1 The Bayes theorem published posthumously as the work of Rev. Thomas Bayes (1701/2-1761) in `Essay Towards Solving a Problem in the Doctrine of Chances' (1764) rediscovered by Lagrange, provides a foundation for some areas of Arti cial Intelligence like Bayesian Reasoning, Bayesian Filtering etc. It had been reformulated in logical terms by Jan Lukasiewicz (1913). Recently, an abstract version couched in mereological terms was formulated and a strengthening of it appeared derived from the Stone representation theorem for complete Boolean algebras. It is our aim to comprehensively present those approaches with emphasis on the abstract setting of mass assignments on mereological universes endiwed with rough inclusions induced by masses of things.
Given a probability distribution on a space of events (cf.[5]) one de nes the conditional probability P (EjH) of the event E modulo the event H as P (EjH) = P P(E(H\H)) . From this the Bayes theorem follows in the presently used form: P (EjH) =
P (HjE) P (E)
P (H)
P (EjH) =
P (HjE) P (E)
Pk
i=1 P (HjGi) P (Gi)
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Commons License Attribution 4.0 International (CC BY 4.0).
(
Jan Lukasiewicz [8] (for english transl. see [1]) considered the logical framework of a collection of inde nite unary formulas over a nite universe of things . For a formula (x) 2 and a thing ! 2 , one says that the thing ! satises the formula (x), in symbols: ! j= (x) when after substitution of ! for all symbols in (x) equiform with x, the formula (x=!) is true (we do not discuss here the criterion of truth assuming it is set).
For a formula (x) 2 , Lukasiewicz had de ned the truth value w:
w( (x)) = jf! 2
j j
: ! j= (x)gj ;
(
I. ( = 0) , [w(a) = 0].
II. ( = 1) , [w(a) = 1].
III. ( ) ) ) [w( ) + w((: ) ^ ) = w( )].
From postulates I, II and III, Lukasiewicz had derived, using the standard
propositional calculus derivations, formulas which turn out to be the basic
formulas of probability calculus like (the numeration is as in original Lukasiewicz
[8], cf. [1]):
(
(14) w1( ) = w( ). (15) w( ^ ) = w( ) w ( ) = w( ) w ( ).
The nal thesis is a rendering of the Bayes theorem as `a special theorem' (cf. [1] p. 31): w( m) w m ( ) . (22) (Wi i = 1) ^ (Wi;j i ^ j = 0) ) [w ( m) = P w( i) w i ( ) i
It is manifest that after substitution of probability of an event for weight of a formula, we obtain true formulas of probability calculus on nite probability spaces. We now proceed to the meerological setting in which we give an abstract formulation for both probability calculus and Lukasiewicz's logical scheme.
Mereology and approximate (rough) mereology The standard version of mereology had been proposed by Stanislaw Lesniewski [6]. The interested reader may as well consult in addition, e.g., Casati and Varzi [2], Pietruszczak [9], or, Polkowski [11].
A mereological universe is a pair (U; ) where U is a collection of things and is a binary relation of being a part of which should satisfy the conditions: 1. For each thing x, it is not true that (x; x). 2. For each triple x; y; z of things, if (x; y) and (y; z), then (x; z).
The relation of a part does induce the relation of an improper part (x; y),
de ned as:
(x; y) , ( (x; y) _ x = y):
(
On basis of the relation , the overlap relation is de ned:
Ov(x; y) , 9z: (z; x) ^ (z; y): (6) Helped by the relation of overlapping, we introduce the third postulate for our model of mereology:
3. For each pair x; y of things, if for each thing z such that (z; x) there exists a thing w such that (w; y) and Ov(z; w), then (x; y).
The important notion in the mereology scheme is that of a class of things; the class operator converts any non-empty collection of things into a thing. The assumption is that classes always exist; by 3. they are unique.
The notion of a class was de ned originally in De nition II in [7]: P is a class of objects a if and only if the following conditions are met: (i) P is an object; (ii) every a is an ingrediens of P ; (iii) for any Q, if Q is an ingrediens of the object P then some ingrediens of the object Q is an ingrediens of some a.
Let us notice that notions of a set and of a class have been the subject of a long going philosophical dispute (cf. Pietruszczak [9] for a discussion; we owe this work the translation into English of De nition II.)
