Logic and Logical Philosophy Volume 27 (2018), 329–350 DOI: 10.12775/LLP.2017.027 Frode Alfson Bjørdal ALL PROPERTIES ARE DIVINE OR GOD EXISTS The Sacred Thesis and its Ontological Argument To the union of Åslaug Hegstad and Lars Eivind Lervåg Abstract. A metaphysical system engendered by a third order quantified modal logic S5 plus impredicative comprehension principles is used to isolate a third order predicate D, and by being able to impredicatively take a second order predicate G to hold of an individual just if the individual necessarily has all second order properties which are D we in Section 2 derive the thesis (40) that all properties are D or some individual is G. In Section 3 theorems 1 to 3 suggest a sufficient kinship to Gödelian ontological arguments so as to think of thesis (40) in terms of divine property and Godly being; divine replaces positive with Gödel and others. Thesis (40), the sacred thesis, supports the ontological argument that God exists because some property is not divine. In Section 4 a fixed point analysis is used as diagnosis so that atheists may settle for the minimal fixed point. Theorem 3 shows it consistent to postulate theistic fixed points, and a monotheistic result follows if one assumes theism and that it is divine to be identical with a deity. Theorem 4 (the Monotheorem) states that if Gg and it is divine to be identical with g, then necessarily all objects which are G are identical with g. The impredicative origin of D suggests weakened Gaunilolike objections that offer related theses for other second order properties and their associated diverse presumptive individual bearers. Nevertheless, in the last section we finesse these Gaunilo-like objections by adopting what we call an apathiatheistic opinion which suggest that the best concepts 'God' allow thorough indifference as to whether God exists or not. Keywords: modal logic; ontological argument; Gödel; higher order logic; philosophy of religion; theology Received January 17, 2017. Revised August 31, 2017. Published online September 4, 2017 © 2017 by Nicolaus Copernicus University 330 Frode Alfson Bjørdal 1. Introduction We employ the word "divine" instead of "positive" as in foregoing literature to expose certain conceptual connections which diverges from the Leibnizian philosophy that influenced Gödel. The term "the sacred thesis" is used as a metonym for the main title. Section 2 gives an account of the syntax and the evaluation semantics of the metaphysical system we presuppose, and it contains philosophical matters as well as a general discussion of higher order modal logic. Section 3 shows how impredicative comprehension with a third order modal logic suffices to define the second order property divine and the first order property God so as to establish the sacred thesis. Section 4 isolates some theorems which relate the work of Section 2 to modern work on modal ontological arguments originating with Gödel's work divulged in 1970 and accounted for most thoroughly in [15]. The conceptual and historical connections evidenced suggest that the use of terms as in the sacred thesis is justified. Section 5 points out how the sacred thesis induces an affiliated ontological argument, and it offers some considerations pertinent to the suggested reasoning and related ontological arguments. The concluding Section 6 proposes an apathiatheistic attitude to whether or not there is a God, and the advocated indifference gives room for admitting the soundness of the advanced ontological argument without committing to epistemological superstition. 2. Preliminaries In this section we seek to make a fairly precise exposition of the system we use, and we relate some references to the literature in order to facilitate comparisons for the reader, and we put forward some philosophical comments. 2.1. Syntax We let the set of symbols A = {∀, ↓, x,X, X, a, A,A,′ ,} be the alphabet of the formal language L, and define the constants, variables and terms of order one, two and three: (I) x (X , X) is a first (second, third) order variable. (II) If y (Y , Y) is a first (second, third) order variable then y′, (Y ′, Y′) is a first (second, third) order variable. (III) a (A, A) is a first (second, third) order constant. All properties are divine or God exists 331 (IV) If b (B, B) is a first (second, third) order constant then b′, (B′, B′) is a first (second, third) order constant. (V) There are no more constants or variables. (VI) All and only first (second, third) order constants and variables are first (second, third) order terms. We define the set of formulas in a Polish manner: (i) If T is a second order and t a first order term then Tt is a formula. (ii) If T is a third order and T a second order term then TT is a formula. (iii) If α is a formula then α is a formula. (iv) If α and β are formulas then ↓αβ is a formula. (v) if y is a first order variable and α is a formula then ∀yα is a formula. (vi) If Y is a