Believing the self-contradictory* Abstract. An argument for the rationality of religious belief in the existence of God is defended. After reviewing three preconditions for rational belief, I show reasons to privilege the criterion of consistency. Taking the inconsistency of the religious belief in God and the belief in the scientific world picture as the impediment to a rational belief in God, I propose that we can overcome this objection by assuming, firstly, that God is a universal class. This allows us to put the problem of God in set-theoretic terms, such that the antinomy that follows from such an assumption can be overcome by assuming that God is not a subject but a strict class that cannot be individuated. I conclude that that the self-contradictory nature of God does not prevent the believer from making a rational, ethical assessment that the contradiction resides in the possibility of using language to explain his existence, but that this does not make belief in the existence of God unjustifiable – on the contrary. In this way, we can say statements that claim God exists are justifiable. 1. Three logics of justification Traditionally, there are three sorts of attitude to religious belief that form the three great interlocutive positions in the discussion of religion: those who believe that God exists, or the believers;1 those who do not believe that God is exists, or the atheists; and those who believe that neither position is justifiable given our human limits, or the agnostics. Among the various thematics into which this discussion flows, one of the commonest, as it connects the abstraction of the topic with practice, is how to make a choice among these three stances. In other words, how do we justify our adherence to one of these three positions? The advocate for religious belief has at least one advantage, in that it is generally agreed that a belief needn't be ascertained to be true in order to be held. I may have good reasons to find 'x' belief plausible without being able to prove it according to some verificational protocol. Thus I am entitled to believe that God exists without having to establish his existence once for all. The lack of a conclusive proof doesn't prevent me from believing something true. But still, I do need reasons to have acquired my belief. Who knows anything about him? We must remember, here, that knowledge is not the point about God, anyway, as he is not simply the object of inquiry. After all, by definition, our entire being and social structure, including inquiry, is grounded on God. That we believe in God does highlight the fact that one can believe as strongly as possible without ever knowing anything about this possibly empty term. However, any belief ought to be justified before we can talk about its being true (or not): a belief must be justifiable in order to be a valid belief, that is, a belief that could be entertained by any reasonable subject. Now what sort of justification do we need to believe correctly? Let us state a first criterion for correct belief in terms of indefensibility: (1a) A given sentence p is justifiable only if the negation of p is indefensible, i.e. there is no evidence for p to be false.2 But this statement is still ambiguous: does the absence of justification against p mean that there cannot ever be such a justification, or, rather, does it simply have a temporal sense: up to the present, none has been found? In order to go beyond a merely falsificationist definition of * I am grateful to Dorota Rybicka for our informal exchange about the present talk, as well as the anynomous referee for his (her) helpful notes. 1"Believers" will be used as a synonym for the theists or positive believers about the sentence p: "God is existing", throughout the paper. Admittedly, the atheists are also believers; but they are negative believers about p. 2 Evidence can be an empirical datum, a thought experiment, a formal proof, or even an intimate experience. 2 justification, we add a further condition to the latter, as follows: (1b) Any evidence for p entails that there is no evidence for its negation, ~p. The condition thus modifies the skeptic view of justification: (1a) entails (1b) from this perspective, whereas a mere falsificationist view doesn't mean that there cannot be evidence for ~p if there are none yet. If we adopt (1a)-(1b) together, such a necessary condition for belief appears to be very stringent: it could satisfy a skeptic, but hardly an agnostic, or even a theist for whom the existence of God can be believed without a conclusive evidence.3 Moreover, an application of (1) would entail that very few sentences could be properly believed, because most of our beliefs rely upon empirical evidence whose falsifiability conditions make their negations