Reasons and Causes

(KrM) x  MA iff $y (xRy  y  A)

Now it is easy to repeat the usual results from modal logic with, however, a new ontological light put on them. Consider, for example, the axiom

(T) A  MA

As is well-known the axiom is characterized by the reflexivity of the alternativity relation R: “x xRx.

It is therefore characterized also by a rather intuitive formula:

(MPR) “x”z_ x MP(z,x)

which says that the universe contains only strongly coherent objects, i. e., that any object is made possible by all its parts, or that nothing introduces internal incoherences in objects under consideration.
This explains, or at least sheds light on, the meaning of the formula (T) which explicitly states that everything which holds must be possible, but implicitly that the ontological reason for this is an internal coherence of objects under con­sideration.

25.2 In the second option we deal with ontological frames instead of relational ones replacing the Kripkean condition (KrM) by the following ontological condition:

(OM) x  MA iff MP(x, A),
A is possible in x iff x makes A possible. [12]

To understand how the above combination semantics [13] works, observe that now the axiom (T) is connected with the condition

(MPR*) If x  A then MP(x, A), i. e.,   MP

saying that everything which verifies A makes it possible.

25.3 Observe that both procedures outlined above can be easily repeated for other ontological or metaphysical modalities, a fortiori makers, defining thus their conjugate logical modalities.

Ontological reasons

26. Ontology, we remember, concerns all possibilities. One of its very basic insights says that possibilities – as possibilities – must be motivated. They are possible, because something made them possible.
Making possible, MP, is therefore the proper operator to define ontological reasons:

(D8) x is an ontological reason for y, if x makes y possible: MP(x,y).

27. The ontological rationalism says that in the realm of ontology nothing is contingent, i. e., without reasons:

(OR) “x$y MP(y,x)

The question of foundation immediately arises: find, investigate, and explain reasons for such and such items. In particular: are these reasons internal or external? The simplest form of the principle of internal reasons says that any item contains its own reason:

(OIR) “x$y_ x MP(y,x)

Its special form is the ontological principle of consistency, or the ontological principle of coherence:

(OC) “x C(x)
Any object is ontologically coherent.

The principle is clearly stronger than the previous principle of ontological rationalism (OR).

28. To sum up, the present ontological framework is, as we saw, subtle enough to characterize ontological reasons.
Is this characterization a solution for our original question?

29. Certainly not. In general, causes are stronger than sufficient reasons which, in turn, are stronger than ontological reasons.
They are reasons (causes) for occurrence or existence in the world, for being a fact or a collection of facts. They are therefore metaphysical or physical items, not purely ontological.
Ontological reasons are sufficient only for being possibilities. Hence from the metaphysical point of view ontological reasons are necessary conditions, not sufficient ones.
To solve our problem we need metaphysical modalities, more specific and powerful than ontological modalities. We should therefore introduce suitable metaphy­sical makers for making facts, i. e., fact makers.

Facts as Complexes and Makers

30. Facts [14] are basic pieces of the world. They are strongly connected with events, existence and reality.
In the everyday way of speaking, a fact is something that has happened or been done, or something known to be true or accepted as true. It is therefore a piece of reality, of what exists or what is true.
Their notion has been thereby taken as basic notion of metaphysics by philosophers trying to be abstract but still close to common sense – Bertrand Russell [15] and Ludwig Wit­tgenstein. [16]

31. Russell never provided an exact definition of a fact. He was instead working with several examples, like “Socrates is dead”, looking rather for an explanation than for a strict, formal or at least preformal, theory.
Nevertheless, his ideas are clear and open for formal development.

31.1 In the Philosophy of Logical Atomism [17], Russell wrote “…When I speak of a fact ….I mean the kind of thing that makes a proposition true or false. …”.
Shortly speaking, facts are truth makers. Taking into account that: (1) the proper bearers of truth and falsity are propositions; and (2) making true is one of paradigmatic metaphysical modalities, we see that Russellian approach is both linguistically related and deeply modal. [18]
To formalize it, we need therefore a formal theory of propositions, which is rather easy to find, and a theory of making true and truth makers.
Let A, B, …. denote propositions. Next, following the approach to makers outlined previously, a metaphysical modality making true, MT, is introduced.
MT(x, A) means that x makes the proposition A either true or false. Now, it is quite easy to define Russellian facts. They are simply truth makers:

(D9) F(x) := $A MT(x, A).

