Logical Proof of Antinomy: A Trinitarian Interpretation of the Law of Identity
Michael C. Rhodes
Department of Theology
Loyola University
1. Introduction
This essay presents Pavel Florensky's paraconsistent logical proof of antinomy and delineates the
relationship of his paraconsistent logic to the notion of the Holy Trinity. My discussion is based on
the seventh chapter of his Pillar and Ground of Truth titled "Letter Six: Contradiction"
(Florensky 1997, 106-123), and on the third chapter of this same text, "Letter Two: Doubt" (Florensky 1997, 14-38).
1.1. Brief Biography
Pavel Florensky (1882-1943) was a Russian Orthodox priest who was trained as a mathematician and
philosopher. When he was 46, the Soviet regime sent him to Nizhny Novgorod as a political exile for
two years; in 1930 he was released under the guise of having been 'rehabilitated,' but three years
later was sentenced to ten years in the Labor Camps, during which time he somehow passed away.
He published on many topics ranging from mathematics and logic to chemistry, philosophy and theology,
some of which is available in translation. The most monumental of his works, however, is his
aforementioned Pillar and Ground of Truth.
2. Logical Proof of Antinomy
The proof purports to show the insufficiency of the reductio ad absurdum formula by
"two equally indubitable proofs" which derive both p and ~p by means of the
reductio formula, forming as a conjunctive proposition (p · ~p)
the 'antinomy P' (cf. Florensky 1997, 112-113).
Florensky's first proof begins with the formula for the reductio ad absurdum.
((~p É p) É ~p)
The antecedent of this conditional (itself a conditional) Florensky transforms by the rule
of material conditional to a disjunctive of the form 'either not-p or p'. But
since the antecedent p is already negated, then the equivalent logical statement is
'either not-not-p or p'. These disjuncts however by the operation of the rule of
'double negation' are equivalent, 'either p or p'. The original formula then has
been transformed so that it now reads: 'If either p or p, then p'.
In symbolic notation, Florensky's reasoning for the derivability of p from ~p is as follows:
| (i) | ~p | By assumption |
| (ii) | ((~p É p) | By implication on (i) |
| (iii) | (~~p v p) | By material conditional on (ii) |
| (iv) | (p v p) | By double negation on (iii) |
| (v) | (p v p) É p | By implication on (iv) |
| \ | (iv) | p By modus ponens on (v) (cf. Florensky 1997, 112) |
Florensky's next proof shows the implication of p from not-p to prove that
both p and ~p are derivable from the reductio formula.
Let ~p be represented through q (i.e. ~p = q) so that the above
reductio formula reads: 'assume not-q, derive q; conclude q':
((~q É q) É q)
Since ~p = q, then the reduction formula equally reads:
'assume not-not-p, derive not-p; conclude not-p'. This substitution yields:
((~~p É ~p) É ~p)
Again, by the rule of 'double negation', this is equivalent to:
((p É ~p) É ~p)
So Florensky's reasoning for the derivability of ~p from p is:
| (i) | ~p = q | By definition |
| (ii) | p | By assumption |
| (iii) | ((~q É q) É q) | By definition on (i) |
| (iv) | ((~~p É ~p) É ~p) | By definition on (i) and (iii) |
| (v) | ((p É ~p) É ~p) | By double negation on (iv) |
| \ (iv) | ~p | By modus ponens on (v) (cf. Florensky 1997, 112) |
In this manner, both p and ~p are derived from their opposites, making together the 'antinomy P', namely p and not-p, or (p · ~p) (Florensky 1997, 113). In symbolic notation, Florensky presents this as follows:
P = (p · ~p) = V
where 'P' is a proposition with two contradictory terms (or a class whose members mutually exclude one another),
and V is the truth truth-operator, which Florensky argues means that for "pure logic" "V in the
definition of P is only an indication of the position of this P, an indication of the relation that is
required toward it. . .it is not a constituent part of the structure of P itself. . .V represents the
constituent elements, the spiritual unity, the suprasensuous [and suprarational] reality of antinomy."
An antinomic proposition, according to Florensky's reasoning therefore, "jointly contains thesis and antithesis,
so that it is inaccessible to any objection. . .[and] above the plane of rationality"
(cf. Florensky 1997, 113).