clearly arguable. To loosen up our conditions in order to allow a greater scope for belief, an alternative, weaker condition for correct belief can be constructed from supposing a criterion of justifiability as defensibility, which should hold for both believers and atheists4: (2) A given sentence p is justifiable if p is defensible, i.e. there is some evidence for p to be true.5 The import of (2) is a "glass half full, glass half empty" story. That God may exist is a sufficient reason to believe in His existence (i.e. believing the sentence p: "God is existing"), according to the believer; that God may be not existing is a sufficient reason to believe in His inexistence, according to the atheist.6 However, it is unlikely that an armed truce like this will bring the religious discussion to a satisfactory close. Alternatively, the mere possibility that God exists does not give the skeptic a sufficient reason to believe in His existence or even his inexistence. In a sense, this is agnosticism straight, in distinction from the soft logic of believers and atheists. At the same time, an anti-skeptic logic of soft justification would turn out to be troublesome if whatever possible is justifiable: any contingent sentence would entail that the believer is entitled to believe both p and its negation, i.e. ~p.7 Correspondingly, most of the atheists and believers seem at this stage in the debate to operate as partial or bad faith agents from a skeptic point of view, reserving a duplicitous double register of justification in which both take any evidence for or against the existence of God to be a sufficient reason to support their belief, but requiring that their opponent give conclusive (i.e. necessary) evidence for or against the existence of God. In other words, they shift the burden of the proof (or conclusive evidence) onto their opponent, only: the agnostic theist would require any atheist being gnostic, and conversely.8 3 By a "theist" is meant whoever believes in the existence of God. No relevant difference is made here between theism and deism, although depicting God as a universal class sounds more like a argument for deism (God as an impersonal logos). Note that being an agnostic believer is not self-defeating: agnosticism merely means that it is not possible to establish the existence of God, and that this needn't lead to atheism, since it is not possible to establish the that there is no God, either. 4 The atheist attitude can be depicted by substituting ~p for p in (2). 5 Let B stand for belief, and E for defensibility (Ep means "there is an evidence for the sentence p"). Then (1) and (2) result in three distinct axioms for two opposite logics of justification, namely: (1a) Bp → ~E~p, (1b) Ep → ~E~p, and (2) Ep → Bp. Ep entails ~E~p for a skeptic, but not for a mere believer: that there is no evidence for ~p (i.e. no evidence against p) does not force the believer to conclude that there is an evidence for p. This means that E proceeds like a strong modal operator  in the strong (conclusive) sense of evidence and like a weak one  in the weak (testimonial) sense: not any evidence for ~p is an argument against p in the latter case, for there can be evidence for both. 6 (1a) entails E~p → ~Bp (by contraposition), but not E~p → B~p. Thus (1) is not a proper logic for believers and atheists, but is adequate for skeptics. 7 Let C stand for contingency. Cp ↔ (Ep ∧ E~p), then, by (2), Cp → (Bp ∧ B~p). And given that the logic of the believer is taken to be normal, B~p → ~Bp. Therefore, (2) entails Cp → (Bp ∧ ~Bp). 8 The logic of the atheist amounts to taking non-belief and negative belief to be equivalent: ~Bp ↔ B~p, so that ~Ep 3 However, faith is a bad faith only if we already subscribed to the skeptic criterion for belief as requiring conclusive evidence, whether for or against a given sentence. Against this strong criterion for evidence in (1), faith seems to be described better in terms of a special logic of merely religious belief that would support (2), while contravening one of the basic axioms of epistemic logic, namely: (3) If p is merely believed, then p is not known.9 Unlike the skeptic, the believer does not require any conclusive evidence for his belief and this holds equally whether the belief is in the existence or inexistence of God. I need not have a definite proof to believe in him, while having some reason to prefer this opinion rather than the contrary one. And yet, (3) is hardly defensible for a believer in the sense that a logic of religious belief cannot go on claiming that logical knowledge is incompatible with religious belief, since a proof of impossibility or necessity should force the believer to either relinquish his belief or endorse nonbelief, respectively. In this respect, St. Anselm