First-order logic and List of Ancien Régime dioceses of France: Difference between pages

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'''French [[Ancien Régime]] [[Roman Catholic]] [[diocese]]s and ecclesiastical provinces''' were heirs of Late Roman [[civitates]] (themselves created out of [[Gaul]]ish tribes) and provinces.

'''First-order predicate calculus''' or '''first-order logic''' is a theory in [[symbolic logic]] that formalizes quantified statements such as "there exists an object with the property that..." or "for all objects, the following is true...". First-order logic is distinguished from higher-order logic in that it does not allow statements such as "for every property, the following is true..." or "there exists a set of objects such that...". Nevertheless, first-order logic is strong enough to formalize all of [[set theory]] and thereby virtually all of [[mathematics]]. It is the classical logical theory underlying mathematics.

[[Image:France ecc 1789 1802.jpg|thumb|right|450px|Ecclesiastical provinces (colored) and dioceses of France in 1789]]

==Historical sketch==

Most of them were created during the first christianization of Gaul, in the 3rd to 5th centuries.

But, at several occasions during the Middle Ages or the Ancien Régime, new dioceses were created, replacing older ones or carved out of them. For instance, fighting against the [[Cathar]] heresy entailed the creation of many new dioceses in the early 14th century. All the same, in 1789, on the eve of the [[French Revolution]], the ecclesiastical map of France still very much recalled that of [[Roman Gaul]]. This explains why many dioceses and provinces did not coincide with French borders, with their head cities lying in present-day [[Belgium]], [[Germany]] or [[Switzerland]].

Like any logical theory, first-order calculus consists of

* a specification of how to construct syntactically correct statements (the well-formed formulas)

* a set of axioms, each axiom being a well-formed formula itself

* a set of inference rules which allow to prove theorems from axioms or earlier proven theorems.

In 1790, this map was entirely revised to fit the new administrative map: dioceses were now to coincide with [[département]]s (the new administrative units). Ancien Régime dioceses all disappeared, then, in 1790. Many former bishoprics remained heads of the new dioceses, but many cities lost their bishop. Even so, in those cities, the former cathedral very often kept its rank as a cathedral church. This explains why many post-Revolutionary episcopal sees bear the name of several cities. For instance, in the département of the [[Drôme]], only the city of [[Valence, Drôme|Valence]] retained its bishop, the former episcopal sees of Die and Saint-Paul-Trois-Châteaux being suppressed, but the bishop retained the title of bishop of Valence, Die and Saint-Paul-Trois-Châteaux.

There are two types of axioms: the logical axioms which embody the general truths about proper reasoning involving quantified statements, and the axioms describing the subject matter at hand, for instance axioms describing sets in set theory or axioms describing numbers in arithmetic.

Here follows a list of Ancien Régime dioceses, as of 1789, on the eve of the Revolution. With the exception of those dioceses which were created in the Late Roman period (before the 6th century), whose date of creation generally cannot be established, we provide the date of creation and, when appropriated of suppression of the bishopric. Dioceses whose sees were not within the borders of the kingdom of France are in brackets.

While the set of inference rules in first-order calculus is finite, the set of axioms may very well be and often is infinite. However we require that there is a general [[algorithm]] which can decide for a given well-formed formula whether it is an axiom or not. Furthermore, there should be an algorithm which can decide whether a given application of an inference rule is correct or not.

==Listing of dioceses by church province==

The well-formed formulas contain

=== Province of [[Aix-en-Provence|Aix]] (Narbonensis Secunda) ===

* variables such as ''x'', ''y'', ... which are place holder for objects of the ___domain under consideration

*[[Archdiocese of Aix]]

* object constants such as 0, 1 or the empty set ø which stand for fixed individual objects in our ___domain

*[[Diocese of Apt]]

* predicate constants such < (lessThan), ∈ (isIn), '=' (equals) which stand for fixed relations between or properties of our objects. These are also called ''first-order predicates'' to distinguish them from predicates that talk about predicates.