For the universe (U; ) of things, we de ne the class V of all things:
V = Clsfx : x 2 U g:
We are now in a position to recall here two fusion operators due to Tarski [18]. These operators are the sum x + y and the product x y de ned as: and x + y = Clsfz : (z; x) _
(z; y)g x y = Clsfz : (z; x) ^ (z; y)g: (7) (8)
We introduce nally the mereological implication x ,! y, valued in things in the universe U and de ned as: x = V
x: x ,! y = x + y: (10) (11) (12) (13) dis(x; y) , :Ov(x; y):
y is de ned as follows: x y = Clsfz 2 U :
(z; x) ^ dis(z; y)g:
It is well-known (see [18]; cf. English transl. in [19]; cf. Pietruszczak [9]. Ch. III) that the mereological universe (U; ) with the universal thing V and operations +; ; carries the structure of a complete Boolean algebra without the zero element. However, contrary to some views, the two are not identical: mereology is de ned for individuals and they are de ned in the Ontology by Lesniewski, so one cannot extract solely mereology from his scheme.
The complement x to a thing x in the universe (U; ) is the di erence V x: The things x; y are disjoint, dis(x; y) in symbols, whenever there is no thing z such that (z; x) and (z; y) (a fortiori, the product of x and y is not de ned).
The implication x ,! y is declared true if and only if x + y = V . 2.1
Rough mereology Approximate mereology (rough mereology, fuzzi ed mereology) [14], [10], [20] takes part relations of mereology and extends them to relations of being a part to a degree . Formally, the rendering of those relations comes in the form of the relation of rough inclusion (see [15] where this notion was introduced); the relation takes as arguments things x; y in the universe U and a real number r 2 [0; 1]. The formula (x; y; r) reads: the thing x is a part of the thing y to a degree of at least r. The relation should obey the requirements: 4. (x; y; 1) , (x; y): 5. (x; y; 1) ) 8z 2 U: 8r 2 [0; 1]: [ (z; x; r) ) (z; y; r)]: 6. [ (x; y; r) ^ s < r] ) (x; y; s).
The reader will nd a discussion of rough inclusions in [10]. 3
Mass-based approximate mereology By a mass assignment on a universe (U; ) we understand a function m : U ! (0; 1] which is subjected to postulates [12]: 7. m( ) = 0, where denotes the empty thing, not in the universe U . 8. x = V , m(x) = 1. 9. x = , m(x) = 0.
Rough inclusions in mass-based mereological universe and the Bayes theorem We continue with the mereological universe (U; ), augmented with a mass assignment m. We de ne a rough inclusion U U [0; 1] with triples of the form (x; y; r), where x; y 2 U and r 2 [0; 1] as arguments: We de ne in addition the inclusion function 1 : U 2 ! [0; 1] returning for each pair of things x; y the maximal degree r such that (x; y; r) holds true, i.e., the maximal degree of inclusion of x into y: (x; y; r) , m(x y) m(y)
r: 1(x; y) = m(x y) m(y) : (14) (15)
The rough inclusion formulas.
24. (y; x) ) [ 1(x; y) = 1]. 25. [ 1(x; y) = 1] ) (y; x). 26. (y; x) , 1(y; x) = 1 , y ,! x. 27. [ (x; y; 1) ) 8z: 1(x; z) 1(y; z). 28. 1( y; x) = 1 1(y; x). 29. [m(x) + m( x y) = m(y)] ) (x; y). 30. 1(x; y) = m(x) 1(y;x) [the Bayes formula].
m(y) and the inclusion function 1 satisfy the following 31. 11((yx;;xy)) = mm((xy)) . (x; y) ) [ 1(y; x) = mm((xy)) ]: 11((zy;;yz)) =
11((zx;;xz)) .
The general form of the Bayes theorem is as follows. The notation +Y for a nite collection of things Y denotes the result of mereological addition of things in Y .
34. [(+i6=j yi yj = ) ^ (+iyi = V )] ) [ 1(z; x) = Pik=m1(mz)(yi1)(x;1z()x;yi) ].