and Descartes' alleged proofs for the existence of God are intended to show that believing is not only justifiable but necessary: I cannot not believe in God, and I believe in Him all the more that I know that it is impossible for Him not to exist. Now given that (3) makes any allegedly religious belief incompatible with logical knowledge10, it should be replaced by another logic including logical knowledge. The believer's argument is premised on a logic that is weaker than (1), but stronger than (2) and (3). Consistency seems to be a necessary condition assuming that the believer in the debate recognizes the Aristotelian Principle of Non-Contradiction (hereafter: PNC). According to the psychological version of the PNC,11 no one may logically believe that God exists and does not exist. This rule equally holds for all our three parties: believers, atheists, and agnostics. As a way to strengthen (2), the participant in the debate must choose between believing p or its negation ~p, even if there is evidence to support both contents. Let us specify that consistency is not only a property of what the sentence says taken in isolation;12 rather, consistency expresses a relation between the given sentence p and the believer's set of initially accepted sentences q. Hence the resulting condition for correct belief: (4) A given sentence p is justifiable only if it is consistent with every other sentence q already accepted by the believer.13 Thus far, defensibility is a sufficient condition and consistency is a necessary condition for correct → B~p. An abductive reasoning can display this partial treatment of evidence in the atheist: pain and injustice are taken to be a sufficient evidence against the existence of God, while the beautiful harmony in snow crystals is not sufficient to establish divine design. The converse is true for the believer. 9 According to the mainstream epistemology, epistemic logic states that knowledge is a justified true belief. Accordingly, belief is one of the necessary conditions for knowledge and the following is given to be an epistemic axiom: Kp → Bp. To the contrary, a religious belief assumes a logic of mere belief that is incompatible with the preceding axiom: if Bp → ~Kp, then Kp → ~Bp (by contraposition). 10 For if we can deduce logical knowledge from every logical truth, then no such truth should be believed according to (3): Kp → B~p (see note 9). 11 Łukasiewicz (2000) stated three formulations of the PNC: an ontological version (no object can have and not have the same property); a logical version (two judgments where the one attaches to the object the very property refused by the other one cannot be true together); and a psychological version (any two convictions related to contradictory judgments cannot coexist in the same mind). (3) concerns the latter version. 12 This is self-consistency; the ensuing difference between contradiction and self-contradiction is detailed in the next section. 13 To simplify the criterion of consistency, we state that it requires the agent not to believe both a given sentence and its negation: (3) Bp → ~B~p. (3) is incompatible with (2), for Ep → Bp, E~p → B~p and, therefore, (Ep ∧ E~p) → (Bp ∧ B~p) entails by (2) that (Ep ∧ E~p) → (Bp ∧ ~Bp). 4 belief. Against those who might still object that (4) is redundant with (2) due to the identity of defensibility and consistency, there is, in fact, a difference between them: a sentence p is defensible whenever it is not self-contradictory, i.e. not contradictory with itself; furthermore, it is consistent whenever it is not contradictory with other sentences q in a given belief set. For if p is inconsistent with a given set of beliefs A but self-consistent, this merely means that p should not be included in A, even if it is compatible with other contexts. However, if p is self-contradictory (or selfinconsistent), this means that p is impossible to be believed in every respect or irrespective of any belief set. Having unpacked the meaning of justification, we can now see that, in the context of the logic of belief, justification combines two interconnected but domain different modalities, namely: defensibility as conceivability, or mental possibility; consistency as logical possibility. Moreover, (2) and (4) entail that everything conceivable is self-consistent: (5) A sentence p is self-defensible only if it is self-consistent, i.e. believing p entails not believing its negation ~p, i.e. whatever is conceivable cannot be self-contradictory.14 If so, I cannot justifiably believe something inconsistent. This minimal condition for grasping a thought is at stake for our main point, which concerns religious belief about the existence of God. To sum up, we have proposed three semantically different modes of justification, which we will label as follows: a skeptical logic with (1), where an evidence must be conclusive to be properly evidential; a relativist logic with (2), where every evidence is defensible and leads to inconsistent beliefs; and a dogmatist logic for (4). The latter will now be the focus of our attention, as it has been favoured as a proper criterion for correct belief.15 As we will see, it seems to entail an atheist position about the religious belief that God is existing. 