2.1. P = (p · ~p) = V as 'God is Consubstantial' and 'God is Trihypostatic'
One such proposition P, Florensky suggests, is 'God is Consubstantial' and 'God is Trihypostatic'.
Taking 'Consubstantial' as the thesis and 'Trihypostatic' as the antithesis, the antinomy P in this case is a
proposition with two contradictory predicates attributed to God.
3. What about the Laws of Thought?
An antinomic proposition P = (p · ~p) taken in a strictly logical sense is either
self-isolating or contradictory. But the assumption that truth is only logical, according to Florensky,
relies on the laws of thought (the laws of identity, non-contradiction[1] and excluded middle).
His proofs, however, necessarily imply that these laws are antinomic. Thus, to affirm that truth
is only of a logical ilk, according to Florensky, is to affirm implicitly that it is not, and
therefore, according to Florensky, truth is unavoidably antinomic.[2]
Florensky's antinomic notion of truth does not formalize the nature of truth as being
'either p or not-p' (law of excluded middle)[3] but rather as a conjunctive synthesis
(p and not-p), which according to Aristotelian, Boolean (algebraic) and mathematical
(logistics, 'class' calculus) forms of logic is a gateway to falsehood via the 'principle of explosion,'
namely ex contradictione quodlibet ('from contradiction anything follows').[4]
Florensky's proofs challenge this 'principle' by grounding antinomy in the Holy Trinity.
Thus, according to Florensky, showing that truth is not of a logical ilk only does not imply
'anything' in general, but Trinitarian thought in particular.[5]
Truth as Antinomy: A Theological Interpretation
The claim that 'from contradiction anything follows' does not hold in a Florenskian paraconsistent logic because of his trinitarian reinterpretation of the law of identity in space-time, a personal interpretation of which leads to an affirmation of the Holy Trinity as the ground of truth. Such a defense against the charge that 'anything follows' begins, according to Florensky's thinking, with an attempt to join together the conditional with the Unconditional.
3.1. Creation, Judgment, Reason
Given the antinomic nature of truth, the existence of creation[6] therefore implies, Florensky postulates, that antinomic-truth is a sort of 'symbol' of Truth, which he maintains suggests the epistemological problem of knowing and speaking the Unconditional by means of the conditional. "How is it possible to construct the unconditional formula of Divine Truth," Florensky queries,
from the conditional material of the human mind?...A
rational formula can be true if and only if it foresees,
so to speak, all objections to itself and answers them.
But to foresee all objections, it is necessary to take not them concretely but their limit. It follows that truth is a judgment that also contains the limit of all its refutations, or (in other words) that truth is a self-contradictory judgment. . .The thesis and antithesis together form the expression of truth. In other words, truth is an antinomy, and it cannot fail to be such (Florensky 1997, 109).
Antinomy provides a way of knowing and speaking the Unconditional by means of the conditional, Florensky maintains, because it expresses the finite limits of reason through contradiction. If a rational formula can 'gather all of life into itself' (i.e. if it can reduce life to itself), both past actualities, present 'becomings' and future contingents, deal with all possible and actual objections, then it is rational, according to Florensky, that a rational formula might be able to fully possess all of life in itself. But according to Florensky, there is no evidence of reason's being able to do this. If it is impossible, then there is no reason to conceive of rational formula in terms of logical certainty. Florensky suggests the notion of 'limit' so that the 'truth claim' of a rational formula is not required to encompass life in terms of logical norms, but to express it (both in a conditional sense as well as in an Unconditional sense) in terms of antinomy. A 'truth claim' of this form contains the limit of all its refutations in which both thesis-p and antithesis-~p remain. Thus, Florensky suggests an interpretation of rational formula in terms of contradiction which accepts its own finitude and conditionality as parameters that suggest not fallacy but Unconditional immediacy.
But what does this mean?
3.2. Spatio-Temporal Identity:
Florensky's position suggests a spatio-temporal interpretation of the law of identity in terms of the conditional aspect of antinomic rationality.
If truth is antinomic, then, Florensky queries, what are we to make of temporal succession and spatiality? He argues that a spatio-temporal actuality both is and is becoming, it is identical to itself both because it is a certain spatio-temporal actuality and because it is becoming a certain spatio-temporal actuality through its relatedness to other spatio-temporal actualities, and so the law of identity is always both grounded and violated. Thus, A is A and A is becoming A through not-A, i.e. through a denial of itself.
