A Computationally Generated Ontological Argument Based on Spinoza s The Ethics: Part 2

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Theodore Blair
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1 A Computationally Generated Ontological Argument Based on Spinoza s The Ethics: Part 2 Jack K. Horner PO Box 266, Los Alamos NM jhorner@cybermesa.com ICAI 2014 Abstract The comments accompanying Proposition (Prop.) 11 ( God... necessarily exists ) in Part I of Spinoza s The Ethics contain sketches of at least three distinct ontological arguments. The first of these is suspiciously short. But worse is true: even the proposition "God exists" (GE), an implication of Prop. 11, cannot be derived from the definitions and axioms of Part I (the "DAPI") of The Ethics; thus, Prop. 11 cannot be derived from the DAPI, either. In a companion paper, I describe an automated first-order model-generator for the DAPI of The Ethics and use it to prove that Prop. 11 is independent of those definitions and axioms. In this paper, I augment the DAPI with some auxiliary assumptions that I believe Spinoza would accept and which sustain an automated derivation of (GE). The analysis demonstrates how an automated deduction system can augment traditional methods of textual exegesis. Keywords: automated deduction, textual exegesis, Spinoza 1.0 Introduction There are two objectives of this paper. The first is to explore some aspects of the validity, that is, whether there is a derivation that follows by inference rules alone from the premises of an argument of interest, of one of Spinoza's arguments for Prop. 11 ([1]). The second is to demonstrate how an automated deduction system can augment traditional methods of textual exegesis. I take no position on whether any proposition in [1] is true. Ontological arguments are arguments for the claim that (GE) God exists based on premises which purport to derive from some source other than observation of the world ([11]), that is, from reason alone. They have a long history in the philosophical literature, extending to at least Anselm ([12]). The comments accompanying Prop. 11 ( God, or substance, consisting of infinite attributes, of which each expresses eternal and infinite essentiality, necessarily exists ) in Part I of Spinoza s The Ethics ([1]) contain sketches of at least three distinct ontological arguments. The first argument sketch of Proposition 11 in [1] (p. 51) is remarkably short. If we deny Proposition 11, Spinoza asserts, then we are claiming that God does not exist. But to deny God's existence, he continues, is to deny that God's

2 essence involves God's existence. That, however, is absurd, ([1], p. 48), he argues, by virtue of Prop. 7 ("Existence belongs to the nature of substances"). I argue in a companion paper that no derivation of Prop. 11 from the DAPI can succeed, because (GE), which is an implication of Prop. 11, cannot be derived from the DAPI. Here, I augment the DAPI with some plausible propositions I believe Spinoza would have accepted, and use that combination to derive (GE). 2.0 Method The method used in this paper to investigate the first argument sketch of Prop. 11 of [1] relies heavily on an automated deduction system (ADS, [2]). It is in some ways similar to the method of [3], which is likely to be the first published use of [2] in "conventional" philosophical literature. [2] and its predecessors have been widely used in mathematical logic applications (see, for example, [8], [9]) for at least two decades. prover9 ([2]) is a software first-order automated deduction framework that searches for derivations of given propositions, given a set of propositions. The script shown in Section 3.0 was executed on a Dell Inspiron 545 with an Intel Core2 Quad CPU Q GHz) and 8.00 GB RAM, running under the Windows Vista Home Premium /Cygwin operating environment. 3.0 Results Figure 1 shows the prover9 script used to derive (GE) from the DAPI conjoined with some plausible auxiliary assumptions formulas(assumptions). % DEFINITIONS % Definition of self-caused. SelfCaused(x) <-> ( EssenceInvExistence(x) & NatureConcOnlyByExistence(x) ) # label("definition I: self-caused"). % Definite of "finite after its kind". FiniteAfterItsKind(x) <-> ( CanBeLimitedBy(x,y) & SameKind(x,y) ) # label("definition II: finite after its kind"). % Definition of substance. Substance(x) <-> InItself(x) & CanBeConceivedThruItself(x) # label("definition III: substance"). % Definition of attribute. Attribute(x) <-> IntPercAsConstEssSub(x) # label("definition IV: attribute"). % Definition of mode. Mode(x) <-> ( ( Modification(x,y) & Substance(y) ) ( ExistsIn(x,y) & ConceivedThru(x,y)) ) # label("definition V: mode").