31.2 Russell gave also another, more general, definition of facts: …I mean by “a fact” anything complex…. [19]
The definition is rather easy to formalize, for the notion of a complex is standard. Without going here into details [20] , let us assume CX is a predicate for complexes.

Now, x is a fact, if it is a complex:

(D10) F(x) := CX(x).

31.3 Both notions of a fact introduced above are clearly interconnected. This connection and notions involved deserve, for sure, further discussion.

32. Wittgenstein was, in fact, working along quite similar lines, in more developed and sophisticated ontology, however.

32.1 One of his approaches to facts, similar to the second approach of Russell, is encoded in a table of Tractarian categories.
Recall that in the Tractatus all complexes are – on the one hand – either possible (M) or impossible (¬M), and – on the other hand – either contingent (K) or noncontin­gent (¬K). Next, two sorts of contingencies (or situations) are distinguished: existing (or real), i. e. facts (F) and nonexisting, purely possible, i. e. counterfacts (-F). There are also two sorts of noncontingencies: necessities and inconsistencies, and two sorts of possibilities: contingencies and necessities.
This classification encodes, in fact, quite a lot of bits of information concerning the logic [21] of the above family of logical modalities. It includes, inter alia, the following definitions:

(D11) Kx := MxM¬x Contingency
x is contingent, if both x and ¬x are possible, or it is possible but not necessary [22]
Fx := Kxx Factuality
x is a fact, if x is contingent and it obtains
-Fx:= Kx¬x Counterfactuality
x is a counterfact, if x is contingent and it does not hold.

32.2 The above list of the Tractarian notions can be completed by two relevant notions resulting from the principle of the world’s completeness: Any contingent situation is either a fact or counterfact.

(D12) CFx := ¬-Fx
x can be a fact, if it is not a counterfact
C-Fx:= ¬Fx
x can be a counterfact, if it is not a fact.

32.3 Immediately from the above definitions we obtain

(14) Both facts and their negations (negative facts) are contingent and possible complexes which, respectively, obtain or do not obtain. I. e. Fx  Kx, Fx  Mx, Fx  M¬x, Fx  x; and -Fx  Kx, -Fx  Mx, -Fx  M¬x, -Fx  ¬x.

Next, by the usual principles of classical [23] modal logic we, among other things, have:

(15) -Fx  F¬x Counterfacts are negative facts,
i. e., x is a counterfact iff it is a fact that x does not hold.

(16) Fx  CFx Facts can be such,
i. e., x is a fact only if it can be a fact.

(17) -Fx  C-Fx Counterfacts can be such,
i. e., x is a counterfact only if it can be a counterfact.

We can also obtain the following characterization of facts and counterfacts:

(18) Fx  KxCFx
Facts are contingencies which can be facts.

(19) -Fx  KxC-Fx
Counterfacts are contingencies which can be counterfacts.

Finally notice that simply by Wittgenstein’s classification:

(20) Fx  Ex, -Fx  -Ex
Facts are existing complexes, counterfacts are non-existing complexes.

33. However, the main Wittgenstein’s achievement as regards facts is his original definition of a fact given in the thesis 2 of the Tractatus:

2 What is the case, a fact, is the existence of states of affairs.

A state of affairs is, in turn, defined in the subsequent thesis:

2.01 A state of affairs is a combination of objects (things).

Combinations are complexes of syntheses. On the other hand, things are analytical simples. Thus states of affairs, denoted by means of a unary functor SA, are com­binations of simples, which are such with respect to a given analysis.

The only point of Wittgenstein’s definition which still needs clarification is – in its context deeply modal – the noun “the existence”. Notice also a characteristic distribution of plural and singular forms in 2: a fact (singular) is connected with the existence (singular) of states of affairs (plural).

33.1 The first formalization of the above definition is rather straightforward.
Variables now are used in the most general way; they denote any item. SA(y) means that y is a state of affairs, and E(y) – that y exists.

(D14) F(x) := $y (SA(y)  x=E(y))
x is a fact iff for some state of affairs, x is its existence.

Notice that all basic components of (D14) are complexes. It is clear for y,
F(x), SA(y) and E(y). Also x, which can a priori be a simple, by the previous discussion of §32 is a complex. Therefore, in (D14) we must understand quantifiers as propositional operators and use a characteristic Tractarian sentential identity. [24]
Next, (D14) made quite clear a need for a plural form used in 2, for quan­tification, which is implicit in 2 but explicit in (D14), obviously recurs to plurality.