This relationship, Florensky argues, is tripartite because if not-A is represented through B, and not-B through C, and not-C through A, then A is A because of B, B is B because of C, and C is C because of A, which according to Florensky grounds the law of identity in a tripartite scheme of spatio-temporal actuality in a conditional sense (cf. Florensky 1997, 36).
A truth claim of this form contains the limit of its refutations in the otherness of that which it is not but with which it stands in some relation, and so by this means expresses rather than encompasses life. But taken alone this scheme is contradictory according to reason. It must be grounded in the Unconditional.
3.3. Personal Spatio-Temporal Identity
Lastly, Florensky's reasoning interprets the law of identity in a personal manner in both a conditional and an Unconditional manner.
The personal conditional dimension of this scheme is 'I', 'He' and 'Thou', according to Florensky, which he suggests implies that truth finds its Unconditional ground in the Holy Trinity: "I is the relation to He through Thou. Through Thou the subjective I becomes the objective He, and, in the latter, I has its affirmation, its objectivity as I." "Truth," therefore, "is the contemplation of Oneself through Another in a Third: Father, Son and Spirit" (Florensky 1997, 37).[7]
The full Trinitarian interpretation of the law of identity could be schematically presented as follows from the spatio-temporal conditional to the personal Unconditional:
| not-A through B | ~I through he | ~Father through Son |
| not-B through C | ~he through you | ~Son through HS |
| not-C through A | ~you through I | ~HS through Father |
| A is A because of B | I is I b/c of he | Father is Father of Son |
| B is B because of C | he is he b/c of you | Son is Son b/c of HS | |
| C is C because of A | you is you b/c of I | HS is HS b/c of Father |
Since the actuality of a thesis, therefore, does not logically necessitate its antithesis, according to Florensky, then "each time it is necessary to become convinced not only of the truthfulness of the thesis p but also to clarify whether it is not half of some antinomy P" (Florensky 1997, 113) for the purpose of avoiding heresy ("hairesis [the Greek term] means choice, tendency, a disposition to something"), i.e. "a rational one-sidedness that claims to be everything" (Florensky 1997, 119). Thus, Florensky's paraconsistent logic being grounded in the Holy Trinity does not allow that 'anything follows', rather it defines specific metaphysical parameters so that heresy as the emphasis of only one half of an antinomy can be avoided, implying, therefore, not that 'anything follows,' but that orthodoxy follows.
An antinomy contra-dicts life in so far as it is limited to the conditions of logic, according to Florensky,
but expresses life insofar as it is grounded in the Tri-hypostatic-Unity: i.e. apart from the Trinity antinomy is
against life as the source of irrationality, but in relation to the Trinity antinomy expresses life and is
the source of rationality that is both conditional and Unconditional. A rational formula, therefore, according to
Florensky, is defined by the relatedness of the conditional to the Unconditional, by its self-denial through that
which it is not, rather than by the laws of logic alone.
4. Conclusion
In this essay I have presented Florensky's form of paraconsistent logic, and have shown how for
him it is grounded in the Trinity. The logical peccadilloes of such a position notwithstanding, Florensky's
position has merit first of all because he authored this theory in the early 20th century, long before the
advent of paraconsistent logics beginning with F. G. Asenjo's work in the 1960's. Secondly, whereas paraconsistent
logics are often motivated by interests in artificial intelligence, Florensky's paraconsistent logic stems from his
Orthodox Christian cosmology, and more particularly from his unique Trinitarian interpretation of identity and
antinomy.
Notes:
[1] Cf. B. Russell Principles of Mathematics (George Allen & Unwin: 1937), 16-18, 20. Russell argues that the principle of non-contradiction is provable from the first nine axioms of symbolic logic.