*[[Diocese of Fréjus]]

* function constants such as +, * which stand for fixed functions taking objects as arguments and returning objects as values

*[[Diocese of Gap]]

* logical connectives such as ∧ (and), ∨ (or), ⇒ (implies), ¬ (not), ∃ (thereExists existence quantifier) and ∀ (forAll or universal quantifier). All of these except for the last two are also used in [[propositional logic]].

*[[Diocese of Riez]]

*[[Diocese of Sisteron]]

=== Province of [[Arles]] (Viennensis Secunda) ===

The object, predicate and function constants will typically depend on the particular ___domain we are talking about.

*[[Archdiocese of Arles]]

*[[Diocese of Marseille]]

*[[Diocese of Orange]]

*[[Diocese of Saint-Paul-Trois-Châteaux]]

*[[Diocese of Toulon]]

out of which (1475):

Instead of giving the lengthy definition of well-formed formulas, we will simply look at some examples from arithmetic together with their ordinary interpretation. Our ___domain here is the set of [[natural number]]s:

===== Province of [[Avignon]] =====

:∀ ''x'' ∃ ''y'' : ''y'' > ''x''

*[[Archdiocese of Avignon]] — became a metropolitan see in 1475.

:(ie: forAll ''x'' thereExists ''y'' suchThat ''x'' greaterThan ''y'' )

*[[Diocese of Carpentras]]

For every number ''x'' there exists a bigger number ''y''. That's true.

*[[Diocese of Cavaillon]]

*[[Diocese of Vaison]]

=== Province of [[Auch]] (Novempopulania) ===

:∃ ''y'' ∀ ''x'' : ''y'' > ''x''

*[[Archdiocese of Auch]] — became head of the province between 7th and 9th century, following the demise of the former metropolitan see, [[Eauze]]

:(ie: thereExists ''x'' forAll ''y'' suchThat ''y'' greaterThan ''x'' )

*[[Diocese of Aire]]

*[[Diocese of Bazas]]

*[[Diocese of Dax]], out of which:

**[[Diocese of Bayonne]] — created late 8th century.

*[[Diocese of Lectoure]]

*[[Diocese of Lescar]]

*[[Diocese of Oloron]]

*[[Diocese of Saint-Bertrand-de-Comminges]]

*[[Diocese of Saint-Lizier]]

*[[Diocese of Tarbes]]

=== Province of [[Besançon]] (Maxima Sequanorum) ===

There exists a number ''y'' which is bigger than every number ''x''. That's not true.

*[[Archdiocese of Besançon]]

*[[Diocese of Basel]]

*[[Diocese of Belley]] — moved to Belley in 537 (former see in [[Nyon]]).

*[[Diocese of Lausanne]]

=== Province of [[Bordeaux]] (Aquitania Secunda) ===

:∀ ''x'' ( (∃ ''y'' : 6 * ''y''=''x'' ) ⇒ (∃ ''y'' : 3 * ''y''=''x'' ) )

*[[Archdiocese of Bordeaux]]

:(ie: forAll ''x'' ( ThereExists ''y'' suchThat 6*''y''=''x'') implies (ThereExists ''y'' : 3*''y''=''x'') )

*[[Diocese of Agen]], out of which:

**[[Diocese of Condom]] — Created 1317.

*[[Diocese of Angoulême]]

*[[Diocese of Périgueux]], out of which:

**[[Diocese of Sarlat]] — Created 1317.

*[[Diocese of Poitiers]], out of which:

**[[Diocese of Luçon]] — Created 1317.

**[[Diocese of La Rochelle]] — Created in 1317 with its see in [[Maillezais]]. Was moved to La Rochelle in 1648.

*[[Diocese of Saintes]]

=== Province of [[Bourges]] (Aquitania Prima) ===

If a number ''x'' is divisible by 6, then it is also divisible by 3. True.