2. Is God a self-contradictory subject? A number of arguments contribute to the notion that the argument for God's existence is selfdefeating. On the one hand, doubting the existence of God has been taken to be self-contradictory by St. Anselm or Descartes: it is not possible to conceive the supreme Being as perfect without assuming his existence necessarily. To contemporary ears, this sort of argument somehow sounds like the performative account of the cogito once thinking of God is taken to be a caused action. According to this view, it is not only that such a conceivable sentence as "God exists" is consistent: its negation is self-inconsistent, so that God cannot be conceived as inexistent.16 However, looked at from the other side, such an argument does not make the atheist position self-defeating because it relies upon a contraposited version of (5): whatever is not self-consistent is not conceivable; God's 14 Note that "conception" differs from "imagination", among the range of mental possibilities: the former concept gives us the formal object of a judgment, while the latter is derived from being somehow being able to picture the object. Descartes' case of the chiliagon marks this difference out: a thousand-sided polygon can be conceived, but not imagined. Hume supports (5) in his "Enquiry concerning Human Understanding": everything conceivable is possible, equating possibility with non-contradiction. Furthermore, (5) could be reasonably strengthened in the form of a biconditional, in the sense that it hardly makes sense to state that something self-consistent may be unconceivable. The question remains open, noting that the converse (5) seems to make sense only if "conceivable" is replaced by "(actually) conceived". About the notions of conception, imagination, logical possibility and their logical interrelations, see Costa-Leite (2011). 15 For a detailed analysis of these three sorts of justification, see Ganeri (2002) and Schang (2010b). 16 About St. Anselm's argument, see his Proslogion seu Alloquium de Dei existentia; about Descartes' proof, see his Metaphysical Meditations, especially Books III and V; about the performativeness of the cogito argument, see Hintikka (1962). 5 inexistence is not self-consistent17; hence God's inexistence is not conceivable. Now God's inexistence is not a self-inconsistent judgment: it is merely inconsistent with the two initial judgments that existence is a predicate and every cause has a supreme cause, respectively. On the other hand, a logical consequence of God's perfection may lead to the contrary conclusion that God's existence is a self-contradictory idea that cannot be conceived accordingly.18 Is God a self-contradictory object? The referent of a sentence is not capable of being selfcontradictory in the logical sense: contradiction refers only to a condition that emerges in language. Thus, when we analyze the term God for 'contradictions' in the language we use about God's existence, we are using the subject at hand as a dummy singular term (a Russellian proper name) and a definite description, in accordance with Russell's theory. For instance, any sentence about a round square is self-contradictory because it states something about a term that is round and not round. Can impossible objects, the referents of the subject, exist? In as much as existence is coextensive with conceivability, there are necessarily no impossible objects. Whatever is selfcontradictory cannot be conceived. As an exception to (5), Łukasiewicz took the case of the Trinity in order to make his point against the so-called universality of the PNC.19 Let φ be a complex sentence composed of three simple sentences p: "God is the Father", q: "God is the Son", and r: "God is the Holy Spirit". Prima facie, φ is a conjunction of three contrary and, a fortiori, contradictory sentences that cannot all be true about the same object, because whoever is the Father is neither the Son nor the Holy Spirit. To put it in Fregean terms: these three names are not, like the morning star and the evening star, different senses for one reference. Each of the three is supposed to express one unique individual. If we employ a Quinean paraphrase to turn these singular terms into predicates, our φ will say that there are three different individuals and that