[2] Cf. Metaphysics III, 3 (1005), trans. Christopher Kirwan (Oxford: 1971), 7-8; George Boole Collected Logical Works vol. II (Open Court Publishing: 1940), 53-4; Lewis Carroll Symbolic Logic (New York: Clarkson N. Potter, 1977), 61-2; Bolzano seems to have been a bit more circumspect Theory of Science, trans Rudolf George (Basil Blackwell: 1972), §45 (George does not translate this section); W. S. Jevons' comment in his Elementary Lessons in Logic (London: Macmillan, 1957) seems to capture the generally accepted perspective on this and the other two laws of thought: "and it is not too much to say that the whole of logic will be plain to those who will constantly use these laws as the key." Concerning this law in particular, he says: "It is the very nature of existence that a thing cannot be otherwise than it is; and it may be safely said that all fallacy and error arise from unwittingly reasoning in a way inconsistent with this law. All statements or inferences taken which imply a combination of contradictory qualities must be taken as impossible and false, and the breaking of this law is the mark of their being false" (118; italics mine). See also Hilary Putnam "What is logic" in Philosophy of Logic (London: George Allen & Uwin Ltd., 1972), 4-5; W. E. V. Quine "Deviant Logic" Philosophy of Logic (Prentice Hall: 1970), 81: "My view of this dialogue is that neither party knows what he is talking about. They think they are talking about negation, '~', 'not'; but surely the notation ceased to be recognizable as a negation when they took to regarding some conjunctions of the form p ( ~p as true, and stopped regarding such sentences as implying all others. Here, evidently, is the deviant logician's predicament: when he tries to deny the doctrine he only changes the subject." To deny the law of non-contradiction (and the law of identity and the law of excluded middle) is to undermine the program of logic, so in this sense it is a 'change of subject'. But for Florensky this 'change' is brought about as a product of logical implication: it reveals not a misunderstanding (as Quine seems to suggest) on the part of the 'deviant' (Florensky in this case) but on the part of the 'non-deviant.' For Florensky, it seems that the logician who insists on this law's logical reliability changes the subject: it is not a question of a consistent usage of 'negation' but of a proper understanding of truth. Kant's discussion in "Introduction" to his Logic trans. R. S. Hartzman and W. Schwarz (Bobbs-Merrill Co., Inc.: 1974) sets the 'principle of contradiction' together with the 'principle of sufficient reason' as the 'internal' and 'external' grounds of logic respectively (57-8). Hegel's logic would be of the 'deviant' sort it seems: see his Logic (Oxford: 1975), 171-3: "In the notion of the circle, centre and circumference are equally essential; both marks belong to it: and yet centre and circumference are opposite and contradictory to each other." See also Heidegger "The Principle of Identity" in Identity and Difference (Harper and Row, 1969), and The Principle of Reason (Indiana: 1996).
[3] Logically equivalent to the law of non-contradiction: 'not p and not-p' is equivalent to 'either p is true or p is false', which is equivalent to 'if p is true, then not p is false' (by the inversion rule known as 'material conditional'). The law of identity (p = p; if p, then p; p and p; p iff p) is transgressed either way.
[4] Also referred to as 'ECQ', this principle maintains "that anything follows from contradictory premises." Cf. Priest, Graham, Tanaka, Koji, "Paraconsistent Logic", The Stanford Encyclopedia of Philosophy (Summer 2004 Edition), Edward N. Zalta (ed.), URL = http://plato.stanford.edu/archives/sum2004/entries/logic-paraconsistent/.
[5] The obvious line of attack on an argument of this sort is by means of showing that the use of double negation can ultimately imply anything: i.e., the 'principle of explosion'. Thus, a position such as Florensky's, it could be argued, misconstrues the notion of 'negation' and simply affirms an absurdity. Cf. W.E.V. Quine "Deviant Logic" in Philosophy of Logic (Prentice Hall: 1970). See also note 4 below.
[6] His ninth letter is devoted to this topic: "Letter Nine: Creation" (190-230).
[7] Florensky addresses the question of participating in the life of the Trinity at the end his "Letter Two: Doubt". He argues that there cannot be less than three hypostasis, but that there can be more than three; but each additional hypostasis would be conditional, not necessary to the self-giving-Subject. Hence, such hypostasis are more precisely referred to as 'deified persons.' The addition of other hypostases to the Trihypostatic Unity is necessary for these hypostases, but not for the Trinity itself; the Trinity could be apart from the addition of these other hypostases, but these other hypostases could not be apart from the Trinity (cf. 38).
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