*[[Archdiocese of Bourges]]

*[[Diocese of Clermont]], out of which:

**[[Diocese of Saint-Flour]] — Created 1317.

*[[Diocese of Limoges]], out of which:

**[[Diocese of Tulle]] — Created 1317.

*[[Diocese of Le Puy]]

out of which (1678):

:&exist; ''x'' : &not; &exist; ''y'' : ''y'' < ''x''

:(ie: thereExists ''x'' suchThat (not ThereExists ''y'' suchThat ''y'' < ''x'' ) )

There exists a number ''x'' such there doesn't exist a smaller number ''y''. True: (for example, take ''x''=0) .

===== Province of [[Albi]] =====

Now one would have to write down 15 logical axioms and 2 inference rules to completely specify the first-order calculus. How can one be sure that those axioms and rules are enough? This is the subject of [[Goedels completeness theorem|G&ouml;del's completeness theorem]]: if you start out with some subject matter axioms and you look at a certain statement, then it is possible to prove that statement using only the subject matter axioms, the 15 logical axioms and the two inference rules ''if and only if'' the statement is true in every ___domain in which the subject matter axioms are true.

*[[Archdiocese of Albi]] — became a metropolitan see in 1678. Out of which:

**[[Diocese of Castres]] — created 1317.

*[[Diocese of Cahors]]

*[[Diocese of Mende]] — moved to Mende in the 6th century (former see was in [[Gévaudan|Javols]]).

*[[Diocese of Rodez]], out of which:

**[[Diocese of Vabres]] — created 1317.

=== Province of [[Embrun, Hautes-Alpes|Embrun]] (Alpes Maritimæ) ===

The [[Peano axioms]] for the natural numbers are sometimes formulated as second-order statements (the induction axiom talks about all "properties" or all "sets of numbers"), but this is not necessary if one is willing to allow infinitely many first-order axioms. A first-order version of the Peano axioms follows.

*[[Archdiocese of Embrun]]

*[[Diocese of Digne]]

*[[Diocese of Entrevaux]] — Actually in the hamlet of Glandèves.

*[[Diocese of Grasse]] — Moved to Grasse in 1244 (former see in [[Antibes]]. Belonged to the province of Aix-en-Provence down to 1057.

*[[Diocese of Nice]]

*[[Diocese of Senez]]

*[[Diocese of Vence]]

=== Province of [[Genoa]] ===

We are using the object constants 0 and 1, the function constants + and *, and the predicate constant =. Here are the axioms:

(Province created in 1133: Northern Corsican sees belonged to this province)

# &forall; ''x'' : &not; (0 = ''x'' + 1)

*Diocese of Mariana — The bishop resides in [[Bastia]]

#* ie: ForAll ''x'' suchThat not (0 = ''x'' + 1)

*Diocese of Nebbio — The bishop resides in [[Saint-Florent, Haute-Corse|Saint-Florent]]

# &forall; ''x'' &forall; ''y'' : &not; (''x'' = ''y'') &rArr; &not; (''x'' + 1 = ''y'' + 1)

#* ie: forAll ''x'' forAll ''y'' suchThat not(''x''=''y'') implies not(''x'' + 1 = ''y'' + 1)

# &forall; ''x'' : ''x'' + 0 = ''x''

#* ie: forAll ''x'' suchThat x'' + 0 = ''x''

# &forall; ''x'' &forall; ''y'' : (''x'' + ''y'') + 1 = ''x'' + (''y'' + 1)

#* ie: forAll ''x'' forAll ''y'' suchThat (''x'' + ''y'') + 1 = ''x'' + (''y'' + 1)

# &forall; ''x'' : ''x'' * 0=0

#* ie: forAll ''x'' suchThat ''x'' * 0=0

# &forall; ''x'' &forall; ''y'' : ''x'' * (''y'' + 1) = ''x'' * ''y'' + ''x''

#* ie: forAll ''x'' forAll ''y'' suchThat ''x'' * (''y'' + 1) = ''x'' * ''y'' + ''x''