second order variable and α is a formula then ∀Y α is a formula. (vii) If Y is a third order variable and α is a formula then ∀Yα is a formula. (viii) Nothing else is a formula. On the basis of the definitions above wherein parentheses are not needed, we use the following metalinguistic conventions to facilitate reading: (A) Instead of t (T , T) followed by a finite number of ′ as per (i) ((ii), (iii)) above we use succeeding letters of the appropriate alphabet for first (second, third) order terms in the exposition. (B) (¬α) is short for ↓ αα, (C) (α ∨ β) is short for ↓↓ αβ ↓ αβ, (D) (α ∧ β) is short for ↓↓ αα ↓ ββ, (E) (α → β) is short for (¬α ∨ β), (F) (α ↔ β) is short for ((α → β) ∧ (β → α)), (G) (∃y)α is short for ¬(∀y)¬α, (H) (∃B)α is short for ¬(∀B)¬α, (I) (∃B)α is short for ¬(∀B)¬α and (J) ♦α is short for ¬¬α. (K) Parentheses which are not needed to disambiguate are supressed for legibility as the occasion allows. We define the set of free first (second, third) order variables in a formula: (a) FV1(Ta) = ∅, 332 Frode Alfson Bjørdal (b) FV1(Ty) = {y}, (c) FV1(α) = FV1(α), (d) FV1(↓ αβ) = FV1(α) ∪ FV1(β), (e) FV1(∀yα) = FV1(α) \ {y}, (f) FV2(TA) = ∅, (g) FV2(TY ) = {Y }, (h) FV2(α) = FV2(α), (i) FV2(↓ αβ) = FV2(α) ∪ FV2(β), (j) FV2(∀Bα) = FV2(α) \ {B}, (k) FV3(AT ) = ∅, (l) FV3(YT ) = {Y}, (m) FV3(α) = FV3(α), (n) FV3(↓ αβ) = FV3(α) ∪ FV1(β), (o) FV3(∀Yα) = FV3(α) \ {Y}. We let α⌈t/y⌉ (α⌈T/Y ⌉, α⌈T/Y⌉) denote the formula obtained by substituting all free occurrences of the first (second, third) order variable y (Y , Y) in α with the first (second, third) order term t (T , T). A first (second, third) order variable y (Y , Y) is substitutable for first (second, third) order variable z (Z, Z) in α just if y (Y , Y) is free in α⌈y/z⌉ (α⌈Y/Z⌉, α⌈Y/Z⌉) precisely where z (Z, Z) is free in α, and a first (second, third) order constant is a closed term so substitutable for any first (second, third) order variable in any formula. The presence of Barcan formulas and their converses of all orders facilitate the definition of prenex normal form of formulas made use of in defining a formula as ∆10 iff it is equivalent to a formula that has no second order or third order quantifiers, and as Σ11 (Π 1 1) iff it is equivalent to a formula with just existential (universal) second order quantifiers to begin with in its prenex normal form. We write that a formula schema is properly parametrized if it just contains second order or third order constants that have already been introduced earlier in a derivation where it occurs. 2.2. Axiom schemas for the logic Q3S5 The superscript 3 on Q indicates that third order quantification is included, and we take the inclusion of the Barcan formulas at any order as default. All properties are divine or God exists 333 A1 α → (β → α) A2 (α → (β → γ)) → ((α → β) → (α → γ)) A3 (¬β → ¬α) → (α → β) K (α → β) → (α → β) 4 α → α B α → ♦α T α → α Q00 ∀xα ↔ ∀xα Q10 ∀Xα ↔ ∀Xα Q20 ∀Xα ↔ ∀Xα Q01 ∀xα → α⌈t/x⌉, provided t is substitutable for x in α. Q11 ∀Xα → α⌈T/X⌉, provided T is substitutable for A in α. Q21 ∀Xα → α⌈T/X⌉, provided T is substitutable for X in α. Q02 ∀x(α → β) → (∀xα → ∀xβ) Q12 ∀X(α → β) → (∀Xα → ∀Xβ) Q22 ∀X(α → β) → (∀Xα → ∀Xβ) Q03 α → ∀xα, provided x is not in FV1(α). Q13 α → ∀Xα, provided X is not in FV2(α). Q23 α → ∀Xα, provided X is not in FV3(α). 2.3. Inference rules and a derivation rule for Q3S5 We write ⊢ α for the statement that α is a thesis of Q3S5, and all axioms of subsection 2.2 (stated without parameters) are theses. We first state the five inference rules for Q3S5: Modus ponens: ⊢ α& ⊢ (α → β) ⇒ ⊢ β Necessitation: ⊢ α ⇒ ⊢ α 1-Generalization: ⊢ α ⇒ ⊢ ∀xα 2-Generalization: ⊢ α ⇒ ⊢ ∀Xα 3-Generalization: ⊢ α ⇒ ⊢ ∀Xα A derivation △ of Q3S5 is a finite sequence of formulas each of which is an instance of an axiom of Q3S5 or resulting from earlier elements of △ by means of one of the five inference rules above or by a use of the derivation rule Existential Instantiation which requires that just if one in constructing a derivation △ reaches a first (second, third) order formula 334 Frode Alfson Bjørdal ∃xα (∃Xα, ∃Xα) and constant b (B, B) has not been used earlier in △ may one existentially instantiate with said constant and in a succeeding step of the derivation △ put α⌈b/x⌉ (α⌈B/X⌉, α⌈B/X⌉). All the inference rules Modus Ponens, Necessitation, 1-Generalization, 2-Generalization and 3-Generalization are as well derivation rules in that they may be appealed to unrestrictedly in any derivation. As pointed out we avail ourselves of the derivation rule Existential Instantiation which we consider proper as a specific instantiation of a constant that it may allow at a stage in a given derivation △ may be prohibited at stages in derivation △′. We illustrate the use of constants in deriving Σ11-comprehension from Π11-comprehension for a given formula ψ where we have that ∃Y∀x(Y x ↔ ∀Zψ). Existentially instantiate with a constant A so that ∀x(Ax ↔ ∀Zψ). Use the constant A as a parameter so we have ∃W∀x(Wx ↔ ∀Z¬Ax). The last quantifier is superfluous as ∃W∀x(Wx ↔ ¬Ax), and we next existentially instantiate with B to obtain ∀x(Bx ↔ ¬Ax); as ∀x(Ax ↔ ∀Zψ), we derive ∀x(Bx ↔ ¬∀Zψ) and so by letting φ be ¬ψ and using existential generalization justified by Q11 we get ∃Y∀x(Y x ↔ ∃Zφ). 