3 % Definition of God. God(x) <-> ( Being(x) & AbsolutelyInfinite(x) ) # label("definition VI: God"). % Definition of absolutely infinite. AbsolutelyInfinite(x) <-> ( Substance(x) & ConstInInfAttributes(x) & ( AttributeOf(y,x) -> ( ExpressesEternalEssentiality(y) & ExpressesInfiniteEssentiality(y) ) ) ) # label("definition VI: absolutely infinite"). % Definition of free. Free(x) <-> ( ExistsOnlyByNecessityOfOwnNature(x) & ( ActionOf(y,x) -> DeterminedBy(y,x) ) ) # label("definition VII: free"). % Definition of necessary. Necessary(x) <-> ( ( ExternalTo(y,x) & DeterminedByFixedMethod(x,y) & DeterminedByDefiniteMethod(x,y) ) & ( IsMethod(Action(y) IsMethodExistence(y) ) ) ) # label("definition VII: necessary"). % Definition of eternity. Eternity(x) <-> ExistConcFollowFromDefEternal(x) # label("definition VIII: eternity"). % AXIOMS % Axiom I. Everything which exists, exists either in itself % or in something else. Exists(x) <-> ( ExistsIn(x,x) (ExistsIn(x,y) & (x!= y) ) ) # label("axiom I"). % Axiom II. That which cannot be conceived through itself must % be conceived through something else. -( ConceivedThru(x,x) ) -> (ConceivedThru(x,y) & (x!= y) ) # label("axiom II"). % Axiom III. From a given definite cause an effect necessarily % follows; and, on the other hand, if no definite % cause be granted, it is impossible that an effect % can follow. DefiniteCause(x) -> ( EffectNecessarilyFollowsFrom(y,x) & ( -DefiniteCause(x) -> -EffectNecessarilyFollowsFrom(y,x) ) ) # label("axiom III"). % Axiom IV. The knowledge of an effect depends on and involves % the knowledge of a cause. KnowledgeOf(x,y) <-> IsACause(x,y) # label("axiom IV: The knowledge of an effect depends on and involves the knowledge of a cause"). % Axiom V. Things which have othing in common cannot be understood, % the one by the means of the other % the one by means of the other; the conception of one % does not involve the conception of the other. HaveNothingInCommon(x,y) -> ( (-CanBeUnderstoodInTermsOf(x,y) ) & (-CanBeUnderstoodInTermsOf(y,x) ) & ( -ConceptionInvolve(x,y) ) & (-ConceptionInvolves(y,x) ) ) # label("axiom V: Things which have nothing in common cannot be understood, the one by means of the other."). % Axiom VI. A true idea must correspond with its idea or object. TrueIdea(x) -> ( CorrespondWith(x,y) & ( IdeateOf(y,x) ObjectOf(y,x) ) ) # label("axiomvi").