# This is an axiom scheme consisting of infinitely many axioms. If ''P''(''x'') is any formula involving the constants 0, 1, +, *, = and a single free variable ''x'', then the following formula is an axiom: &nbsp;&nbsp;( ''P''(0) &and; &forall; ''x'' : ''P''(''x'') &rArr; ''P''(''x'' + 1) ) &rArr; &forall; ''x'' : ''P''(''x'')

#* ie: ( ''P''(0) and (forAll ''x'' suchThat ( ''P''(''x'') implies ''P''(''x'' + 1) ) ) ) implies (forAll ''x'' suchThat ''P''(''x'') )

=== Province of [[Lyon]] (Lugdunensis Prima) ===

Axioms 1, 2 and 7 are the traditional Peano axioms while axioms 3-6 serve to define addition and multiplication. If one omits the function constant * and axioms 5 and 6 and allows in scheme 7 only formulas without *, then one gets the very interesting [[Presburger arithmetic]].

*[[Archdiocese of Lyon]], out of which:

**[[Diocese of Saint-Claude]] — Created 1742.

*[[Diocese of Autun]]

*[[Diocese of Langres]], out of which:

**[[Diocese of Dijon]] — Created 1731.

*[[Diocese of Chalon-sur-Saône]]

*[[Diocese of Mâcon]]

=== Province of [[Mainz]] (Germania Prima) ===

----

*(Diocese of [[Speyer]])

'''References:'''

*[[Diocese of Strasbourg]]

* [http://www.ltn.lv/~podnieks/ Introduction to mathematical logic]

*other dioceses wholly in [[Germany]]

=== Province of [[Narbonne]] (Narbonensis Prima) ===

*[[Archdiocese of Narbonne]], out of which:

**[[Diocese of Alet]] — Created 1317.

**[[Diocese of Carcassonne]] — Created late 6th century.

**[[Diocese of Montpellier]] — Created late 6th century. Moved to Montpellier in 1536 (former see in [[Maguelonne]]).

**[[Diocese of Perpignan]] — Created late 6th century. Moved to Perpignan in 1602 (former see in [[Elne]]).

**[[Diocese of Saint-Pons]] — Created 1317.

*[[Diocese of Agde]]

*[[Diocese of Béziers]]

*[[Diocese of Lodève]]

*[[Diocese of Nîmes]], out of which:

**[[Diocese of Alès]] — Created 1694

*[[Diocese of Uzès]]

out of which (1317):

===== Province of [[Toulouse]] =====

*Archdiocese of Toulouse — Became a metropolitan see in 1317. Out of which:

**[[Diocese of Lavaur]] — Created 1317

**Diocese of [[Lombez]] — Created 1317

**Diocese of [[Montauban]] — Created 1317

**Diocese of [[Pamiers]] — Created 1295, out of which:

***Diocese of [[Mirepoix, Ariège|Mirepoix]] — Created 1317

***Diocese of [[Rieux-Volvestre|Rieux]] — Created 1317

**Diocese of [[Saint-Papoul]] — Created 1317

=== Province of [[Reims]] (Belgica Secunda) ===

*[[Archdiocese of Reims]], out of which

**[[Diocese of Laon]] — Created late 5th century.

*[[Diocese of Amiens]]

*[[Diocese of Beauvais]]

*[[Diocese of Châlons-en-Champagne]]

*[[Diocese of Senlis]]

*[[Diocese of Soissons]]

*[[Diocese of Noyon]]

*[[Diocese of Thérouanne]] — Suppressed 1553, out of which:

**[[Diocese of Boulogne]] — Created 1567.

**[[Diocese of Saint-Omer]] — Created 1559.

out of which (1559):

===== Province of [[Cambrai]] =====

*[[Archdiocese of Cambrai]] — Became a metropolitan see in 1559.

