2. The Universe in Mereology with Ontology
The concept of the universe which Sobociński explained in a letter to Bocheński was already introduced by Leśniewski in his early mereology, where he put forward the following definition
I use the expression ’universe’ to denote the class of objects.
and proved theorems on the existence and uniqueness of the universe:
Some object is the class of non-contradictory objects. [...]
The class of non-contradictory objects is the universe. [...]
If P is the universe, and is the universe, then P is .
] (159–160), [5
] (L2: 31–32)
The question of the provability of theorems XLIII and XLIV requires a commentary on the ontological commitment of mereology: if it requires that the domain of “objects” should be nonempty. In the proofs of XLIII and XLIV the existential assumption is used that there is at least one “object” (or “non-contradictory object”(there is no explicit definition of a non-contradictory object but certainly it is dependent on the notion of an object)) (comments on provability of XLIII and XLIV are to be found in [6
] (128–129). However, we would not agree with the opinion that including XLIII and XLIV as theses shows that “Leśniewski was not clear as to the logical foundations of his system”. We would rather say that he simply changed the opinion about the ontological commitment of his theory). Indeed, Leśniewski in his early studies believed that the sentence “no object contains contradictions” is true and may be proved ([4
] (46) ([5
] (L1: 226))). If he could use a strong interpretation of universal negative sentences (as he declared in [4
] (231), [5
] (L3: 264)), it follows that there exists at least one object (non-contradictory object) (theorem I “No object is a part of itself.” [4
] (131) ([5
] (L2: 9)) of early mereology already implies the existence of an object). In his later works, however, Leśniewski changed his opinion on the matter and decided not to assert conclusively whether any objects exist at all [4
] (232) ([5
] (L3: 265)). Ontology with mereology in their later version have models with an empty domain of individuals where formulas XLIII and XLIV are not true (in this sense Leśniewski’s system is not ontologically committed to any object (we follow Urbaniak referring to [7
Sobociński essentially took over the notion of the universe from Leśniewski, but did not formulate any existential theses about it. Perhaps he assumed the later version of ontology with mereology. In his correspondence with Bocheński, he included many more theses about the universe than Leśniewski did, considering them interesting for philosophical reasons.
We will reconstruct Sobociński’s exposition in Leśniewski’s assumed system.
We expound Sobociński’s approach in mereology based on ontology using the same method as the one employed by Sobociński himself in [8
The assumed theory is expressed in the first-order language, whose vocabulary comprises: individual variables:
; name-forming functor
); inherence two-place predicate ε
); logical connectives:
and parentheses (, ). The terms of our language are individual variables and expressions
, where τ
is a term. Atomic formulas are of the shape
, where τ
are terms. Other formulas are built in a standard way. (In the original manuscript Sobociński used the style of notation from Principia Mathematica
. Like Leśniewski, he applied two types of variables A
(which are of the same sort). The same notation is used in [8
], with one exception: in the manuscript there is
(from the Polish “część” meaning “part”). Cf. also [9
Mereology with ontology OML is characterized by:
|-||theorems of first-order logic (QL) |
|-||axiom of ontology|
|-||axioms of mereology|
|()||(ingredient, in the manuscript el)|
(We delete the redundant part of the conjunction occurring on the right side of the definition of
We use also the following definitions:
| || (exterior, in the manuscript zw)|
| ||(sum of exterior objects)|
The definitions listed above are special cases of three definitional schemata, which enable us to introduce predicates (symbolized by F
), function constants (f
) and individual constants (n
contain the same free variables as the left sides of the equivalences.) (cf. [10
Primitive rules of OML are MP: and Gen: .
Let us now in our notation retype formulas listed by Sobociński with his original comments (we also change notation of symbols in the quoted commentaries).
| ||(x is the Universe (objects).)|
| ||(The Universe is not a part of anything.)|
| ||(If x is the Universe and x is an element [ingredient of] (B), then B is the Universe.)|
| ||(If some x is W, then any Kl(x) is W.)|
| ||(If , then Kl(x) is identical with W.)|
| ||(Definition of W using the term “pt”.)|
| ||(If every object is ing(x), then .)|
| ||(Definition of W using the term “ing”.)|
| ||(Nothing is exterior to W.)|
| ||(Definition of W using “extr”.)|
| ||(Definition of W using “+”; addition of anything to W is not possible.)|
In our derivations of formulas written down by Sobociński we will use the following OML theses:
(T12 expresses the Weak Supplementation Principle
accepted by Simons in [11
] (p. 28) as a mereological axiom. For possible connections between this principle and other mereological assumptions cf. [3
] (pp. 71–72).)