4 % Axiom VII. If a thing can be conceived as non-existing, its % essence does not involve its existence. CanBeConceivedAsNonExisting(x) -> -EssenceInvExistence(x) # label("axiom VII"). % AUXILIARY ASSUMPTIONS (JKH added). Substance(x) -> Being(x) # label("auxiliary assumption 1: if x is a substance, x is a being"). (InItself(x) & CanBeConceivedThruItself(x)) -> (Modfication(y,x) -> PriorTo(x,y)) # label("auxiliary assumption 2: if x is in itself and can be conceived through itself, then if y is a modification of x, x is prior to y"). KnowledgeOf(x,y) -> CanBeUnderstoodInTermsOf(y,x) # label("auxiliary assumption 3: if x is knowledge of y, then y can be understood in terms of x"). InItself(x) -> SelfCaused(x) # label("auxiliary assumption 4: if x is in itself, x is self-caused"). HasInfiniteAttributes(x) -> EssenceInvExistence(x) # label("auxiliary assumption 5: if x has infinite attributes, x has the property that x's essence involves x's existence"). AbsolutelyInfinite(x) -> HasInfiniteAttributes(x) # label("auxiliary assumption 6: If x is absolutely infinite, then x has infinite attributes"). Being(x) -> HasEssence(x) # label("auxiliary assumption 7: If x has being, then x has essence"). (EssenceInvExistence(x) & HasEssence(x)) -> Exists(x) # label("auxiliary assumption 8: if the essence of x involves the existence of x and x has essence, then x exists"). end_of_list. formulas(goals). % Propositions XI. God exists. God(x) -> Exists(x) # label("proposition XI. God exists"). end_of_list. Figure 1. prover9 ([2]) script for generating a derivation of (GE) from the DAPI conjoined with some plausible auxiliary assumptions. For a detailed description of the prove9 syntax and semantics, see [2]. Figure 2 shows the derivation obtained by executing the script shown in Figure 1. ============================== Prover9 ===============================

5 Prover9 (32) version A, February Process 1080 was started by Owner on Owner-PC, Tue Dec 3 09:49: The command was "../bin/prover9". ============================== end of head ===========================... ============================== PROOF ================================= % Proof 1 at 0.06 (+ 0.01) seconds. % Length of proof is 27. % Level of proof is SelfCaused(x) <-> EssenceInvExistence(x) & NatureConcOnlyByExistence(x) # label("definition I: self-caused") # label(non_clause). [assumption]. 3 Substance(x) <-> InItself(x) & CanBeConceivedThruItself(x) # label("definition III: substance") # label(non_clause). [assumption]. 6 God(x) <-> Being(x) & AbsolutelyInfinite(x) # label("definition VI: God") # label(non_clause). [assumption]. 7 AbsolutelyInfinite(x) <-> Substance(x) & ConstInInfAttributes(x) & (AttributeOf(y,x) -> ExpressesEternalEssentiality(y) & ExpressesInfiniteEssentiality(y)) # label("definition VI: absolutely infinite") # label(non_clause). [assumption]. 21 InItself(x) -> SelfCaused(x) # label("auxiliary assumption 4: if x is in itself, x is selfcaused") # label(non_clause). [assumption]. 24 Being(x) -> HasEssence(x) # label("auxiliary assumption 7: If x has being, then x has essence") # label(non_clause). [assumption]. 25 EssenceInvExistence(x) & HasEssence(x) -> Exists(x) # label("auxiliary assumption 8: if the essence of x involves the existence of x and x has essence, then x exists") # label(non_clause). [assumption]. 26 God(x) -> Exists(x) # label("proposition XI. God exists") # label(non_clause) # label(goal). [goal]. 28 -SelfCaused(x) EssenceInvExistence(x) # label("definition I: self-caused"). [clausify(1)]. 30 -InItself(x) SelfCaused(x) # label("auxiliary assumption 4: if x is in itself, x is selfcaused"). [clausify(21)]. 35 -Substance(x) InItself(x) # label("definition III: substance"). [clausify(3)]. 40 -AbsolutelyInfinite(x) Substance(x) # label("definition VI: absolutely infinite"). [clausify(7)]. 64 -God(x) Being(x) # label("definition VI: God"). [clausify(6)]. 65 -God(x) AbsolutelyInfinite(x) # label("definition VI: God"). [clausify(6)]. 66 God(c1) # label("proposition XI. God exists"). [deny(26)]. 72 -AbsolutelyInfinite(x) InItself(x). [resolve(40,b,35,a)]. 80 AbsolutelyInfinite(c1). [resolve(66,a,65,a)]. 97 -InItself(x) EssenceInvExistence(x). [resolve(30,b,28,a)]. 113 InItself(c1). [resolve(80,a,72,a)] Being(x) HasEssence(x) # label("auxiliary assumption 7: If x has being, then x has essence"). [clausify(24)]. 129 Being(c1). [resolve(66,a,64,a)] EssenceInvExistence(x) -HasEssence(x) Exists(x) # label("auxiliary assumption 8: if the essence of x involves the existence of x and x has essence, then x exists"). [clausify(25)]. 138 EssenceInvExistence(c1). [resolve(113,a,97,a)] Exists(c1) # label("proposition XI. God exists"). [deny(26)]. 188 HasEssence(c1). [resolve(129,a,127,a)] HasEssence(c1) Exists(c1). [resolve(138,a,136,a)]. 192 $F. [copy(191),unit_del(a,188),unit_del(b,153)]. ============================== end of proof ========================== Figure 2. prover9 derivation of (GE) from the script shown in Figure 3. For further detail on the syntax and semantics of prover9, see [2]. The derivation shown in Figure 2 uses only three of the auxiliary assumptions of the script shown in Figure 1 (the other auxiliary assumptions are used in the derivation of

