On Porphyrian Tree
On Porphyrian Tree
On Porphyrian Tree
Janusz Kaczmarek
Abstract
The paper formalises the structure of Porphyrian tree (PTS) and introduces
operation of determination (D), which renders the Wolffs idea of generating
attributes of being by means of essentialia. The analysis of their mutual depen-
dencies has lead to the following conclusion: operation D is generated by PTS
and, conversely, some Porphyrian structure is determined by operation D.
Introduction
The paper has found its inspiration in the analyses of being as presented
by Porphyrius in his substance classification, the so-called Porphyrian tree.
Therein the contradictory properties divide kinds into species yielding at
the very bottom of classification the natural species as man in general etc.
I claim that the natural species are determined and characterised by both
essential and attributive properties, essentialia and attributes, respectively.
This makes us adherents of Wolffs view (compare Wolffs Philosophia
prima and Gilsons [3]) according to which the essence of being derives
from the properties that are:
1) mutually compatible (not contradictory);
2) independent, i.e. determined neither by essential nor any other
properties;
3) generating all attributes of a given being.
38 Janusz Kaczmarek
Remark 1. The function E0 satisfying (1E) has property 1/2 (in short:
E0 P1/2). Likewise in case of A0 and condition (1A), i.e. A0 P1/2.
STEP j + 1: (for j < k). Let us assume that step j is defined and it
yields m pair: <Ej,1 , Aj,1 >, . . . , <Ej,m ,Aj,m >. Let Tj+1 Tj . Consider
any non-empty set
EA(j) {<Ej,1 , Aj,1 >, . . . , <Ej,m , Aj,m >}
40 Janusz Kaczmarek
such that for any <Ei , Ai > EA(j) there exists a set EA(j +1)i composed
of p (p 1) pair
<Ej+1,1 , Aj+1,1 >, . . . , <Ej+1,p , Aj+1,p >
satisfying the conditions (4) (12). That is: for any n such that 1np:
(4) Ej+1,n : Tj+1 {0, 1/2, 1} and Aj+1,n : Tj+1 {0, 1/2, 1},
(5) Ej+1,n and Aj+1,n have the property P1/2,
(6) (Ej,i )1 01 (Ej+1,n )1 ,
(7) (Aj,i )1 01 (Aj+1,n )1 ,
(8) (Ej+1,n )1 01 (Aj+1,n )1 ,
For any n0 and n00 , 1 n0 p, 1 n00 p and n0 6= n00 :
(9) ((Ej+1,n0 )1 01 (Ej+1,n00 )1 or (Ej+1,n00 )1 01 (Ej+1,n0 )1) ,
(10) ((Aj+1,n0 )1 01 (Aj+1,n00 )1 or (Aj+1,n00 )1 01 (Aj+1,n0 )1 ),
(11) For any <E0 ,A0 > EA(j + 1)i and any <E,A> EA(i), i
j+1, such that <E,A>6=<E0 ,A0 > and any t T:
(11.1) t A1 ({0}) E1 ({0}) t 6 E01 ({0}),
(11.2) t A1 ({1}) E1 ({1}) t 6 E01 ({1}).
(12) (under the assumption given in (11)):
(12.1) t E1 ({0}) t 6 A01 ({0}) E01 ({0}),
(12.2) t E1 ({1}) t 6 A01 ({1}) E01 ({1}).
Definition 1. A structure satisfying the conditions given in steps from
0 to k is a Porphyrian Tree Structure, shortly: PTS structure.
Definition 2. Any pair <E, A> occurring in PTS will be called an
incomplete object. The set of all incomplete objects is denoted by PTS.
Definition 3. Let <E, A> be an incomplete object occurring in step j,
1 j k. If there exists no object <E0 , A0 > such that
E1 01 E01 and A1 01 A01
then <E, A> is called natural species of j-order.
Definition 4. Any object <E, A> occurring in step j is called
a j-order object (in short: <E, A> ORD(j).
Definition 5. A set of all natural species of j-order, for all j k, will
be referred to as the set of natural species and denoted by SN (species
naturale).
Definition 6. The set G = PTS SN is called a set of genera.
On the Porphyrian Tree Structure and an Operation of Determination 41
Definition 7. For a given objects <E, A> ORD(j) and <E0 , A0 >
ORD(j+1) such that
object <E, A> is called the nearest genus of <E0 , A0 > and <E0 , A0 > its
nearest species (compare some definitions in [5]).
Commentary. The construction in j+1 step may be described along
following lines: In step j there appear m pairs of which not all may be
chosen to build a set of objects in step j+1, i.e. pi new pairs are built
from pair i. All functions in j+1 step are defined on Tj+1 Tj . If we
assume that Tj+1 = Tj (=T), for j 0, the resulting construction may
be called L ukasiewiczs since he claimed that any property can be predi-
cated about any object (see [4], where he uses examples like the column
is sad). Conditions 6 and 7 guarantee that the essences (represented by
E) and attributes (represented by A) of beings on lower levels (obtained in
following steps) are richer in proper sense, respectively, and Condition 8
that essentialia generate themselves and some additional new properties.
