & ⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∃*xψ ↔ ∃*xχ)) | ||
Theorem | mobid 1932 | Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.) |
⊢ Ⅎxφ & ⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∃*xψ ↔ ∃*xχ)) | ||
Theorem | mobidv 1933* | Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∃*xψ ↔ ∃*xχ)) | ||
Theorem | mobii 1934 | Formula-building rule for "at most one" quantifier (inference rule). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.) |
⊢ (ψ ↔ χ) ⇒ ⊢ (∃*xψ ↔ ∃*xχ) | ||
Theorem | hbmo1 1935 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) |
⊢ (∃*xφ → ∀x∃*xφ) | ||
Theorem | hbmo 1936 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (∃*yφ → ∀x∃*yφ) | ||
Theorem | cbvmo 1937 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) |
⊢ Ⅎyφ & ⊢ Ⅎxψ & ⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃*xφ ↔ ∃*yψ) | ||
Theorem | mo23 1938* | An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.) |
⊢ Ⅎyφ ⇒ ⊢ (∃y∀x(φ → x = y) → ∀x∀y((φ ∧ [y / x]φ) → x = y)) | ||
Theorem | mor 1939* | Converse of mo23 1938 with an additional ∃xφ condition. (Contributed by Jim Kingdon, 25-Jun-2018.) |
⊢ Ⅎyφ ⇒ ⊢ (∃xφ → (∀x∀y((φ ∧ [y / x]φ) → x = y) → ∃y∀x(φ → x = y))) | ||
Theorem | modc 1940* | Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.) |
⊢ Ⅎyφ ⇒ ⊢ (DECID ∃xφ → (∃y∀x(φ → x = y) ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))) | ||
Theorem | eu2 1941* | An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) |
⊢ Ⅎyφ ⇒ ⊢ (∃!xφ ↔ (∃xφ ∧ ∀x∀y((φ ∧ [y / x]φ) → x = y))) | ||
Theorem | eu3h 1942* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.) |
⊢ (φ → ∀yφ) ⇒ ⊢ (∃!xφ ↔ (∃xφ ∧ ∃y∀x(φ → x = y))) | ||
Theorem | eu3 1943* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) |
⊢ Ⅎyφ ⇒ ⊢ (∃!xφ ↔ (∃xφ ∧ ∃y∀x(φ → x = y))) | ||
Theorem | eu5 1944 | Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) |
⊢ (∃!xφ ↔ (∃xφ ∧ ∃*xφ)) | ||
Theorem | exmoeu2 1945 | Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.) |
⊢ (∃xφ → (∃*xφ ↔ ∃!xφ)) | ||
Theorem | moabs 1946 | Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.) |
⊢ (∃*xφ ↔ (∃xφ → ∃*xφ)) | ||
Theorem | exmodc 1947 | If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.) |
⊢ (DECID ∃xφ → (∃xφ ∨ ∃*xφ)) | ||
Theorem | exmonim 1948 | There is at most one of something which does not exist. Unlike exmodc 1947 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.) |
⊢ (¬ ∃xφ → ∃*xφ) | ||
Theorem | mo2r 1949* | A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.) |
⊢ Ⅎyφ ⇒ ⊢ (∃y∀x(φ → x = y) → ∃*xφ) | ||
Theorem | mo3h 1950* | Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that y not occur in φ in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (New usage is discouraged.) |
⊢ (φ → ∀yφ) ⇒ ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y)) | ||
Theorem | mo3 1951* | Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that y not occur in φ in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) |
⊢ Ⅎyφ ⇒ ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y)) | ||
Theorem | mo2dc 1952* | Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.) |
⊢ Ⅎyφ ⇒ ⊢ (DECID ∃xφ → (∃*xφ ↔ ∃y∀x(φ → x = y))) | ||
Theorem | euan 1953 | Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (∃!x(φ ∧ ψ) ↔ (φ ∧ ∃!xψ)) | ||
Theorem | euanv 1954* | Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.) |
⊢ (∃!x(φ ∧ ψ) ↔ (φ ∧ ∃!xψ)) | ||
Theorem | euor2 1955 | Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (¬ ∃xφ → (∃!x(φ ∨ ψ) ↔ ∃!xψ)) | ||
Theorem | sbmo 1956* | Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ([y / x]∃*zφ ↔ ∃*z[y / x]φ) | ||
Theorem | mo4f 1957* | "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.) |
⊢ Ⅎxψ & ⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ ψ) → x = y)) | ||