We now pass to a discusssion of existence of a system fyig satisfying the premises of formula 34. We prove (ine ecively) the existence of such a system with use of the Stone duality theorem (Stone [17]). 5
We explore the fact that the mereological space (U; ) carries the structure of a complete Boolean algebra without the null element and we refer to the Stone Representation Theorem [17] for complete Boolean algebras. We recall that a lter on a Boolean algebra B is a collection F of elements of B with properties: (i) if x; y 2 F then x y 2 F ; (ii) if x 2 F and x y then y 2 F ; (iii) the null element not in F . A lter maximal with respect to containment in the collection of all lters is called an ultra lter. It is well-known that each ultra lter H has the following de ning properties (see [?]): (iv) for each thing x 2 U , either x 2 F or x 2 F ; (v) H is prime, i.e., if x + y 2 H then either x 2 H or y 2 H.
The Stone theorem states that a complete Boolean algebra B is isomorphic to the space of closed - and - open sets in a compact 0-dimensional Hausdor space. This space is the space of all ultra lters on B (the Stone space St(B)) and it is topologized by admitting sets S(x) for x 2 B as the closed-and-open base, where S(x) is the set of all ultra lters in St(B) which contain x.
It has been shown as the consequence to the Stone theorem cf. [13], that: In the mereological space (U; ), there exists a nite set of elements fx1; x2; : : : ; xkg for some k with the property that each thing x 2 U admits the representation x = +ix xi. The set fxi : i kg is called a base in the space (U; ).
One can produce an orthogonal base fyi : i kg by letting yi = xi Qj<i( xj ) for i k [13]; then:
yi yj = whenever i 6= j.
Accordingly, the rough inclusion m has by virtue of 13 the property, 35. For each thing x 2 U , m(x) = Pk
i=1 m(x yi).
In consequence, the mass-based inclusion function 1 acquires the form 1(x; z) =
Pi k m(x z yi)
Pi k m(x yi) : (16)
From (17), we obtain the ultimate form of the Bayes theorem by taking as the set Y in 34 any orthogonal base fyi : i kg and expressing terms for 1 as in (17). 6
Appendix. Topological notions used and some proofs . A topological space is a pair (U; ), where U is a set and is a family of subsets of U which is closed on arbitrary unions and nite intersections. It follows that both the empty set ; and the set U are members of , whose elements are called open sets. Complements to open sets are closed sets. An open covering of a space U is a family ! such that S ! = U . A topological space (U; ) is Hausdor when for each pair x; y of distinct elements of U there exist open disjoint sets X; Y such that x 2 X; y 2 Y . An open base for a topological soace is a family such that each non{empty open set X is a union of some sub{family X . A topological space (U; ) is zero-dimensional when there exists in it an open base consisting of sets which are closed as well. A topological space (U; ) is compact when each open covering contains a nite subfamily which is also a covering of U. A convenient paraphrase of this condition is couched in terms of closed sets: a family K of closed sets is nitely centered when each nite non-empty sub{family of K has a non{empty intersection; the compactness condition can be stated as follows: a topological space is compact when each nitely centered family of closed sets has a non{empty intersection. The reader will nd a detailed discussion of topological spaces in [3].
For a Boolean algebra (see Sikorski [16]) B, by a lter on B we understand a collection F of elements of B such that (i) if x; y 2 F then x y 2 F , (ii) if x 2 F and x y) then y 2 F , (iii) ; 2= F . For a mereological space (U; ), condition (ii) translates as (ii)' if x 2 F and (x; y) then y 2 F and condition (iii) translates as (iii)' 2= F . An ultra lter is a lter which is not contained properly in any other lter. A Boolean algebra B is complete when each subset C B has the least upper bound L, i.e. (i) x L for each x 2 C and (ii) if an element M satis es (i) then L M . It is well known that the mereological space (U; ) with mereological operations of sum, product and complement, the unit element V and augmented with is a complete Boolean algebra (see [18], [19]).
For a Boolean algebra B, the Stone space S(B) consists of ultra lters on B. The Stone topology st(B) on the Stone space S(b) is induced by the open base consisting of sets S(x) = fF : F an ultra lter on B and x 2 F g for all x 2 B.
The fundamental Stone theorem [17] states that Theorem 1 (M. Stone [17]). The Stone space S(B) with the Stone topology st(B) on a complete Boolean algebra B is a compact Hausdor zero-dimensional topological space.
We recall a proof, for completeness' sake.