Now we are immediately able to notice that the definition of the universe (W) is equivalent to def-n for W because of ().
Actually, formula brings problems because:
added to OML causes a contradiction.
From we have . Because the OML thesis is , we get and with : . But the OML thesis is .
Perhaps Sobociński’s original comment to : “The Universe is not a part of anything” should be understood as weaker than but only as the implication
Formula is derivable in OML+W and the same is to be said about other formulas listed by Sobociński:
S1 and – are theses of OML extended by W.
We formulate the above fact giving the following proofs:
As we have said, Leśniewski’s ontology has interpretations in an empty set of individuals and this is not changed in the case of OML. This is why the counterparts of Leśniewski’s theorems XLIII and XLIV are not theses of OML+W but only their weaker versions:
It is derivable in OML+W that
Let us sketch the following model. We take a set of individuals D. The power set of D is a domain of a valuation of individual variables, is a semantical counterpart of the inherence predicate - it is a certain set of order pairs, where the first element of every pair is a singleton made of an inividual and the second element is any of its supersets; is an operation which for every singleton assigns a set of all parts of the element of this singleton. We can sketch the following model for which falsifies formula : , such that . In such a model all axioms of OML are true and .
However, we may easily obtain the counterpart of Theorem XLV on the uniqueness of the universe:
In OML+W the following formula is derivable
Sobociński himself formulated theses about the universe also in [8
]. He noted that everything which is exterior to W
is a contradictory object (in the sense of ⋀) and that universe W
by itself corresponds to a Boolean-algebraic unit element.
Let us add further theses expressed by Sobociński:
In OML+W the following formulas are derivable
|S12.||(cf.  (221))|
|S14.|| ( (93, A18, A19))|
The universe may possess more interesting properties in atomistic mereology which we obtain from OML by adding the following axiom:
and the definition of an atom:
| ||(x is an atom if every of its ingredient is identical with x)|
(We can use also more intuitive definition: (x is an atom if x is an object which does not have parts).)
OML+ theses are
|S15at.|| (T11, T8, , ) || |
|S16at.|| ||(cf.  (V2, 96))|
In other words, if there is at least one object in the atomic universe, everything which is the universe (or, should we say, ‘universal’) is a compound of atoms. Moreover, the universe is identical with the mereological class of all atoms.
Regardless of how the universal set of individuals is structured—moving still within OML—we may identify yet another feature of the mereological class of all objects. As it can be demonstrated, in OML, the following formula is the thesis:
To prove (***) we use , , and ; the part of predecessor is essential in a proof of ← (from and we get: and from we have ).
Finally we note:
Formula follows from (***) in OML+.
The formula (***) says that for every z which fulfills we can consider a ‘local’ (restricted to z) universe which is a mereological class of all z with the same property as is expressed for W in .
3. Universes in ZFM
Sobociński was convinced about the advantage of mereology over Zermelo’s set theory in application to philosophical issues. He expressed this conviction in his letter to Bocheński giving a theological example:
If somebody takes a position of Zermelo’s set theory, he can draw conclusions that are grossly in relation to theological opinions, eg. we assume that exists, and so the object exists [...] and this object , ect. , and they are all different, concretely existent objects! [...] In mereology it is not the case, because , and so it is only a different way of speaking and this is always permitted [...] Pay attention to the consequences of Zermelo’s system: before the creation of anything, there have been existent already an infinite amount of other objects !!!
The question which we want to put now is: how does this preference occur when the philosophical notion of the universe is considered? In other words: in which sense does the notion called by Sobociński “the Universe” (we would say: the world, totum) identified with W
in OML have more significant philosophical content than its set theoretical counterpart? We will analyze this issue in the frame of a richer system than OML, which gives the possibility of speaking about both types of multitudes: mereological collections and distributive sets—in the ZFM theory proposed by A. Pietruszczak ([3
] (pp. 172–181)).
ZFM is expressed in a first order-language with the following primitive symbols: Z
), ∈ (for being an element
), = (first-order identity) and ⊏— symbol for a part relation. ZFM is built on first-order predicate logic with identity with proper axioms of Zermelo-Fraenkel and the following axioms for ⊏:
The idea of the interpretation of elementary mereology in the set theoretical frame is obviously realizable because of Tarski’s well-known observation concerning close connection between the so-called mereological structures (which are models of elementary mereology) and complete Boolean algebras (an extensive description of this topic is given e.g., in [3
] (especially Chapter 3 (pp. 91–107))).