Next, Conditions 9 and 10 assure that there are not two beings on the same
level such that all essential properties (attributes, respectively) of one are
essentialia (attributes) of the other. It means that on ontic side different
essences generate different properties. Condition 11 assures that if a prop-
erty t is essential for an object <E, A> obtained in step j than t is not
attributive (in proper sense) and Condition 12 warrants that an object <E,
A> received in step j does not possess attributes that would be essential
properties of an object <E0 , A0 >.
Remark 2. In our approach the domains of (functions) E and A ob-
tained in the successive steps of the construction should be narrower,
i.e. Tj+1 Tj and Tj+1 6= Tj (here we differ from L ukasiewicz). Ap-
parently the premise that any property can be predicated about an object
(for example that the man is even or that the man is not even) is not
proper. Similarly, if a plant in general has an indefinite property like four
leaves, then it does not seem that one can predicate four leaves about
a man. On the contrary, definite properties of genus (like being alive) are
the properties of all species (like a man in general or a horse in general)
which are under that genus.
42 Janusz Kaczmarek
2. Operation of Determination
2.1. Preliminaria
Let
T 6= , m(T) = 0
and, moreover, let us consider a set F of functions such that for any f F:
(2.1) f: T {0, 1/2, 1}
(2.2) f P1/2.
Let us denote: 3 = {0, 1/2, 1}.
3. Interdependencies
Theorem 3.1. (PTS structure fixes the operation of determination)
For any PTS structure, the set EA = {<ei , ai >:<ei , ai > PTS}
and the set E = {ei : where ei = id1 (<ei , ai >) for some <ei , ai > EA}
an operation D defined as
ai , if f = ei , for some ei E,
D(f ) =
f, oth.
is an operation of determination.
Proof. We have to show that D satisfies (T1) (T3) and (D). Proofs for
(T1) - (T3) are evident. Here let us present the proof for (D).
Let E = {e1 , . . .,en } be the set given like in the theorem. Let us
consider any ei E, t e1 T
i ({0, 1}) and g 3 . We will show that (D1)
1
holds. Let us assume that t ei ({0}) and t 6 g1 ({0}).
10 . If g 6 E, then t 6 (D(g))1 ({0}), because D(g) = g; thus (D1)
holds.
20 . If g E, then g = ej 6= ei (since t 6 g1 ({0})) for j {1, . . . , n}
{i}.
Let us additionally assume (by contradiction) that t (D(g))1 ({0}).
Then one of the following cases holds:
A) ei = id1 (<em ,am >) for some <em ,am > ORD(j1 ) and g =
id1 (<ep ,ap >) for some <ep ,ap > ORD(k) and k j1 ,
or
B) g = id1 (<em ,am >) for some <em ,am > ORD(j2 ) and ei =
id1 (<ep ,ap >) for some <ep ,ap > ORD(k) and k j2 .
Ad. A) We have: t (D(g))1 ({0}) g1 ({0}). Thus by (11.1) t 6
1
ei ({0}); contradiction.
Ad. B) If ei ORD(k) for k j2 , then the premise of (12.1) for ei
holds. We have: t e1 i ({0}), hence, in particular, for g:
t 6 (D(g))1 ({0}) g1 ({0}); contradiction.
The proof for (D2) goes analogously by (11.2) and (12.2). 2
Theorem 3.2. (An operation of determination fixes PTS structure)
Let E = {E1 , . . .,En } be a set of functions from T into {0, 1/2, 1}
fulfilling the conditions:
(E1) Ei P1/2, for i {1, . . .,n},
(E2) (E1 1
0 ({0}) E0 ({1})) 6= ,
44 Janusz Kaczmarek
4. Final remarks
The above considerations belong to the paradigm of analytic philosophy.
K. Fine (see [1] and [2]) is its another representant. Our proposal, as we
hope, is a method of presenting the well-known philosophical problem of
being and its constitutive elements (essentialia and attributives), by means
of formal tools. Therefore, we intend to treat this paper as a starting-point
of further investigations. Namely, the tools introduced here may be used
to:
1) formulate some ontological conclusions about essentialia and at-
tributives of incomplete objects,
2) build simple sentential language (with individual constants and
predicates) and a model based on PTS - structure for the language.
References
[1] Kit Fine, The Logic of Essence, J. Philos. Logic 24 (1995),
pp. 241273.
[2] Kit Fine, Semantics for the Logic of Essence, J. Philos. Logic
29 (2000), pp. 543584.
[3] Etienne Gilson, Letre et lessence, Librairie Philosophique
J. Vrin 1962.
[4] Jan L ukasiewicz, O zasadzie sprzecznosci u Arystotelesa.
Studium krytyczne, [On the Principle of Contradiction in Aristotle.
A Critical Study,] PAU Krakow 1910.
[5] Jan Salamucha, Z historii jednego wyrazu (istota), Tygodnik
Powszechny nr 7 (1946), pp. 34, (see also in his:) Wiedza i wiara.
46 Janusz Kaczmarek
Department of Logic
L
odz University
Kopcinskiego 16/18
90232 L odz
Poland
e-mail: kaczmarek@filozof.uni.lodz.pl