6 propositions not essential to the purposes of this paper): Auxiliary assumption 4: If x is in itself, x is self-caused Auxiliary assumption 7: If x is a being, then x has essence Auxiliary assumption 8: If the essence of x involves the existence of x and x has essence, then x exists. It is difficult to imagine any serious objection to any of these assumptions within the context of Spinoza's philosophy (in that context, they are arguably definitional), and I believe Spinoza would have accepted them. 4.0 Discussion and conclusions Sections 2.0 and 3.0 motivate at least two observations: 1. A companion paper shows that (GE) cannot be derived from the DAPI. Figure 2 shows that (GE) can be derived from the DAPI conjoined with a few plausible assumptions. 2. As [3] notes, an ADS can be a useful tool for augmentation of traditional methods of exegis. An ADS is not a panacea, of course. Indeed, there is no guarantee that an ADS will produce results of interest in a time we care to wait. That said, an ADS is often astonishingly good at providing insights that may not otherwise be obvious. 5.0 Acknowledgements This work benefited from discussions with Ed Zalta, Paul Oppenheimer, Paul Spade, Tom Oberdan, and Joe Van Zandt. For any infelicities that remain, I am solely responsible. 6.0 References [1] Spinoza. The Ethics (published posthumously, 1677). In Benedict de Spinoza. On the Improvement of the Understanding, The Ethics, Correspondence. Unabridged trans. by RHM Elwes (1883). Dover reprint, [2] McCune WW. prover9 and mace4. February [3] Oppenheimer P and E Zalta. A computationally-discovered simplification of the ontological argument. Australasian Journal of Philosophy 89/2 (2011), [4] Horner JK. spinoza_ethics, an automated deduction system for Part I of Spinoza's The Ethics Available from the author on request. [5] Chang CC and HJ Keisler. Model Theory. North-Holland [6] Kant. The Critique of Pure Reason. Trans. by N Kemp Smith. St. Martin's [7] Church A. Introduction to Mathematical Logic. Part I. Princeton [8] Journal of Automated Reasoning. Springer. [9] Horner JK. An automated deduction of implication-type-restricted Foulis-Holland theorems from orthomodular quantum logic. In eight parts. Proceedings of the 2013 International Conference on Artificial Intelligence. Las Vegas NV, July CSREA Press pp [10] Clarke AC. Rendezvous with Rama. Spectra

7 [11] Oppy G. Ontological Arguments. Stanford Encyclopedia of Philosophy [12] Anselm. Prosologion. Circa Trans. by SN Deane. In St. Anselm: Basic Writings. Second Edition. Open Court [13] Horn A. On sentences which are true of direct unions of algebras. Journal of Symbolic Logic 16 (1951), [14] Hennessy JL and DA Patterson. Computer Architecture: A Quantitative Approach. Fourth Edition. Morgan Kaufmann. 2007

In Part I of the ETHICS, Spinoza presents his central

TWO PROBLEMS WITH SPINOZA S ARGUMENT FOR SUBSTANCE MONISM LAURA ANGELINA DELGADO * In Part I of the ETHICS, Spinoza presents his central metaphysical thesis that there is only one substance in the universe.