Theorem | mo4 1958* | "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ ψ) → x = y)) | ||
Theorem | eu4 1959* | Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃!xφ ↔ (∃xφ ∧ ∀x∀y((φ ∧ ψ) → x = y))) | ||
Theorem | exmoeudc 1960 | Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.) |
⊢ (DECID ∃xφ → (∃xφ ↔ (∃*xφ → ∃!xφ))) | ||
Theorem | moim 1961 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.) |
⊢ (∀x(φ → ψ) → (∃*xψ → ∃*xφ)) | ||
Theorem | moimi 1962 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.) |
⊢ (φ → ψ) ⇒ ⊢ (∃*xψ → ∃*xφ) | ||
Theorem | moimv 1963* | Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.) |
⊢ (∃*x(φ → ψ) → (φ → ∃*xψ)) | ||
Theorem | euimmo 1964 | Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.) |
⊢ (∀x(φ → ψ) → (∃!xψ → ∃*xφ)) | ||
Theorem | euim 1965 | 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.) |
⊢ ((∃xφ ∧ ∀x(φ → ψ)) → (∃!xψ → ∃!xφ)) | ||
Theorem | moan 1966 | "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.) |
⊢ (∃*xφ → ∃*x(ψ ∧ φ)) | ||
Theorem | moani 1967 | "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.) |
⊢ ∃*xφ ⇒ ⊢ ∃*x(ψ ∧ φ) | ||
Theorem | moor 1968 | "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.) |
⊢ (∃*x(φ ∨ ψ) → ∃*xφ) | ||
Theorem | mooran1 1969 | "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ ((∃*xφ ∨ ∃*xψ) → ∃*x(φ ∧ ψ)) | ||
Theorem | mooran2 1970 | "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (∃*x(φ ∨ ψ) → (∃*xφ ∧ ∃*xψ)) | ||
Theorem | moanim 1971 | Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.) |
⊢ Ⅎxφ ⇒ ⊢ (∃*x(φ ∧ ψ) ↔ (φ → ∃*xψ)) | ||
Theorem | moanimv 1972* | Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.) |
⊢ (∃*x(φ ∧ ψ) ↔ (φ → ∃*xψ)) | ||
Theorem | moaneu 1973 | Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.) |
⊢ ∃*x(φ ∧ ∃!xφ) | ||
Theorem | moanmo 1974 | Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.) |
⊢ ∃*x(φ ∧ ∃*xφ) | ||
Theorem | mopick 1975 | "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) |
⊢ ((∃*xφ ∧ ∃x(φ ∧ ψ)) → (φ → ψ)) | ||
Theorem | eupick 1976 | Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing x such that φ is true, and there is also an x (actually the same one) such that φ and ψ are both true, then φ implies ψ 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.) |
⊢ ((∃!xφ ∧ ∃x(φ ∧ ψ)) → (φ → ψ)) | ||
Theorem | eupicka 1977 | Version of eupick 1976 with closed formulas. (Contributed by NM, 6-Sep-2008.) |
⊢ ((∃!xφ ∧ ∃x(φ ∧ ψ)) → ∀x(φ → ψ)) | ||
Theorem | eupickb 1978 | Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) |
⊢ ((∃!xφ ∧ ∃!xψ ∧ ∃x(φ ∧ ψ)) → (φ ↔ ψ)) | ||
Theorem | eupickbi 1979 | Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!xφ → (∃x(φ ∧ ψ) ↔ ∀x(φ → ψ))) | ||
Theorem | mopick2 1980 | "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 1519. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ ((∃*xφ ∧ ∃x(φ ∧ ψ) ∧ ∃x(φ ∧ χ)) → ∃x(φ ∧ ψ ∧ χ)) | ||
Theorem | moexexdc 1981 | "At most one" double quantification. (Contributed by Jim Kingdon, 5-Jul-2018.) |
⊢ Ⅎyφ ⇒ ⊢ (DECID ∃xφ → ((∃*xφ ∧ ∀x∃*yψ) → ∃*y∃x(φ ∧ ψ))) | ||
Theorem | euexex 1982 | Existential uniqueness and "at most one" double quantification. (Contributed by Jim Kingdon, 28-Dec-2018.) |
⊢ Ⅎyφ ⇒ ⊢ ((∃!xφ ∧ ∀x∃*yψ) → ∃*y∃x(φ ∧ ψ)) | ||
Theorem | 2moex 1983 | Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.) |
⊢ (∃*x∃yφ → ∀y∃*xφ) | ||
Theorem | 2euex 1984 | Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (∃!x∃yφ → ∃y∃!xφ) | ||
Theorem | 2eumo 1985 | Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.) |
⊢ (∃!x∃*yφ → ∃*x∃!yφ) | ||
Theorem | 2eu2ex 1986 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |
⊢ (∃!x∃!yφ → ∃x∃yφ) | ||
Theorem | 2moswapdc 1987 | A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.) |