God is each of these. If so, then the Trinity is to the effect that there is a x that is y and z while y is not z, so that the premises x = y, x = z and y ≠ z imply that x ≠ x. Contradiction: it is impossible to believe such a sentence as φ, according to (5). While admitting this reading, Łukasiewicz begs the point in saying that contradiction may not be a necessary condition for meaningfullness or, rather, that it should be bracketed for certain sorts of religious beliefs that go beyond any rational understanding. However, there is another and obvious way to state that φ is not self-contradictory without violating any property of classical logic, bringing up the issue of predication and the meaning of "being", or in other words, the relation between identity and membership. As to whether φ should be a contradictory sentence to be rejected by (5), let us return to the basic formulation of opposition in terms of predication with classes and their corresponding sentences (i.e. the basic form 'S is P' for p). What sort of Russellian proper name should God ever be, assuming that the Trinity really differs from a dummy proper name? Any two sentences p and q are said to be contrary if and only if they cannot be true together and can be false together. They are said to be contradictory if and only if they cannot both be true and cannot both be false. Accordingly, p, q and r should be contrary to each other: none of the pairs {p,q}, {p,r} and {q,r} can be true together, but they can all be false. A sample of contrary sentences has been displayed by Keynes (1884) in order to characterize the Aristotelian square of opposition within a logic of classes. Thus the universal pairs {A,E}, {A',E}, {A,E'} and {A',E'} are taken to be cases in point, where: A: "Every S is P", A': "Every not-S is not-P", E: "No S is P". and E': "No not-S is not-P".20 Which of these pairs is a counterpart of φ? Following the Quinean paraphrase of 17 It could be replied to this account that existence might not be taken to be a predicate, as famously claimed by Kant. But this does not undetermine my point at all, insofar as "being existing" can be safely replaced there "being existing as such and such". I thank the anonymous referee for emphasizing this technical point. 18 About St. Anselm's argument and its ensuing antinomy, see Vuillemin (1971). 19 See Łukasiewicz (2000), pp. 70-1. 20 Such a translation of classes into universal sentences yields the following relations in first-order predicate logic: Sentences Logic of classes First order logic A: SaP S∈P ∀x (S(x) → P(x)) 6 singular terms, φ states that whatever S-izes is P and only one x P-izes. Although they are usually said to be contrary, there are two opposite conditions for these pairs to be true: {A,E} and {A',E'} are true pairs if and only if no x is S, whereas {A',E} and {A,E'} are true pairs if and only if every x is P. The first condition fulfills the atheist position in as much as it is construed in Christian terms, meaning that S is an empty class or, equivalently, that God does not exist if He is both the Father and something else (i.e. the Son and the Holy Spirit). [Wouldn't the atheist position be, rather, that he couldn't be a Father at all and be God? As being a father is defined in biological terms, having to do with passing on DNA, and God is not physical?] Reply of the author: My point is that the atheist claims that God does not exist, irrespective of the reasons why he claims so; the materialist-minded (or biological) argument suggested above by the anonymous reader can be neglected, accordingly. Conversely, the second condition can satisfy the believer's claim if it entails that P is a universal class [instead of the referent of a Russellian proper name] Note of the author: does any Russellian proper name have to be a particular subject by definition? I haven't assumed it thus far, and I take a universal class to be a possibly Russellian proper name so long as it denotes an object rather than expressing a Fregean sense. and requires for God to be everything (every x). The third pair {A,E'}: (6) ∀x((S(x) → P(x)) ∧ (~S(x) → P(x)) is a plausible candidate for the truth of φ, therefore.21 Letting P for "God" and S for e.g. "the Father", it is true that the Father is God and not only the Father but the Son and the Holy Spirit too, which are not S. By the same token, that God is P and not S prevents one from believing something self-contradictory with φ: the intersection (S∈P ∩ S'∈P) clearly differs from (S∈P ∩ S∉P) and implies that nothing contradictory is said about God in the Trinity. But there is still a problem. Assuming that S and P are two unique classes with only one member, that God is the Father and someone other (the Son and the Holy Spirit) entails that two individuals