33.2 However, ambiguity concerning existence still remains. Indeed, the existence of y, i. e. E(y), can be understand either as a state of affairs which exists, i. e. existing combination of simples, or as the source or reason of this existence.
Taking into account the results of another Tractarian approach, outlined previously, we can express the first alternative as follows:

(D14.1) F(x) := $y(SA(y)  x=Kyy).

The second alternative leads, however, to a rather unusual formalization by means of the most typical metaphysical modality making existent, ME:

(D14.2) F(x) := $y(SA(y)  ME(x,y)).

Facts here are existence makers, i. e. reasons for existence or rather causes of existence.
Notice the similarity of this approach to the first approach of Russell which, as we observed, is also metaphysical but much more linguistically oriented. Definitions (D9) and (D14.2) say, in fact, the same: facts are makers of something in the world. The only difference is that where Russell is using propositions, Wittgenstein uses states of affairs.
34. To resume, the Russellian and Wittgensteinian view of facts points out that they are basic furniture of the world and next tries to pick up effects of facts charac­terizing them as makers of, respectively, either proposition’s truth and falsity or the existence of (existing) states of affairs.

Making Facts

35. However, from a metaphysical point of view the reverse question concerning reasons for being a fact is more fundamental.
To study it we must first introduce a metaphysical modality making fact, MF, and next define facts as items made to be facts by something.

(D15) F(x) := $y MF(y,x)
x is a fact, if some item makes it to be such.

Clearly, the present approach needs an axiomatization of the modality involved: MF.

36. To stress the point again, Russell and Wittgenstein were dealing with effects of facts, whereas my proposal is to study facts as results.
The proposal is obviously more ontological in spirit, for it follows the ontological question applied to facts: How and why facts are possible?

37. Now we are going to discussion of the most natural candidates for axiomatiza­tion, first, of making facts itself and, next, with results of §§16 and 17 in mind, of foundation and causation treated as successive strenghtenings of MF.

37.1 Ask first about connection between making fact and the basic ontological modality making possible.
Obviously, factuality implies possibility, i. e., possibility is necessary, but usually not sufficient, for existence (being real or being a fact). Hence we should accept the following axiom of facts’ possibility:

(FP) MF(x,y)  MP(x,y)
If x makes y a fact, then x makes y possible.

The axiom connects in the most obvious way our present metaphysical modality with the basic ontological modality discussed previously and offers in such a way quite a lot of consequences to making facts.

37.2 The following axiom of soundness is also the most natural to state:

(S) MF(x,y)  F(y)
Making facts produces facts, or what is made to be a fact is indeed a fact.

We will see that (S) implies a quite nice characterization of items resulting by making facts.

37.3 Consider now the axiom of transitivity:

(T) MF(x,y)MF(y,z)  MF(x,z)
If x makes y a fact which, in turn, makes z a fact, then x itself makes z a fact.

It sounds quite intuitive. Taking, however, into account the results of §16.2 we must distinguish several cases, in particular the case when MF is global and the case when MF is local.
On the other hand, by (S), two latter variables occurring in the axiom (T), i. e. y and z, must be facts and only the first variable – x need not be such. Hence all but at most one elements of transitive MF – chains must be facts.
Transitivity thereby seems to be reasonable only in the realm of facts.

37.4 Pass next to the axiom of irreflexivity. In §16.1, we collect arguments that MF is irreflexive:

(IR) ¬MF(x,x)

at least in the case, when making facts means to cause them.

37.5 On the other hand, recalling the discussion of §16.3 we should at least consider the axiom of asymmetry:

(AS) MF(x,y)  ¬MF(y,x)

as characteristic for causation; and accept the axiom of nonemptiness as a guarantee that MF is not trivial:

(NE) $x$y MF(x,y).

37.6 Turn now to our starting problem concerning the claim of the ontological rationalism (Nihil sine ratione) and the claim of the metaphysical determinism (Nihil sine causa).
Both principles, among other things, answer in the positive questions of closure: Is the world closed with respect to reasons or, respectively, causes? In the other words, is the world closed on foundation or, respectively, causation?
Formalizing appropriate components of the above principles we obtain the axiom of metaphysical rationalism:

(MR) “x(F(x)  $y MF(y,x))
Being a fact is not without reason, i. e., for any fact there is an item which makes it a fact.

Finally, the following axiom of closure says that only facts make facts:

(C) MF(x,y)  F(x).