*[[Diocese of Arras]]

*[[Diocese of Tournai]])

*other dioceses in present-day [[Belgium]], created in 1559.

===== Province of [[Mechelen]]/Malines =====

*Province and diocese of Mechelen created in 1559.

**(Diocese of [[Ypres]]) — Created 1559.

*other dioceses in present-day [[Belgium]] and the [[Netherlands]], created in 1559.

=== Province of [[Pisa]] ===

(Province created out of the Province of [[Rome]] in 1092: central and Southern Corsican sees belonged to this province)

*[[Diocese of Ajaccio]]

*Diocese of [[Aléria]] — The bishop resides in [[Cervione]]

*Diocese of Sagone — The bishop resides in [[Vico, Corse-du-Sud|Vico]]

=== Province of [[Rouen]] (Lugdunensis Secunda) ===

*[[Archdiocese of Rouen]]

*[[Diocese of Avranches]]

*[[Diocese of Bayeux]]

*[[Diocese of Coutances]]

*[[Diocese of Évreux]]

*[[Diocese of Lisieux]]

*[[Diocese of Sées]]

=== Province of [[Tours]] (Lugdunensis Tertia) ===

*[[Archdiocese of Tours]]

*[[Diocese of Angers]]

*[[Diocese of Le Mans]]

*[[Diocese of Nantes]]

Some dioceses of this province were part, in the 9th and 10th centuries, of an autonomous but

===== Short-lived Province of [[Dol-de-Bretagne|Dol]] =====

*Diocese of Dol — Created 6th century, was an autonomous archbishopric for about 1 1/2 centuries after the mid 9th century.

*Diocese of [[Quimper]] — Created 6th century.

*Diocese of [[Rennes]] — Out of which a short-lived diocese of [[Redon, Ille-et-Vilaine|Redon]] in the 15th century.

*Diocese of [[Saint-Brieuc]] — Created 6th century.

*Diocese of [[Saint-Malo]] — Moved to Saint-Malo in the 12th century (former see was in [[Alet]]).

*Diocese of [[Saint-Pol-de-Léon]] — Created 6th century.

*Diocese of [[Tréguier]] — Created 6th century.

*Diocese of [[Vannes]]

=== Province of [[Sens]] (Lugdunensis Quarta) ===

*[[Archdiocese of Sens]]

*[[Diocese of Auxerre]]

*[[Diocese of Nevers]]

*[[Diocese of Troyes]]

out of which (1622):

===== Province of [[Paris]] =====

*[[Archdiocese of Paris]] — Became a metropolitan see in 1622.

*[[Diocese of Chartres]], out of which:

**[[Diocese of Blois]] — Created 1697.

*[[Diocese of Meaux]]

*[[Diocese of Orléans]]

=== Province of [[Moûtiers|Tarentaise]] (Alpes Grææ et Pœninæ) ===

*[[Archdiocese of Tarentaise]] — Its see was in [[Moûtiers]].

*other sees in present-day Italy ([[Aosta]]) and Switzerland ([[Sion, Switzerland|Sion]]).

=== Province of [[Trier]] (Belgica Prima) ===

*([[Archdiocese of Trier]])

*[[Diocese of Metz]]

*[[Diocese of Toul]], out of which:

**Diocese of [[Nancy]] — Created 1777.

**Diocese of [[Saint-Dié]] — Created 1777.

*[[Diocese of Verdun]]

=== Province of [[Vienne, Isère|Vienne]] (Viennensis Prima) ===

*[[Archdiocese of Vienne]]

*[[Diocese of Die]]

*[[Diocese of Geneva]]) — In the 16th century, following the Reformation, the see was moved to [[Annecy]] but kept its name.

*[[Diocese of Grenoble]]

*[[Diocese of Maurienne]]

*[[Diocese of Valence]]

*[[Diocese of Viviers]]

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