2.4. Semantics 2.4.1. Philosophical remarks In order to account for the meaning of the axiom schemas just given and the derivation and inference rules, we will elaborate upon the evaluation semantics for modal logic introduced by the author in [3] as this simplifies some matters. A translation to an account in a more standard possible worlds semantics may be undertaken, but the evaluation semantics we give has some independent interest which we point out. Notice well that the evaluation semantics we propose differs from Carnap's state description semantics and it does not as the latter validate the formula ♦α for any atomic formula α; for more on this arguably unwelcome feature of Carnap's state description semantics, confer [9]. It is true that a variety of entities can take the place of possible worlds in accounting for modal logics, and indeed a lot of theories have been offered even to clarify the meaning of the term "possible world". However, it is vitally important that the semantical apparatus presupposed also does justice to the way we speak as well as to the ontological econAll properties are divine or God exists 335 omy of our commitments. As I see it, the evaluation semantics is quite advantageous on these scores as it simplifies matters ontologically and operates with just one domain of discourse. The evaluation semantics does thus not in the semantics commit to anything like possible worlds, which have shown themselves prone to lead philosophical and related discourse astray. Instead the valuations of the evaluation of any model are intuitively just the properties, states or situations the world may be in; model theoretically the valuations are just properties of formulas or ordered pairs from variables and objects with an appropriate set theoretical relation to the domain of discourse. As I see it there is nothing ontologically untoward by a metalogical commitment to properties which are not instantiated by the world, and we do not intend the evaluation semantics to account for what it means to be a property. Accordingly the evaluationist semantics induces a replacement of the slogan that necessary truths are to be identified with "truth in all possible worlds" with the motto that "a necessary truth is a statement that is true in all possible states of the world". 2.4.2. Evaluation models A novelty with the evaluation semantics is the unusual conditions rhombicity and quadrangularity for order 1 through 3 which justify the Barcan formula and its converse at that order. Pay also attention to the fact that we do not invoke the power set of the domain of discourse in our statement of the semantics. We suppress the semantical consideration of constants in derivations in this exposition. In the next subsection we illustrate how the semantics validates a selection of axioms and inference rules of the third order logic Q3S5, and we further below introduce a moderate level of second order and third order comprehension. Let an evaluation model M be a triple 〈D,E,R〉 where D is the domain of discourse, E is an evaluation and R ⊂ E2 is the accessibility relation on E. An evaluation E is a set of valuations, and the latter we denote by V , V ′ and its kins. Valuations are monadic properties or attributes or states which only hold of some formulas of L and some ordered pairs 〈y, d〉 (〈Y, δ〉, 〈Y,∆〉 )where y (Y , Y ) is a first, (second, third) order variable of L and d (δ, ∆) is intuitively the value the valuation assigns the said variable relative to the domain of discourse D: If V ∈ E and y is a first order variable of L then V(〈y, d〉) only if d ∈ D. If V ∈ E and Y is a second order variable then V(〈Y, δ〉) only if δ ⊂ D. If V ∈ E andY is a third order variable then V(〈Y,∆〉) only if ∀x(x ∈ ∆ ⇒ x ⊂ D). 336 Frode Alfson Bjørdal Valuations are to be functional in that for V ∈ E and y a first order variable V(〈y, d〉) and V(〈y, d'〉) only if d is d', so we write V (y) = d for V (〈y, d〉); we for the same reason write V (Y ) = δ for V (〈y, δ〉) and V (Y) = ∆ for V (〈Y,∆〉). For any first (second, third) order variable y (Y , Y) we write V y©V ′ (V Y©V ′, V Y©V ′) to express that V and V ′ at most differ in that V (y) 6= V ′(y). To save space we use Σ and Π for existential and universal generalisation at the metalevel, and for the same reason we use ∼ for negation, ⇒ for implication. ⇔ for biimplication and & for conjunction at the metalevel. With these notions we require for any model M = 〈D,E,R〉 and V ∈ E and first order variable y, second order variable Y , third order variable Y and formulas α, β ∈ L: 1-denotation: Σd(d ∈ D & V (y) = d) 2-denotation: Σδ(δ ⊂ D & V (Y ) = δ) 3-denotation: Σ∆(Πy(y ∈ ∆ ⇒ (y ⊂ D)) & V (Y) = ∆) Higher atoms: V (YY ) ⇔ (V (Y ) ∈ V (Y)) Lower atoms : V (Y y) ⇔ (V (y) ∈ V (Y )) NOR: V (↓ αβ) ⇔ (∼V (α) & ∼V (β)) 1-Generality: V (∀yα) ⇔ ΠV ′(V y©V ′ ⇒ V ′(α)) 2-Generality: V (∀Y α) ⇔ ΠV ′(V Y©V ′ ⇒ V ′(α)) 3-Generality: V (∀Yα) ⇔ ΠV ′(V Y©V ′ ⇒ V ′(α)) Apodicticity: V (α) ⇔ ΠV ′(VRV ′ ⇒ V ′(α)) 1-Rhombicity: (VRV ′&V ′ y©V ′′) ⇒ ΣV ′′′(V y©V ′′′&V ′′′RV ′′) 1-Quadrangularity: (V y©V ′&V ′RV ′′) ⇒ ΣV ′′′(VRV ′′′&V ′′′ y©V ′′) 2-Rhombicity: (VRV ′&V ′ Y©V ′′) ⇒ ΣV ′′′(V Y©V ′′′&V ′′′RV ′′) 2-Quadrangularity: (V Y©V ′&V ′RV ′′) ⇒ ΣV ′′′(VRV ′′′&V ′′′ Y©V ′′) 3-Rhombicity: (VRV ′&V ′ Y©V ′′) ⇒ ΣV ′′′(V Y©V ′′′&V ′′′RV ′′) 3-Quadrangularity: (V Y©V ′&V ′RV ′′) ⇒ ΣV ′′′(VRV ′′′&V ′′′ Y©V ′′) 1-Indifference: (V (α) & x /∈ FV(α)) ⇒ ΠV ′(V x©V ′ ⇒ V ′(α)) 2-Indifference: (V (α) & X /∈ FV(α)) ⇒ ΠV ′(V X©V ′ ⇒ V ′(α)) 3-Indifference: (V (α) & X /∈ FV(α)) ⇒ ΠV ′(V X©V ′ ⇒ V ′(α)) If 1-Rhombicity and 1-Quadrangularity had not been assumed and we had deleted references to principles of higher order, we would have generated a Kripke style semantics as in [19] for first order modal logic with the peculiar restrictions on the use of open sentences. As I see All properties are divine or God exists 337 it, 1-Rhombicity and 1-Quadrangularity are plausible principles, but the associated first order Barcan Formula and its converse are not needed in the derivation of the Sacred Thesis below. I mention en passant that I welcome the necessitist consequences of adopting the first order Barcan formula and its converse. The necessitist point of view that all things exist necessarily has been most extensively defended by Timothy Williamson in [30] and foreshadowed by [31], and it was also implicit in [20]. The author independently came up with a necessitist point of view in the lecture [7]. The argument for necessiticm in the evaluationist framework is different, and appeals to there being such a plenitude of valuations in the evaluation so that 1-Rhombicity and 1-Quadrangularity obtain. 2.4.3. Semantical validations We first establish the basic truth functional conditions for valuations given the definitions (B) to (F) in section 2.1 and the conditions for NOR in section 2.4.2 and our metalinguistic conventions. We presuppose the truth functional logic of the meta language and take previous steps into account as we proceed: NOT V (¬α) ⇔ V (↓ αα) ⇔ (∼V (α) & ∼V (α)) ⇔ not V (α) OR V (α ∨ β) ⇔ V (↓↓ αβ ↓ αβ) ⇔ (∼V (↓ αβ) & ∼V (↓ αβ)) ⇔ not V (↓ αβ) ⇔ not (∼V (α) & ∼V (β)) ⇔ (V (α) or V (β)) AND V (α ∧ β) ⇔ V (↓↓ αα ↓ ββ) ⇔ (∼V (↓ αα) & ∼V (↓ ββ))) ⇔ (∼V (¬α) and ∼V (¬β)) ⇔ (V (α) and V (β)) IF V (α → β) ⇔ V (¬α ∨ β) ⇔ (not-V (α) or V (β)) IFF V (α ↔ β) ⇔ (V (α → β) and V (β → α)) In validating the axioms of Q3S5 we in each case assume that there is an evaluation model to the contrary and derive a contradiction, and we assume that the relation R in the model M = 〈D,E,R〉 is an equivalence relation. We now validate the axiom schemas of Q3S5 as provided in 2.2: A1-3 These follow from NOT, OR, AND, IF and IFF. K Suppose a valuation V has V (α) and V ((α → β)) and not V (β). From the latter, V (¬β) so that V (♦¬β) and so ΣV ′(V RV ′ & ∼V ′(β)), and we let V ∗ be a representative so that V RV ∗ & ∼V ∗(β)). From the former ΠV ′(V RV ′ ⇒ V ′(α)) and ΠV (V RV ′ ⇒ V ′(α → β)) so that by truth functionality of valuations ΠV ′(V RV ′ ⇒ V ′(β)), and 338 Frode Alfson Bjørdal so by instantiation with the representative V ∗ we get (V RV ∗ ⇒ V ∗(β)) which contradicts V RV ∗ &mathop∼V ∗(β). 4 Assume V ((α)) and not V ((α)). From these we derive that ΠV ′(VRV ′ ⇒ V ′(α)) and ΣV ′, V ′′(VRV ′ & V ′RV ′′ & ∼V ′′(α)). As R is transitive it follows that ΣV ′(VRV ′ & ∼V ′(α)), so we have a contradiction. B Assume V (α) and not V (♦α). It follows that V (α) and V (♦¬α), so that V (α) and ΣV ′(VRV ′&ΠV ′′(V ′RV ′′ ⇒ ∼V ′′(α)))). By a choice of instantiation in the latter we have that VRV ′ &V ′RV ⇒ ∼ V (α))), and that V (α). As R is supposed to be symmetric we derive that V ′RV and the contradiction that V (α) and ∼V (α). T Assume to the contrary that V (α) and not V (α)). Frome these we have that ΠV ′(VRV ′ ⇒ V ′(α)) and not V (α), and so by instantiation VRV ⇒ V (α) and not V (α). As R is reflexive we have a contradiction. Q0 0 a) Assume V (∀xα) and not V (∀xα). From the former we deduce ΠV ′(V x©V ′ ⇒ ΠV ′′(V ′RV ′′ ⇒ V ′′(α))) and from the latter we derive that ΣV ′(VRV ′ & ΣV ′′(V ′ x©V ′′ & ∼V ′′(α))). Let V ∗ and V ∗∗ be representatives so that VRV ∗ & V ∗ x©V ∗∗ & ∼V ∗∗(α)). By 1-Rhombicity we find a V ∗∗∗ such that (V x©V ∗∗∗ & V ∗∗∗RV ∗∗ & ∼V ∗∗(α)). If we now instantiate with with V ∗∗∗ and then with V ∗∗ in ΠV ′(V x©V ′ ⇒ ΠV ′′(V ′RV ′′ ⇒ V ′′(α))) aabsurdity follows. b) Assume V (∀xα) and not V (∀xα). A contradiction is derived as in a) by invoking 1-Quadrangularity. Q1 0 As Q0 0 , but invoking 2-Rhombicity and 2-Quadrangularity. Q2 0 As Q0 0 , but