Foundation and Causation

38. In the previous chapter ten possible axioms were introduced. Three of them, i. e. FP, S and NE, are very general and characteristic for the pure theory of making facts; the next three, i. e. T, IR and AS, are well-known general relational conditions; whereas the last two, i. e. MR and C, express our fundamental ideas concerning the domain of facts and some basic relations governing it.
Through combination of these axioms we can obtain a rich family of appropriate theories. [25] Five of them deserve a special attention:

PMF – the pure theory of making facts, axiomatized by FP, S and NE;
PTF – the pure theory of foundation, which is a strengthening of PMF obtained from it by assuming in addition the axiom of metaphysical rationalism MR;
TF – the theory of foundation, which extends PTF by acceptance of T;
WTC – the weak theory of causation, which is obtained from TF by adding axioms IR and C; and finally
TC – the theory of causation, resulting from WTC by replacing IR by AS.

Taking into account that

(21) AS  IR – asymmetry implies irreflexivity;

we see that each successive theory is stronger than previous ones.
In particular, the theory TC is the strongest theory in our framework.

39. Where is the cutting point differentiating between foundation and causation, or between reasons and causes?

39.1 Notice, first, the points which are common. Both relations are here considered as a par­ticularization of the more general relation MF. This is rather clear in the case of causes which, for sure, work in the domain of facts, but seems in general to be doubtful in the case of reasons.
We like, however, to consider reasons of facts and for them the use of MF is rather natural. Therefore, MF is a proper relation to be common base of making fact, foundation and causation.
In both cases, we also accept the rationalistic principle MR,because from the very beginning we are interested in comparing ontological rationalism to metaphysical determinism.
Thus PTF is the common base for the theory of foundation and the theory of causation.

39.2In accordance with the Leibnizian idea of ontology as more general than metaphysics we, in fact, consider causation as strengthening of foundation. Thus the points of difference, we are looking for, are these axioms which are present in both theories of causation, i. e. WTC and TC, but absent in the strongest considered theory of foundation, i. e., TF.
Two axioms are particularly relevant here: IR and C. Although both are accepted for causation, they are refuted for foundation.
Indeed, we can easily imagine facts which are ontological reasons for themselves. Nothing similar, however, happens for (meta)physical causes, for causation is intimately connected with time succession, hence natural causes must occur earlier than their effects.
The axiom of closure is even more important on this matter. It expresses the fundamental idea concerning the (physical) world, that the world is causally closed: effects of facts are facts, which is pretty clear, and facts makers must be facts.

39.3 Answering the question which is behind the title of this paper: Making a fact is to be its reason. Causes are fact makers which are facts.

A Few Observations and Theorems

40. Consider first the basic theory PMF.

40.1 The axiom FP connects making fact with making possible, theory of which is known to be fruitful. [26] Therefore, it provides PMF with a number of results, including applications outlined previously in §§ 22 – 29.
Due to the limitation of space, I skip the discussion here passing it to another occasion.

40.2 Among immediate consequences of the axiom S, you can find everything which for MF follows from the previous discussion of facts in §32. Inter alia:

(22) Only contingent items can be made facts: MF(x,y)  Ky.

(23) Only possible items can be made facts: MF(x,y)  My.

(24) If something is made a fact, then it can be a fact: MF(x,y)  CF(y).

Also

(25) Only complexes can be made facts: MF(x,y)  CX(y).

On the other hand, by applying respectively either Russellian idea of a fact or its second Wittgenstein’s idea as developed in (D14.2) we easily obtain:

(26) Only truth makers can be made facts: MF(x,y)  MT(y, A); and

(27) Only makers of the existence of states of affairs can be made facts:
MF(x,y)  $ST(z) ME(y.z).

Observe that the last two theorems are implicitly existential formulas, for they are equivalent respectively to:

(26*) $A (MF(x,y)  MT(y, A)), and

(27*) $ST(z) (MF(x,y)  ME(y,z)).

40.3 At least one effect of the axioms S and NE deserves retaining in memory:

(28) The family of all facts, denoted simply FACTS, is nonempty: FACTS  .

Of course [27] , FACTS equals the world.

41. Notice at once that assuming S and C, i. e., working in a weakening of WTC, we have:

(29) Both arguments of MF must be facts: MF(x,y)  F(x)F(y).

If, in addition, we suppose MR then we can prove that NE is, in fact, equivalent to nonemptiness of the world:

(30) S, MR, C _ NE iff FACTS  .