are one and the same: "S is P" means an identity ("S = P"), and this is more than merely saying a membership "S∈P" since identity ("S = P") means two-sided membership ("S∈P and P∈S"). Accordingly, the pair {A,E'} should be reformulated in stating that "Every S is every P, and every not-S is every P".22 It results in the following conjunction of biconditionals: (7) ∀x((S(x) ↔ P(x)) ∧ (~S(x) ↔ P(x)), the truth-condition of which cannot be merely satisfied by the truth of every P: it is true only if A': S'aP' = PaS S'∈P' ∀x (~S(x) → ~P(x)) E: SeP = SaP' P∈S' ∀x (S(x) → ~P(x)) E': S'eP' = S'aP P'∈S ∀x (~S(x) → P(x)) {A,E}: SaP and SeP S∈P ∩ S∈P' ∀x((S(x) → P(x)) ∧ (S(x) → ~P(x)) {A',E}: S'aP' and SeP S'∈P' ∩ S∈P' ∀x((~S(x) → ~P(x)) ∧ (S(x) → ~P(x)) {A,E'}: SaP and S'eP' S∈P ∩ S'∈P ∀x((S(x) → P(x)) ∧ (~S(x) → P(x)) {A',E'}: S'aP' and S'eP' S'∈P' ∩ S'∈P ∀x((~S(x) → ~P(x)) ∧ (~S(x) → P(x)) 21 Let us assume that v(P(x)) = T. For every x, if v(S(x)) = T then v(S(x) → P(x)) = v(~S(x) → P(x)) = T; if v(S(x)) = F then v(S(x) → P(x)) = v(~S(x) → P(x)) = T. Therefore v((6)) ≠ F, and v((6)) = T only if v(S(x)) = T. Hence A and E' can be true together and are not contraries. The first condition is related to the well-known topic of traditional logic: the so-called "existential import", stating that the Aristotelian square is valid only if no class is empty. For a validation of the square with or without a nonempty model, see Schang (2010b) and another paper with Saloua Chatti (in preparation): "Import, or not? On the meaning of negation in the Aristotelian square". 22 This formulation has been put forth by Sir William Hamilton, in his doctrine of quantified predicates. 7 every x is S, every x is P, ... and no x is P. By obversion, that no x is P entails that every x is not-P and this amounts to a self-contradictory sentence (x∈P ∩ x∉P). No such sentence is possible to be believed, unless we come to admit that some sentences are both true and false.23 But to do this is only presupposing what is forbidden by the clause of consistency (5) for every correct belief. 3. Is God a proper class? The problem can be stated as follows: how can God be everything without entailing a selfcontradiction and, accordingly, being an impossible thing? [Isn't the problem rather that you are saying both that God is every x and that God is a unique class?] Note of the author: the former entails the latter, i.e. God's being a unique class entails that God is a self-contradictory object; see the biconditional form of (7) in this respect. A radical "solution" to this problem is to claim that nothing can be properly said about God, while everything is said by him (or his name). This means that God cannot be treated as a proper subject S in a predication like "S is P", since no class can include Him fully without restricting His domain of application and negating His "perfection" as it stands. A corollary of this assumption is that God cannot be individuated: no definite collection of properties can be attached to Him in order to determine His identity; determinatio est negatio, as claimed by Spinoza,24 so that the very process of predication is unable to express anything consistent about God. But could anything be said without subsuming a subject under a given concept? That God may not fall under any other class than himself has been echoed repeatedly in Hegel's philosophy about Eastern thoughts: according to the latter, contradiction doesn't oppose two exclusive but, rather, inclusive or complementary sentences that may be accepted together if we sort them according to a context in which each plays a role in a sub-context. Likewise, it has been argued elsewhere that some Indian non-classical logics admit a case where one sentence could be asserted and denied simultaneously or, conversely, that it can be neither asserted nor denied when it deals with some extra-rational entities like Atman or Brahman.25 No wonder, if so, given that everything and nothing can be said about God: nothing, as a subject; everything, as a predicate. Whatever the final word may be about this logical debate between the promoters of Aristotle and Heraclitus, set theory is the culprit that leads us to atheism if we put our belief into only well built sets; no sentence could make sense without resorting to a subsumptive relation between a class P and its element S. At the same time, the view that God is an unexpressible and universal class also avoids the wellknown set-theoretical paradox that normally follows from it: if God is taken to be the set of all the sets, then it leads to the famous Russell's Paradox that disallows such a universal set because it must include itself while, as the brackets of the set, not include itself.26 A way out to this self23 Such is the case with the Logic in Paradox in Priest (1979), where some sentences are assigned the paradoxical truth-value {T,F}. Then v(S(x)) = {T,F} entails that (7) is true: if v(P(x)) = T, then v((S(x) ↔ P(x)) = v(~S(x) ↔ P(x)) = {T,F}; the same if v(P(x)) = F, so that (7) is always true and false (and, hence, true). 