By De nition of a lter S(x) \ S(y) = S(x y), hence, st(B) has properties of a base. Each set S(x) is clopen: S(x) = S(U ) n S( x). S(U ) is compact: let B, a collection of sets of the form of S(x) for x 2 W U , be nally centered, i.e., for each nite sub-collection X=fx1; x2; : : : ; xkg of W , there exists an ultra lter F with X F . Let us consider a set G = W [ fz 2 U : there exists x 2 W with (x; z)g. Then G extends to an ultra lter H and H 2 T B, i.e., S(U ) is compact. S(U ) is Hausdor : let F 6= G for ultra lters F; G. Assume, for the attention sake, that x 2 F n G for some thing x. Hence, x 2 G and F 2 S(x), G 2 S( x), and, S(x) \ S( x) = ;.
The Stone theorem implies that there exists a nite set = fx1; : : : ; xkg of elements of the space (U; ) with the property that S(U ) = SfS(xi) : i = 1; : : : ; kg. One proves that is a base in U in the sense that for each x 2 U , we have the representation for x in the form x = +ix xi; (18) where +i denotes the mereological sum of all elements xi f ori = 1; : : : ; k.
We recall a proof from [12]. Consider an arbitrary thing y with (y; x). Let F (y) be an ultra lter containing y; hence, x 2 F (y). Let xi 2 K be such that F (y) 2 S(xi). Then, y xi 6= and i 2 I(x). As (y xi; x xi), it follows (x; +ix xi) by M3. Contrariwise, assume for an arbitrary thing z that (z; +ix xi), hence, (z; x xi) for some i 2 I(x) and thus (z; x); by M3, (+ix xi; x) and nally x = +ix xi.
We let yi = xi (+j<iyj ; then, the formula x = +ix yi; (19) holds true (cf. [12]).
The collection fyi : i = 1; : : : ; kg is an orthogonal base in U . It serves as the collection of pairwise disjoint elements needed in the general Bayes formula. 7
We have presented a survey of some approaches to the Bayes theorem in frameworks of logic and mereology. An extended version will appear in [13]. 6. Lesniewski, S.: Foundations of the General Theory of Sets (in Polish). Moscow (1916)(cf. Surma et al. (eds.). Lesniewski, Stanislaw. Collected Works I, II. Springer Netherlands, Dordrecht (1992)). 7. Lesniewski, S.: On the foundations of mathematics (in Polish). Przeglad Filozo czny XXXI, pp 261-291 (1928) (cf. Surma et al. (eds.). Lesniewski, Stanislaw.
Collected Works I. Springer Netherlands, Dordrecht (1991)). 8. Lukasiewicz, J.: Die Logischen Grundlagen der Wahrscheinlichkeitsrechnung.
Krakow (1913). 9. Pietruszczak, A.: Metamereology. The Nicolaus Copernicus University Scienti c
Publishing House, Torun, Poland (2018). DOI: 10.12775/3961-4
10. Polkowski, L.: Approximate Reasoning with Parts. An Introduction to Rough
Mereology. Springer-Verlag, Berlin Heidelberg. ISRL vol. 20 (2011). DOI:
10.1007/978-3-642-22279-5
11. Polkowski, L.: Mereology in Engineering and Computer Science. In: Calosi, C.,
Graziani, P. (eds.): Mereology and the Sciences. Springer Synthese Library vol.
371, pp 217-292 (2015). DOI: 10.1007/978-3-319-05356-1
12. Polkowski L.: Introducing mass-based rough mereology in a mereological universe
with relations to fuzzy logics and a generalization of the Lukasiewicz logical
foundations of probability. Fundamenta Informaticae 166 (
Informaticae 172 (2020), in print.
14. Polkowski, L., Skowron, A.: Rough mereology: A new paradigm for
approximate reasoning. Int. J. Approx. Reasoning 15(
869, 1994 pp. 85-94. ISBN 3-540-58495-1. 16. Sikorski, R.: Boolean Algebras. Springer-Verlag, Berlin (1960). 17. Stone M.: The theory of representations for Boolean algebras. Trans. Amer. Math.
Soc. 40: 37-111 (1936). 18. Tarski, A.: Zur Grundlegen der Booleschen Algebra I. Fund. Math. 24, pp 177-198 (1935). 19. Tarski, A.: On the foundations of Boolean Algebra. In: Woodger, J. K. (ed.): Tarski, A. Logic, Semantics, Metamathematics. Papers from 1923 to 1938. Clarendon Press, Oxford UK (1956). 20. Varzi, A.: "Mereology", The Stanford Encyclopedia of Philosophy (Spring 2019 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/spr2019/entries/mereology/>.