Observe also that in PTF we indeed can define facts, like it was intended in (D15), as objects made to be facts, i. e.

(31) S, MR _ F(x)  $y MF(y,x).

Determinism

42. Consider now the weaker theory of causation, i. e, WTC. It is easy to see that in it, hence also in WT, we can prove the following principle of determinism:

(D) Any fact is made fact by a fact, or any fact is caused by another fact:
“y (F(y)  $x (F(x)MF(x,y)).

Indeed, by MR we know that any fact has its own reason which, by C, must be a fact.

43. Finally, notice the following rather striking consequence of determinism:

(32) Assume NE, S, IR, T and D. Then the world, i. e. FACTS, is infinite.

Proof. By NE, the domain of MF is nonempty. Now, by the rest of axioms, we can start the well-known Schütte’s relation machinery to produce the desired infinity result.
To be more exact, by (29), in presence of NE and S, we have that FACTS  . Fix now an arbitrary x1  FACTS. By D there exists its cause, say x2. Hence MF(x2 , x1). Clearly, x2  FACTS, for x2 is a cause, and x1  x2 , for otherwise MF(x1, x1) which is forbidden by IR.
Again apply D to x2 . Let x3 be its cause. We claim that x3  FACTS and MF(x3 , x2), which is pretty sure, as well as that x3  x2 and x3  x1.
Indeed, if x3 = x2 then MF(x2 , x2), contrary to the irreflexivity axiom IR.
Suppose now that x3 = x1. By this assumption and by our construction we have that MF(x1 , x2) and MF(x2 , x1). Applying now the transitivity axiom T we obtain that
MF(x1 , x1), which again contradicts IR.
Obviously, we can repeat the above reasoning again and again, constructing in such a way an infinite descending chain of facts, i. e., (xi )iNat such that MF(xi+1 , xi) for any i  1.

As a matter of fact, we proved more than stated above. Namely, under assumptions of the theorem (31), we have:

(33) FACTS is an infinite family and its causal relation MF is not well – founded. [28[

44. The above consequence of determinism is, at least to the present author, rather surprising and striking.
Determinism is a rather popular doctrine, especially among scholars dealing with the classical, Newtonian picture of the world.
Now, if somebody is ready to accept the five axioms involved, which are rather intuitive, then he or she is in fact dealing with infinite and not well-founded world.
However, most of empirically oriented people [29] are very cautious in claims concerning the number of items of the world.

45. Surely, we can stop the above infinity result by refutation one of the axioms involved.
But S and NE are indeed very intuitive. On the other hand, refuting irreflexivity axiom IR we challenge the strict connection between causation and time ordering, which is rather well established doctrine. Next, refuting transitivity T we broke our basic conviction about the mechanism of the world. [30] Finally, by refutation of D we stop a very traditional paradigm of the mechanism, of at least a part, of the world.

46. Perhaps, we should preserve rationalism MR, but refute the closure principle C. It means that at least some facts are made by objects which are outside the world.
The idea is not so strange as, at the first look, it seems to be. It was, for example, sucessfully used by Leibniz in the very crucial point of his Monadology. [31]