24 B. Spinoza, Opera IV, "Letter to Jelles", 240; see also Łukasiewicz (2000), 60. Another way to put it is that any sentence must be truly negated to be informative, following Carnap's theory of information: tautology and contradiction say nothing, accordingly. 25 A sentence that can be both asserted and denied is said to be "avaktavya", whose translation is variously rendered as "unassertable", "undescriptible", "unsayable", or "unexpressible". About these Indian non-classical logics and the status of contradiction within these, see e.g. Tripathi (1968) and Schang (2009a,2010). 26 Russell's Paradox is an indirect consequence from Cantor's Paradox (1891), according to which the set of all subsets of A (the powerset of A) has a strictly greater cardinality than A itself: Card(℘(A)) > Card(A). Now if A = U is the set of all the sets, then Card(A) > Card(℘(A)). Contradiction. If we weaken A by stating that a universal set does not contain itself as a member, it follows from it that, if x is a member of itself, then it is not a member of itself by definition: x∈x → x∉x.; and, conversely, if x is not a member of itself, then it is a member of itself by definition: x∈x ← x∉x. Therefore x is a member of itself if and only if it is not a member of itself: x∈x ↔ x∉x. This settheoretical formulation may be turned into a semantic version in terms of sentential truth and falsity: the sentence p, 8 contradiction occurs in John von Neumann's distinction between a set and a class: unlike a set, a proper class cannot be included within another set and figure in the left-sided part of a membership relation. In other words, everything said in connection with God should be said, not about Him, but in His name: "God is love", e.g., must be replaced with "Love is divine". Such a reversion would be trivial in a usual logic of classes S and P, insofar as S and P can be interchangeably treated as subjects or predicates by conversion;27 but it is not whenever class and proper class are separated and accounted for by the very special nature of God. It can be finally argued that, pace Priest,28 the distinction between sets and classes is an appropriate solution to the inconsistency of God's perfection. The price to pay for a proper belief on this account is that God be not predicable, therefore not expressible by any other predicate that must be included into himself as a perfect, i.e. proper class.29 [Here, since you are using Christianity as the archetype of religious belief, you might mention that in the Judeo-Christian tradition, there is a thematic of disallowing the pronunciation of God's name. One can construe this as a response to the fact that God is not a Russellian proper name, and taking God to be that kind of name is to misunderstand God's referential status.] 4. Conclusion: an ethics of justified belief After reviewing three logics of justification and opting for one feasible but consistent view of proper belief, it seems as if the meaning of the claim that God is perfect is not, pace? per Anselm and Descartes, that God is a necessary existent, but that talk about God shows that we have to go outside the limits of language in order not to utter self-contradictory statements about God. Must he be made silent, accordingly? Yes, if we don't want to fall into misleading predications about his nature. But our incapacity to grasp a thought that is only intelligible outside of our language doesn't negate the content of such a dummy thought, it merely posits a relationship between it and our language, putting into relief the limit of the latter. In a nutshell, we are entitled to believe in God; but in silence, or in such a way that the words don't purport to rule the true and the false. The point here is not that any proper logic for understanding God should violate the rule of noncontradiction by allowing true contradictions. Rather, a believer would be entitled to reply that God created everything and rejected a world in which contradictories could be true together for His thinking creatures.30 "p is false", is true if and only if p is false. 