invoking 3-Rhombicity and 3-Quadrangularity. Q0 1 Suppose V ((∀x)α) and not V (α⌈t/x⌉), where t is substitutable for x in α. We then have ΠV ′(V x©V ′ ⇒ V ′(α)) and not V (α⌈t/x⌉). Let V x©V ′ and V ′(x) = V (t), which entails that V ′(α) and that V ′(α) ⇔ V (α⌈t/x⌉). A contradiction follows as we also assumed not V (α⌈t/x⌉). Q1 1 As Q0 1 . Q2 1 As Q0 1 . Q0 2 Suppose V (∀x(α → β)), V (∀xα) and not V (∀xβ)). It follows that ΠV ′(V x©V ′ ⇒ V ′(α → β)), ΠV ′(V x©V ′ ⇒ V ′(α)) and ΣV ′(V x©V ′ & ∼V (β)). Let V x©V ∗ & not V ∗(β), and instantiate in the first to get V x©V ∗ ⇒ V ∗(α → β) and V x©V ∗ ⇒ V ∗(α). By detachment and repetition we have V ∗(α) and V ∗(α → β) and not V ∗(β), so a contradiction follows by the truth functionality of V ∗. All properties are divine or God exists 339 Q1 2 As Q0 2 . Q22 As Q 0 2. Q0 3 By 1-Indifference. Q1 3 By 2-Indifference. Q2 3 By 3-Indifference. 2.5. The comprehensive metaphysical system Q3S5Π 1,2 1,1 The metaphysical system Q3S5Π1,21,1 comes about by adding the following impredicative parametrized second and third order comprehension schemas 2C and 3C to the logic Q3S5, as we to make space employ conventional vector notation to abbreviate that we may have several succeeding variables: 2C ∀~x∀ ~X∃Y ∀y(Y y ↔ ∀~Zβ(~x, y, ~X, ~Z)) for β any ∆10 formula. 3C ∀~x∀ ~X∀~X∃Y∀Y (YY ↔ ∀~Zβ(~x, Y, ~X,~X, ~Z)) for β any Σ11 formula. 2.6. Higher order modal logics Higher order modal logic was appealed to in the work of Montague to account for phenomena in the semantics of natural language in a series of very influential papers [23, 24, 25] which were also reprinted in the collection [29]. The monograph [14] of Montague's student Gallin develops Montague's Intensional Logic further. In the useful survey article [26] Muskens also relate the work of Bressan on higher order modal logics in the monograph [8] which was motivated by axiomatizing classical mechanics. In a standard semantics for a second order logic the second order quantifiers are taken to range over the full power set ℘(D) of the domain D presupposed for the firt order quantifyers. In a Henkin semantics it is instead presupposed that the second order quantifiers range over a general structure which extends pre-structures built from the domain D such that all second order comprehension principles are validated. Our investigation shall be carried through with a third order monadic modal predicate logic extending a second order monadic modal predicate logic which in its turn extends a first order modal predicate logic. We do not need first order UI or EG in the derivation of the Sacred Theses. So we might have presupposed a free logic as discussed 340 Frode Alfson Bjørdal towards the end of 2.4.2, though extended with higher order principles as indicated. We mention that Hájek in [16, 17] uses moderate cautious comprehension whereas Sobel appeals to full comprehension in the criticism of Gödel, and also Anderson uses full second order comprehension in his rectification of Gödel's proof. Also Hájek's "vorsichtige Komprehension" is impredicative. 3. Deriving the Sacred Thesis The terms K-axiom, T -axiom, B-axiom, 4-axiom and 5-axiom have their expected meaning as from the literature (cfr. also section 2.2); we suppress the appendage "schema" for brevity, and overlook that the axiom schemas so named are not independent of each other. We assume that gothic majuscules stand for monadic third order predicate letters, and α⌈M/N⌉ denotes the formula obtained from replacing all occurrences of third order predicate letter N in formula α with third order predicate letter M. We consider formula α modally operative in C iff ∀H(AH → BH) entails that (α⌈A/C⌉ → α⌈B/C⌉). If we disregard the contribution modal operators make to complexity and comprehensive strength it turns out that just an instance of Π21-comprehension to obtain a third order property D and an instance of Π11-comprehension with D as parameter to isolate second order property G are the impredicative principles needed for the crucial definitions needed to obtain the sacred thesis. Use the modal Π21-comprehension principle of Q 3S5Π1,21,1 to obtain: (0) ∃X∀X(XX ↔ ∀Y((∀x(∀Y (YY → Y x) → Xx) → YX) → YX)) Existentially instantiate with the constant D: (1) ∀X(DX ↔ ∀Y((∀x(∀Y (YY → Y x) → Xx) → YX) → YX)) It follows that: (2) ∀X∀Y((∀x(∀Y (YY → Y x) → Xx) → YX) → (DX → YX)) In third order S5 the following is a thesis: (3) ∀X∀Y((∀x(∀Y (YY → Y x) → Xx) → YX) ↔ (∀x(∀Y (YY → Y x) → Xx) → YX)) So by a hypothetical syllogism the following is a thesis of third order S5: (4) ∀X∀Y((∀x(∀Y (YY → Y x) → Xx) → YX) → (DX → YX)) All properties are divine or God exists 341 By necessitation and the second and third order converse Barcan formulas: (5) ∀X∀Y((∀x(∀Y (YY → Y x) → Xx) → YX) → (DX → YX)) By the K-axiom and an application of the 4-axiom it follows that: (6) ∀X∀Y((∀x(∀Y (YY → Y x) → Xx) → YX) → (DX → YX)) On account of (3) and (6): (7) ∀X∀Y((∀x(∀Y (YY → Y x) → Xx) → YX) → (DX → YX)) By the modal operativity of ∀x(∀Y (BY → Y x) → Xx) in B: (8) ∀X∀Y((DX → YX) → (∀x(∀Y (DY → Y x) → Xx) → ∀x(∀Y (YY → Y x) → Xx))) From (7) and (8) by a hypothetical syllogism: (9) ∀X∀Y((∀x(∀Y (YY → Y x) → Xx) → YX) → (∀x(∀Y (DY → Y x) → Xx) → ∀x(∀Y (YY → Y x) → Xx))) By a truth functional argument on (9): (10) ∀X∀Y((∀x(∀Y (YY → Y x) → Xx) → YX) → (∀x(∀Y (DY → Y x) → Xx) → YX)) By universal instantiation and change of order: (11) ∀x(∀Y (DY → Y x) → Xx) → ∀Y((∀x(∀Y (YY → Y x) → Xx) → YX) → YX)) By the T -axiom: (12) ∀x(∀Y (DY → Y x) → Xx) → ∀Y((∀x(∀Y (YY → Y x) → Xx) → YX) → YX) By (1) and (12) and universal generalization and necessitation: (13) ∀X(∀x(∀Y (DY → Y x) → Xx) → DX) Use third order comprehension with D as parameter and existential instantiation to define F so that: (14) ∀X(FX ↔ ∀x(∀Y (DY → Y x) → Xx)) From (13) and (14) we have: (15) ∀X(FX → DX) 342 Frode Alfson Bjørdal By the modal operativity of ∀x(∀Y (YY → Y x) → Xx) in Y we have: (16) ∀x(∀I(FI → Ix) → Xx) → ∀x(∀I(DI → Ix) → Xx) By (14) and (16) we have: (17) ∀x(∀Y (FY → Y x) → Xx) → FX By the 4-axiom of third order S5 we have: (18) ∀x(∀Y (FY → Y x) → Xx) → FX By (18) and an instance of (1) it follows: (19) ∀X(DX → FX) Combining (15) and (19) we have: (20) ∀X(DX ↔ FX) Using (14), this is to say: (21) ∀X(DX ↔ ∀x(∀Y (DY → Y x) → Xx)) Define, by using second order comprehension with parameter D: (22) ∃X∀x(Xx ↔ ∀Y (DY → Y x)) Existentially instantiate with second order constant G: (23) ∀x(Gx ↔ ∀Y (DY → Y x)) From substitution of (23) in (21): (24) ∀X(DX ↔ ∀x(Gx → Xx)) From (24) and the fact that ∀x(Gx → Gx) is a thesis: (25) DG By the 4-axiom: (26) DG From (24) and the fact that ♦∃x(Gx ∧ ¬Xx) → ♦∃x(Gx): (27) ¬DX → ♦∃x(Gx) From (23) by instantiation, simplification and permutation: (28) (DG → ∀x(Gx → Gx)) All properties are divine or God exists 343 From (26) and (28) and the K-axiom: (29) ∀x(Gx → Gx) From (29) and quantifier rules: (30) (∃xGx → ∃xGx) The following is a theorem of all quantified modal logics: (31) (∃xGx → ∃xGx) From (30) and (31): (32) (∃xGx → ∃xGx) From (32) and the 4-axiom: (33) (∃xGx → ∃xGx) From (33) using an equivalent of the K-axiom: (34) ♦∃xGx → ♦∃xGx From the B-axiom: (35) ♦∃xGx → ∃xGx From (34) and (35): (36) ♦∃xGx → ∃xGx From (27) and (36): (37) ¬DX → ∃xGx As X is arbitrary, by generalization: (38) ∀X(¬DX → ∃xGx) From the quantifier rules: (39) ∃X¬DX → ∃xGx By connective rules and interdefinability we have: (40) ∀XDX ∨ ∃xGx 4. Bonds and theorems This is not the occasion to expound the extensive literature initiated by the circulation of Gödel's handwritten note, and such an endeavor 344 Frode Alfson Bjørdal would at any rate fall outside the scope of the author's competence. In [28] Sobel showed that the assumptions of Gödel in [15] lead to modal collapse in the sense that α ↔ α becomes a thesis with Gödel for all formulas α; but confer Hájek's qualification referred to in the next section. In [1] Anderson presented a rectified ontological argument that avoids the problem pointed out by Sobel. We list a translation of Gödel's Definition 1 (DG1), Anderson's first definition (DA1) and the first four of Anderson's five axiomatic assumptions (AA1-4) into our idiolect with divine instead of positive and a use of overline for denoting the complement of a property; we stress that just AA3 is a proper axiom: DG1 Gx , ∀I(DI → Ix) DA1 Gx , ∀I(DI ↔ Ix) AA1 DA → ¬DĀ AA2 DA ∧ ∀x(Ax → Bx) → DB AA3 DG AA4 DA → DA In our idiolect Gödel presupposed an axiom AG1 stating that DA ↔ ¬DĀ instead of Anderson's AA1, and the translation of Gödel's AG2 to AG4 would be as AA2 to AA4. Anderson and Gödel further presupposed axiomatically that some defined property akin to necessary existence is divine, and below we refer to these respective assumptions as AA5 and AG5. We use a modification of Gödel's DG1 and Anderson's DA1: D Gx , ∀I(DI → Ix) Theorem 0. (i) The biconditional of definition D,(ii) AA2, (iii) AA3, and (iv) AA4 are theses of S5Π1,21,1. Proof. (i) By step (23) of Section 3. (ii) From step (24) of Section 3 and the transitivity of strict implication. (iii) By step (25) of Section 3. (iv) By step (21) and other steps of Section 3 and the 4-axiom of Q3S5Π1,21,1 We leave it as an exercise to establish: Theorem 1. Q3S5Π1,21,1 + ∃X¬DX ⊢ AA1 and Q3S5Π1,21,1 + AA1 ⊢ ∃X¬DX All properties are divine or God exists 345 We next establish that Q3S5Π1,21,1 + ∃X¬DX does not have all sentences as consequences: Theorem 2. Q3S5Π1,21,1 + ∃X¬DX has a model. Proof. Let the set E of valuations be a singleton set {V }, the domain D = D0∪D1∪D2 whereD0 = {g} is the domain for the first order quantifiers, D1 = {{g}, ∅} is the domain for the second order quantifiers and the domain for the third order quantifiers is D2 = {∅, {∅}, {∅, {g}}, {{g}}}. The accessibility relation R on E is {〈V, V 〉}. We leave it as an exercise to verify that a model with these ingredients serves. Theorems 0 to 2 justify us in thinking of the derivation