47. Anyway, one thing is clear. Determinism needs further, careful and subtle, discussion.

References

Fox, J. F.: 1987, “Truthmaker”, Australian Journal of Philosophy 65, 188 – 207.
Leibniz, G. W.: 1969, Philosophical Papers and Letters, Translated and edited by Loemker L. E., D. Reidel Publ. Company, Dordrecht, p. 737.
Mulligan, K., Simons, P. & Smith, B.: 1984, “Truth – Makers”, Philosophy and Phenomenological Research 44, 287 – 331.
Perzanowski, J.: 1985, “Some Observations on Modal Logics and the Tractatus”, in Chisholm R., eds. Philosophy of Mind/Philosophy of Psychology. Proceedings of the 9th International Wittgenstein Symposium, 19th – 26th August 1984, Kircherg am Wechsel, Verlag Hölder-Pichler-Tempsky, Wien, p. 544 – 550.
Perzanowski, J.: 1989, Logiki modalne a filozofia, Jagiellonian University Press, Kraków, p. 159.
Perzanowski, J.: 1990, “Ontologies and Ontologics”, in ¯arnecka-Bia³y E., ed. Logic Counts, Kluwer Academic Publishers, Dordrecht-Boston-London, p. 23 – 42.
Perzanowski, J.: 1991, “Modalities – Ontological”, in Burkhardt H. & Smith B., eds. Handbook of Metaphysics and Ontology, Philosophia Verlag, München,
p. 560 -562.
Perzanowski, J: 1992, “Combination Semantics – An Outline”, in Balat M. & Daledalle – Rhodes J., eds. Signs of Humanity, vol. I, Mouton de Gruyer, p. 437 – 442.
Perzanowski, J.: 1994a, Facts (in preparation).
Perzanowski, J.: 1994b, Makers (in preparation).
Russell, B.: 1918, “The Philosophy of Logical Atomism”, The Monist, reprinted in Russell 1956, p. 177 – 281.
Russell, B.: 1919a, “On Propositions: What They Are and How They Mean?”, reprinted in Russell 1956, p. 285 – 320.
Russell, B.: 1919b, Introduction to Mathematical Philosophy, G. Allen and Unwin Ltd., London
Russell, B.: 1956, Logic and Knowledge, G. Allen and Unwin Ltd., London, p. 382.
Scheffler, U.: 1993, “On the Logic of Event Causation, I. Fundamental Reflections”, Logic and Philosophy, 1
Suszko, R.: 1968, “Ontology in the Tractatus of Wittgenstein”, Notre Dame Journal of Formal Logic, 9, p. 7 – 33.
Urchs, M.: 1986, Kausallogik, Leipzig
Wittgenstein, L.: 1961, Tractatus Logico – Philosophicus, translated by D. F. Pears and B. F. McGuinness, Routledge & Kegan Paul, London, p. 89.

Notes:

[1] Cf. Wittgenstein 1961.
[2] I. e. connection between reasons and their results (founded objects)
[3] Either a reason to or a reason for
[4] I. e. first – order
[5] For more advanced discussion cf. Urchs 1986 and Scheffler 1993.
[6] Cf. Perzanowski 1989 and Perzanowski 1991
[7]Which is clarified and exploited by the relational semantics of logical modalities
[8] For a more developed analysis of makers cf. Perzanowski 1994b.
[9] Cf. Perzanowski 1989.
[10] Cf. thesis 2.012 of Wittgenstein 1961: In logic nothing is accidental: if a thing can occur in a state of affairs, the possibility of the state of affairs must be written into the thing itself
[11] Classical connectives are treated here after the well-known Tarskian manner.
[12] Notice that propositions are objects, i. e., items of the ontological space therefore they can be arguments of MP.
[13] Cf. Perzanowski 1989 and Perzanowski 1992.
[14] For development of the present outline of a theory of facts cf. Perzanowski 1994a and Perzanowski 1994b.
[15] Cf. Russell 1918 and Russell 1919a, reprinted in Russell 1956.
[16] Cf. Wittgenstein 1961.
[17]Cf. Russell 1918.
[18] What is rather ironic, when we take into account Russell’s well-known strong antimodal attitude. It was, I think, the main essential obstacle which put a stop for progress in Russell’s own work on mathematization of philosophy.
Wittgenstein, we will see, was much more free in using philosophical modalities.
Notice, however, that Russell’s idea of truth makers took serious attention between open-minded analytic philosophers; cf. Mulligan, Simons & Smith 1984 and Fox 1987.
[19] Cf. Russell 1919a.
[20] This requires a suitable theory of analysis and synthesis
[21] It is, mutatis mutandis, the modal calculus D. For details of its decoding cf. Perzanowski 1985.
[22] Variables x, y, z,…. run over complexes, i. e. sentence-like items occurring in the logical space. Hereafter, reading of variables and their negations occupying sentences’ position in formulas is like reading of propositions: x means that x holds or x obtains, whereas ¬x means that x does not hold or x does not obtain.
[23] I. e. congruential or extensional.
[24] Formalized by R. Suszko as a strong modal equivalence. Cf. Suszko 1968.
[25] They are, of course, based on the classical quantifier logic.
[26] Cf. Perzanowski 1989.
[27] At least for Wittgensteinians, cf. Tractatus thesis 1.
[28] A relation R is said to be well – founded, if there is no infinite descending chain of elements with respect to R.
[29] Cf. Russell 1919b.
[30] But global use of T in the domain of causation seems to be somehow mysterious.
[31] Cf. Leibniz 1969, pp. 643 – 653.

JERZY PERZANOWSKI

Department of Logic
N. Copernicus University
Fosa Staromiejska 3
87-100 Toruñ

Department of Logic
Jagiellonian University
Grodzka 52
31-044 Kraków