27 Such a reversion between subjects and predicates has been urged by Ludwig Feuerbach as an atheist argument to the effect that "Man is to Man the supreme being". According to Max Stirner, "So Feuerbach instructs us that, 'if one only inverts speculative philosophy, i.e. always makes the predicate the subject, and so makes the subject the object and principle, one has the undraped truth, pure and clean'. [Anekdota II, 64]. Herewith, to be sure, we lose the narrow religious standpoint, lose the God, who from this standpoint is subject; but we take in "exchange for it the other side of the religious standpoint, the moral standpoint. E.g., we no longer say 'God is love', but 'Love is divine'. If we further put in place of the predicate 'divine' the equivalent 'sacred', then, as far as concerns the sense, all the old comes back again. According to this, love is to be the good in man, his divineness, that which does him honor, his true humanity (it 'makes him Man for the first time', makes for the first time a man out of him). So then it would be more accurately worded thus: Love is what is human in man, and what is inhuman is the loveless egoist". (M. Stirner, The Ego and its Own, New York, Tucker, B. (ed.), 1907, 61). Feuerbach assumes hereby that whatever exists does so as a subject and must be individuated. We do not. 28 According to Priest, von Neumann's distinction and Tarski's metalanguage "are not solutions. A paradox is an argument with premises which appear to be true and steps which appear to be valid, which nevertheless ends in a conclusion which is false. A solution would tell us which premise is false or which step invalid; but moreover it would give us an independent reason for believing the premise or the step to be wrong. If we have no reason for rejecting the premise or the step other than that it blocks the conclusion, then the 'solution' is ad hoc and unilluminating." (Priest (1979, 220). 29 This account of God as an unsayable entity nicely matches with the Judeo-Christian tradition as disallowing the very pronunciation of God's name. For one can construe this as a response to the fact that God is not a Russellian proper name, while taking God to be that kind of name is to misunderstand God's referential status. I thank again the anonymous referee for this relevant note. 30 Such an ethical account of the PNC is given in Lukasiewicz (2000): there is no logical proof for the PNC, which is 9 A final distinction is to be made between God as the cause of every creature and God as entertained by some of these creatures (i.e. us, the believing agents): that no one could think about him self-consistently does not imply that he does not exist, because he might still have wanted to restrict our understanding in accordance to his almighty and good deliberation. In other words: assuming that the order of truth is subjected to the order of good, whatever cannot be true for us might be true for Him because, unlike fallible creatures, his perfect will leads him not to act badly, but, in relation to creatures of an imperfect will (who cannot, for instance, will their all of their conceptions to be imaginable) should restrict the understanding of those who can. Here we have self-consistent grounds for claiming that our limited understanding is not a sufficient condition to decide about whatever is entitled to exist or not. I'm not entitled to believe anything properly about God, for want of any consistent judgment about Him. But the failure of the believer to convince the atheist conclusively that rational belief that God exists is still compatible with the belief in God and his attributes, including, most pertinently, as the one who limited our language for a good reason. Here is a sufficient evidence for being both rational and a theist. References Costa-Leite, A. (2010). "Logical properties of imagination". Abstracta – Linguagem, Mente e Ação, Special Issue VI. Hintikka, J. (1962). "Cogito ergo sum: inference of performance?". Philosophical Review, 71, 3-32. Keynes, J.N. (1884). Studies and Exercises in Formal Logic, London: MacMillan. Łukasiewicz, J. (2000). Du Principe de contradiction chez Aristote, Paris: L'Eclat. Priest, G. (1979). "The Logic of Paradox". Journal of Philosophical Logic, 8, 219-41. Russell, B. (1905). "On denoting". Mind, 14, 479-93. Schang, F. (2010a). "Two ancient dialectical logics: saptabhaṅgī and catuṣkoṭi". Journal of the Indian Council of Philosophical Research, 27, 45-75. Schang, F. (2010b). "Questions and Answers about Oppositions", forthcoming in New perspectives on the square of opposition, Béziau, J.-Y. & Payette, G. (eds), Peter Lang, Bern (2010). Tripathi, R.K. (1968). "The concept of avaktavya in Jainism", Philosophy East and West, 18, 18793. Vuillemin, J. (1971). Le Dieu d'Anselme et les apparences de la raison, Paris: Aubier Montaigne. viewed as a necessary condition for agents to live together; otherwise, mistakes and lies couldn't be avoided and would lead to self-destruction. Now if an ethical principle purports to restrict the range of possibility by will, then God can be entitled to think beyond non-contradictory things because He needn't be restricted in his perfect will.