of the salient conclusion of Q3S5Π1,21,1 + ∃X¬DX in such terms as have been used in the literature on modal ontological argument in the tradition from [15]. We define a = b in a Leibnizian-Russellian manner as given by the formula ∀F (Fa → Fb). It is straightforward to show that identity so taken is symmetric given the impredicative second order comprehension principle. Identity of second order predicates can be isolated analogously by an appeal to our third order impredicative comprehension. In the following we make use of Church's lambda operator so that λxFx is an alternative name for the property F . Theorem 3 (The Monotheorem). If Gg and Dλx(x = g), then ∀x(Gx → x = g). Proof. Given (24) in Section 3 we have Dλx(x = g) ↔ ∀x(Gx → λx(x = g)x). As we assume Dλx(x = g) we have ∀x(Gx → λx(x = g)x), so that by beta abstraction we have ∀x(Gx → x = g). 5. Pinpoints and diagnoses We will in this paper not scrutinize possible objections towards the use of the logical and metaphysical apparatus invoked, and now just mention en passant that the use of systems such as our third order modal logic has been pointed to e.g. in [14, 30]. It nevertheless bears mentioning that we have invoked a rather impredicative system that some philosopers with a predicativist bent are prone to find dubious. Precisely the quasi circularity with impredicative principles such as we have invoked has been an aversion to predicativists, and those in the mathemathical minority who want to press 346 Frode Alfson Bjørdal predicativist points of views may raise similar doubts concerning other ontological arguments. Feferman in [11] gives a good introduction to the predicativist point of view and its tradition back to Cohen, Poincaré and Russell at the beginning of the previous century. It is popular to disagree with predicativist strictures for being too prohibitive as concerns self-reference. In my opinion, a respectable framework for thinking about modal matters should also deal properly with modal paradoxes, as isolated by Montague in [22] and others, in some comprehensive way that also deals with the set theoretical paradoxes and thus accounts more appropriately for the foundation of mathematics. Some readers may at this point find it relevant to consult the author's take on the paradoxes in his alternative set theory £ as so far set out in [4, 5] and in some preceding work. The definiens of definition D in Section 4 is employed crucially in step (1) of Section 3, and the definiens of Anderson's definition DA1 could not have been so presupposed for the formula ∀x(∀I(GI ↔ Ix) → Hx) is not modally operative in G as the latter has negative occurrences in the former. In [16, 17] Hájek improves upon an, as it turns out, just slightly insufficient model theoretic argument by Magari in [21] to the effect that Gödel's AG1, AG2 and AG3 already suffice for the derivation of the theistic main theorem. Magari's idea is confirmed as applied to Anderson's argument for Hájek shows that Anderson's AA3 and AA4 are superfluous in the presence of full second order modal comprehension with the underlying logic. Hájek also shows that such a comprehensive Andersonian argument is interpretable in Gödel's original set of axioms with a cautious comprehension principle, and that such a cautious version of Gödel's argument does not lead to the modal collapse which Sobel derived presupposing full comprehension in [28]. The reader is sent to Hájek's articles for more details. In [2] the author showed that a simplification using just a part of Gödel's first axiom and some version of his last axiom suffices for the theistic conclusion, and it was noted that a combination with Hájek's independent result can be obtained to simplify matters further. This article delivers upon the promise of such an improved result. We mention that [2] was referred to by André Fuhrmann in [13], and he correctly related the point that we in [2] take the property of being Godlike as primitive unlike Gödel's more Leibnizian approach of taking the second order property positive (with us divine) as primitive. This All properties are divine or God exists 347 issue as to what is more primitive of divine and God disappears in our third order framework. One may use the negation of the first disjunct of the sacred thesis as premise in an ontological argument for the necessary existence of a God, and we say that that argument is the ontological argument supported by the sacred thesis. The said ontological argument is naturally analyzed by a fixed point strategy, and we summarily state some upshots. A least fixed point is atheist, and atheists are committed to the point of view that all properties are divine in the sense of our definition. There may be theistic fixed points which commit to properties which are not divine, and such fixed points have simple models as in Theorem 2. The content of Theorem 3, the Monotheorem, is that if there is a God and being identical with