Our aim will be now to interpret the formalism of Sobociński in ZFM and to reconsider his definition of the universe.
We take mereology expressed in a slightly different language than OML. We use a first-order language with two primitive predicates ε and . The second one may be understood in OML as a part relation by:
We call this version of mereology OML
and characterize it by all theorems of first-order logic (QL), specific axiom of ontology
and the following counterparts of
We accept all counterparts of the OML definitions mentioned above.
Primitive rules are as of OML.
Actually, we want to define in ZFM predicates ε , and the notion of the universe that depends on them.
We start with the extension of ZFM by the following equivalence introducing predicate M
Predicate M is applied to every object z which is also a set, every mereological sum of each its subset is an element of z, every part of every element of z is an element of z.
In ZFM+M it is derivable that .
From axioms of ZF we get:
. We name this set ∅.
|∅ fulfills M because and .|
Let us fix any element fulfilling M.
Depending on this choice we define predicate
We also know that , because .
We take two axioms more:
We consider an interpretation function of the OML
language in a fragment of the ZFM+
language which we name
(we follow [13
] (pp. 61–65)). For every formula A
of the OML
language we define formula
belonging to the ZFM+
language in the following way: (i
) every subformula of A
of the shape
we retype with a modification, respectively:
) every subformula B
we retype with prefix:
Let us take the name ZFM for the considered extension of ZFM.
Now we can observe that
For every axiom A of OML ZFM.
To prove we need only (ε). and are derivable using (), and . To prove and we use the following: (here we use essentially the second and third part of the conjunction occurring on the right side of equivalence M).
Because of the interpretation theorem [13
] (pp. 62–63), for any chosen
we can speak about theory
) which consists of all
interpretations of theorems of OML
. Of course:
Let us now come back to our notion of the universe considered by Sobociński.
We introduce constant
representing the universe dependent on
We take the symbol ⨆:
. The abstract operator is a metatheoretical symbol used just as in [3
] (p. 175).
Now we can speak about different ‘Universes’, depending on the chosen . Remember that in OML we have already considered ’local’ universes which fulfilled condition (cf. (***), Fact 7)). Now every universe is ‘global’ and we could speak about ‘local’ universes which are certain subsets of .
Let us give selected examples of chosen z.
Example 1. At first we consider an extension of
ZFM : the Unitary Theory of Individuals and Sets (UTIS) described in  (pp. 172–181).
In UTIS ur-elements called individuals
are considered in the following sense:
To get UTIS from ZFM we add two axioms concerning the existence of individuals which form a set:
From the extensionality axiom we know that
and we name the set of all individuals
. The set
. We can prove in UTIS both
] (FT’, 181)) and
Now we choose and from () and () we obtain and . Just because and , we also know that . By the way, although is composed of individuals, they do not need to be atoms.
Let us stay in ZFM and take . We know that . Now and
In ZFM we can prove the existence of the set which fulfills M. Now we choose .
In this case: ).
As we wrote at the beginning of our article, Sobociński claimed that the mereological tools are more suitable for philosophical investigations than set theoretical ones. Actually, the connotations linked with the term “the Universe” and expressed in quoted theses are not dependent on some specific properties of individuals or their mereological whole. As it can be seen their proofs may be presented using only two steps. In the first step we use the fact that the mereological class of all objects is an object (if there is at least one object). Then we take and get implications with consequents of the same structure as the appropriate theorems of Sobociński, with a restricted quantification to z and with antecedent . Now we obtain — via dictum de omni and taking .
ZFM gave us the possibility of looking at Sobociński’s approach from a wider perspective, but also showed that the questioned philosophical expectations linked with would be too high. Although we could find the intended interpretation of the notion of W described in Example 1, we found also some undesirable cases: in Example 2 the universe is empty and in Example 3 the universe consists of distributive sets. After all, Example 1 also is far insufficient to realize the idea of ‘the Universe created’ by expressed in Sobociński’s quoted reflection. In this case, would need to be singled out from the set of all individuals and to stay in some causal relation to other individuals. The given characteristics of W of course does not depend on any such a construction and can be treated at most as the starting point of next philosophical investigation.