⊢ (DECID ∃x∃yφ → (∀x∃*yφ → (∃*x∃yφ → ∃*y∃xφ))) | ||
Theorem | 2euswapdc 1988 | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.) |
⊢ (DECID ∃x∃yφ → (∀x∃*yφ → (∃!x∃yφ → ∃!y∃xφ))) | ||
Theorem | 2exeu 1989 | Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) |
⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) → ∃!x∃!yφ) | ||
Theorem | 2eu4 1990* | 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 ∃!x∃!yφ. See 2exeu 1989 for a one-way implication. (Contributed by NM, 3-Dec-2001.) |
⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ (∃x∃yφ ∧ ∃z∃w∀x∀y(φ → (x = z ∧ y = w)))) | ||
Theorem | 2eu7 1991 | Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.) |
⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ ∃!x∃!y(∃xφ ∧ ∃yφ)) | ||
Theorem | euequ1 1992* | Equality has existential uniqueness. (Contributed by Stefan Allan, 4-Dec-2008.) |
⊢ ∃!x x = y | ||
Theorem | exists1 1993* | 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.) |
⊢ (∃!x x = x ↔ ∀x x = y) | ||
Theorem | exists2 1994 | A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ ((∃xφ ∧ ∃x ¬ φ) → ¬ ∃!x x = x) | ||
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 mpto1 1311) 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 φ, ψ, and χ 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 φ is ψ " are consistently translated as ∀x(φ → ¬ ψ). These can also be expressed as ¬ ∃x(φ ∧ ψ), per alinexa 1491. We translate "all φ is ψ " to ∀x(φ → ψ), "some φ is ψ " to ∃x(φ ∧ ψ), and "some φ is not ψ " to ∃x(φ ∧ ¬ ψ). 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 ∈ A, or x ⊆ 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 ∈ 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 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 1999, celaront 2000, cesaro 2005, camestros 2006, felapton 2011, darapti 2012, calemos 2016, fesapo 2017, and bamalip 2018. 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. | ||
Theorem | barbara 1995 | "Barbara", one of the fundamental syllogisms of Aristotelian logic. All φ is ψ, and all χ is φ, therefore all χ is ψ. (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 ∀x(x ∈ 𝐻 → x ∈ 𝑀) (all men are mortal) and ∀x(x = 𝑆 → x ∈ 𝐻) (Socrates is a man) therefore ∀x(x = 𝑆 → x ∈ 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 1523. 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.) |
⊢ ∀x(φ → ψ) & ⊢ ∀x(χ → φ) ⇒ ⊢ ∀x(χ → ψ) | ||
Theorem | celarent 1996 | "Celarent", one of the syllogisms of Aristotelian logic. No φ is ψ, and all χ is φ, therefore no χ is ψ. (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.) |
⊢ ∀x(φ → ¬ ψ) & ⊢ ∀x(χ → φ) ⇒ ⊢ ∀x(χ → ¬ ψ) | ||
Theorem | darii 1997 | "Darii", one of the syllogisms of Aristotelian logic. All φ is ψ, and some χ is φ, therefore some χ is ψ. (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.) |
⊢ ∀x(φ → ψ) & ⊢ ∃x(χ ∧ φ) ⇒ ⊢ ∃x(χ ∧ ψ) | ||
Theorem | ferio 1998 | "Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No φ is ψ, and some χ is φ, therefore some χ is not ψ. (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.) |
⊢ ∀x(φ → ¬ ψ) & ⊢ ∃x(χ ∧ φ) ⇒ ⊢ ∃x(χ ∧ ¬ ψ) | ||
Theorem | barbari 1999 | "Barbari", one of the syllogisms of Aristotelian logic. All φ is ψ, all χ is φ, and some χ exist, therefore some χ is ψ. (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.) |
⊢ ∀x(φ → ψ) & ⊢ ∀x(χ → φ) & ⊢ ∃xχ ⇒ ⊢ ∃x(χ ∧ ψ) | ||
Theorem | celaront 2000 | "Celaront", one of the syllogisms of Aristotelian logic. No φ is ψ, all χ is φ, and some χ exist, therefore some χ is not ψ. (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.) |
⊢ ∀x(φ → ¬ ψ) & ⊢ ∀x(χ → φ) & ⊢ ∃xχ ⇒ ⊢ ∃x(χ ∧ ¬ ψ) |
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