a God is a divine property then also monotheism is true. On account of the given fixed point diagnosis one may resist the objection by [27] towards Gödelian ontological arguments that arbitrary second order properties may be substituted for divine. It bears noticing that the sacred thesis and its ontological argument do not depend upon there being some maximum for properties like being positive or being perfect, as with Anselm or Leibniz and others. Nevertheless, somewhat Gaunilo-like objections remain and corresponding to the sacred thesis there is e.g. a similar diabolical thesis and related theses for other second order properties and their associated presumptive bearers. Statements as the sacred thesis and kindred theses do not by themselves carry ontological commitments, however. Notice that by our results the fixed point analysis propounded here applies as well to related modal ontological arguments. 6. Apathiatheistic appendage Our attitude to arguments such as the ontological argument supported by the sacred thesis carries information concerning our attitude to ratiocination or ontology which may be of some wider interest. As I see it, the word "God" is well nigh maximally imprecise so that it seems rather unhelpful to be a militant atheist without clarifying what interpretation one has of the word; by the same token, it is not so useful to be a zealous theist. We are certainly all militant atheists with respect to some interpretations of the word "God" and no serious person is a theist with respect to more than a small range of interpretations of the word. Notice 348 Frode Alfson Bjørdal that one initially better takes "God" to express a predicate so as to not favor monotheism and so as to avoid certain befuddlements concerning supposedly non-referring names. I have, as opposed to the theophilic and theophobic attitudes considered by Magari in [21], come to adopt what I call an apathiatheistic opinion according to which the best concepts 'God' cum understandings of reality are such that the question as to whether there is a God or not is academic in a sense similar to the question as to whether there are holes or just holed things. If God is such an object as apathiatheism requires then there is, as I see it, not sufficient untowardness ratiocinatively or ontologically with accepting the soundness of the ontological argument supported by the sacred thesis for this to be ruled out. From an apathiatheistic point of view atheist thinkers may as well engage in religious discourse as theological anti realists who take "God" as a theoretical term, and I do not see that apathiatheism is in conflict with reasonable theologies. Arguably, the most important religious question is not whether there is a God but whether something ultimately rectifies the unsayable sufferings in the world; I have scrutinized this and related religious and philosophical questions in [6]. However these matters are, from my apatiatheistic point of view the world may be as if there is a God even though there is no God and it may be as if there is no God even though there is a God. Acknowledgements. Let me express my gratitude to the anonymous referees for useful suggestions which improved the paper. I am also thankful to Anil Gupta as he many years ago clarified an elementary question I had on matters relating to impredicative definitions, and I am grateful to Sara Negri for spurring me on to write this paper; hopefully having a comprehensive account published is better than having half finished versions in circulation. I am also thankful to Giovanni Sambin for providing me with a copy of [21]. Let me last, but not least, express gratitude to Åslaug Hegstad and my nephew Lars Eivind Lervåg for getting married, as I in a speech to them in their wedding committed to dedicate such a paper as this to them. References [1] C. A. Anderson, "Some emendations of Gödel's ontological argument", Faith and Philosophy VII, 3 (1990): 291–303. All properties are divine or God exists 349 [2] F. A. Bjørdal, "Understanding Gödel's ontological argument", pages 214– 217 in T. 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Blacburn et al. (eds.) Handbook of Modal Logic, Elsevier 2006. DOI: 10.1016/S15702464(07)80013-9 [27] G. Oppy, "Gödelian ontological arguments", Analysis 56, 4 (1996): 226– 230. DOI: 10.1093/analys/56.4.226 [28] J. H. Sobel, "Gödel's ontological proof", pages 241-261 in J. J. Thomson (ed.), On Being and Saying, MIT Press 1987. [29] R. Thomason (ed), Formal Philosophy, Selected papers of Richard Montague, Yale University Press 1974. [30] T. Williamson, Modal Logic as Metaphysics, Oxford University Press 2013. DOI: 10.1093/acprof:oso/9780199552078.001.0001 [31] T. Williamson, "Bare possibilia", Erkenntnis 48 (1998): 257–273. Frode Alfson Bjørdal University of Oslo; Universidade Federal do Rio Grande do Norte Filosofiseksjonen ved IFIKK; Programa de pós-graduaçao em filosofia frode.bjordal@ifikk.uio.no