A Characterization of Mason’s Synthesis
Charakteryzacja syntezy mularskiej
II Polskie Warsztaty Logiczno – Filozoficzne
Toruń, 12. IX 1996 r.
JERZY PERZANOWSKI
0. Cartesian Paradigm of Knowledge, in Leibniz’s words:
Duae sunt methodi, synthetica seu per artem combinatoriam, et analytica…. [cf. L. Couturat “La Logique de Leibniz”, Math. I, 26 c]
Forma sive ordo…consistet in conjunctione duarum maximarum invendi artium, Analyticae et Combinatoriae… [cf. Foucher de Careil, VII, 173]
The idea has been taken seriously in science, in particular in chemistry. Observe lack of an appropriate philosophical theory, general enough and commonly accepted.
1. Three approches towards a general theory of analysis and synthesis.
The most general theory of analysis and synthesis is combination ontology: GAS = CO.
I Order approach (natural and quite abstract).
I.1 Ontological spaces.
Let OB (the family of all items (objects)) be given. The universe of discourse UÍOB.
The space of analysis: áU, <ñ where < is the relation to be simpler than;
The space of synthesis: áU, Ìñ where Ì is the relation to be a component of,
The space of analysis and synthesis: áU, <, Ìñ
Questions: Axioms for < ? Axioms for Ì ? Axioms connecting < with Ì ? Etc.
I.2 Simples and maximals. Supports and substance.
Analytical simples: sa(x) means x is simple with respect to <
Synthetical simples: ss(x) means x is simple with respect to Ì
Simples: s(x) means x is simple in an appropriate sense
(1) At least five kinds of simples have to be distinguished.
Maximals ("possible worlds") — suitable simples with respect to appropriate converse relations, i. e., <-1 or Ì-1.
Let (x] denote respective (analytical or synthetical) order ideal generated by x.
Substance: analytical – the family of all simples of an analysis – Sa, synthetical – the family of all simples of a synthesis – Ss, in general – the family of all simples in general – S.
Substance (or support) of x
Analytical: Sa(x) := SaÇ(x]
Synthetical: Ss(x) := SsÇ(x]
General: S(x) := SÇ(x].
Notice quite a lot of questions concerning use of substance in synthesis and analysis. It plays, in particular, a very fundamental role in the case of founded (combination) ontologies.
Order approach offers an external description of synthesis. It doesn't explain its mechanism.
II Operator approach.
It is based on investigation of two operators: analyser α and synthetiser σ.
Analyser α produces for a given x the family of all its pieces (parts); whereas synthetiser σ collects for any x the family of all objects which can be synthetised from x (its substance):
α(x) := {y: y is obtained from x by α}
Q. Is α(x) the family of all parts of x ?
σ(x) := {y: y can be obtained by a combination involving x, or S(x)}.
Many questions concerning α and σ. One of them is, in fact, the topic of the present talk.
Operator approach gives also an external (and extensional) description of synthesis. It opens, however, a way to its internal description.
III Internal, or modal, approach.
To describe (at least necessary) conditions of a successful synthesis it is convenient to use two basic ontological modalities: making possible – MP( , ), and making impossible – MI( , ). It is a rather complex story, for another occasion.
Let me mention here only axioms of reasonable syntheses, which connects the second and the third approaches:
(RS1) yÎσ(x) ® MP(x, y)
(RS2) yÎσ(x) ® ¬MI(x, y).
2. Boolean Mason’s Syntheses.
Boolean syntheses are connected with ontology of the realm of Mind. They are deeply combinatorial, with set-theoretical representation:
(BR) x<y iff [x)Í[y).
On the other hand, a typical mason's constructions enlarge themselves by using more and more bricks.
Bricks of Nature are simples. Connecting now Boolean and mason's ideas together we obtain the following principles of Boolean Mason's Representation:
Let R denote one of two basic ontological relations, either <
or Ì.
(BMR1) xRy ® S(x)ÍS(y)
(BMR2) S(x)ÍS(y) ® xRy
(BMR) xRy « S(x)ÍS(y).
Boolean mason's syntheses satisfy, by definition, at least one of the above conditions.
Q. Find reasonable characterization of Boolean mason's syntheses.
3. An outline of an appropriate fragment of the operator theory of synthesis.
3.1 Global conditions.
DF. σ(x) := σ(S(x)). For XÍOB: σ(X) = σ(È{S(x): xÎX}).
Combinations are made up of their stuff.
(T) R is transitive.
(AS) R is asymetric: xRyÙyRx ® x=y
(REG) ƹXÍS ® σ(X) = {C(y): S(y)ÍX}
Regularity: Synthesis based on X produces combinations built up from X.
(MON) XÍYÍS ® σ(X)Íσ(Y)
Monotonicity: More material more products.
(COMP) C(x) ® S(x) = È{S(y): yRx}
Compactness: Compact synthesis uses only simples of its com ponents.
(SE) yÎσ(x)ÙxRy ® x=y
Economy: Bigger are not synthetizable from smaller.
(ES) S(x)=S(y) ® x=y
Extensionality: Extensional combinations are fully determined by their stuff.
3.2 Locality conditions
(SL) xÎσ(x), i. e., xÎσ(S(x))
x is synthetizable from its own substance. Weak form:
(wSL) C(x) ® xÎσ(x)
The same, under proviso that x is a complex.
3.3 Axioms for fusion
(F0) °: áx, yñ ® x°y
Fusion operator is a function
(F1) xR(x°y), yR(x°y)
(F2) xRy « (x°y)=y
(F3) S(x°y)=S(x)ÈS(y). In a weaker form:
(F3w) S(x°y)ÍS(x)ÈS(y)
3.4 Observations
(2) REG _ wSL
(3) F1, BMR1 _ F3 « F3w
(4) T _ BMR1
(5) F2, F3 _ BMR1
(6) ES, BMR1, REG _ SE
In a Boolean semimason’s synthesis extensionality plus regularity imply economy.
(7) AS, BMR, REG _ SE
3.5 Characterizations
(8) F0-F3, wSL, SE _ BMR
(9) F0-F3, ES _ BMR
(10) F0-F3, REG, AS _ BMR « SE
In regular and asymetric domains with fusion boolean mason’s syntheses are just economical ones.
(11) F0-F3, AS _BMR « ES
In asymetric domains with fusion boolean mason’s syntheses are just extensional ones.
JERZY PERZANOWSKI
