Formal Reconstructions of Anselm's Ontological...

Formal Reconstructions of Anselm’s OntologicalArgument

Gunther Eder, Esther Ramharter

December 15, 2015

Eder, Ramharter Formal Reconstructions of Anselm’s Ontological ArgumentDecember 15, 2015 1 / 31

Content

1 Arguments for the Existence of God

2 Ontological Arguments

3 Anselm’s Ontological Argument and its Formal Reconstructions

4 Criteria for Formal Reconstructions

5 Central Concepts in Anselm’s Argument

6 A Modal-free Reconstruction

7 Two Modal Reconstructions

8 What can we learn?

Types of arguments for the existence of God

Different categorisations for arguments for the existence of God:

A priori – a posteriori arguments

Teleological arguments (Aquinas) – ontological arguments(Anselm)

Kant: physico-teleological - cosmological - ontological

Compelling arguments (e.g. deductive arguments) – ‘plausibilityarguments’/pragmatic arguments (Pascal’s Wager)/probabilityarguments

Some Ontological Arguments and Their Purposes

Anselm – the first and best of all? (to understand/reconstruct whatwe already believe/know)

Descartes (God as means to knowledge)

Leibniz (interested in everything; an emendation: it is necessaryto prove that the existence of God is possible)

Godel (just for fun or serious?)

On the other hand...

Proofs that ontological arguments (arguments for the existence ingeneral) are impossible

Proofs for the non-existence of God

What is Anselm’s Ontological Argument?

Pavel Tichy: much better than Descartes’ version, because it does notpresuppose that we know God’s essence

Anselm defines God as ‘something than which nothing greater can beconceived’

Ch. II of Proslogion concludes that such a being exists in realityCh. III of Proslogion concludes that such abeing cannot be thought notto exist

Some interpreters argue that only the argument in Ch. III is conclusive.But this is contrary to what Anselm himself says; we therefore focus onreconstructions of Ch. II.Nevertheless, we hold that it is important not to ignore Ch. III whenreconstructing Ch. II.

Proslogion II

Now we believe that You are something than whichnothing greater can be conceived.Or can it be that a thing of such nature does not exist, sincethe fool has said in his heart, there is no God [Ps. 13: 1;52:1]?But surely, when this same Fool hears what I am speaking about,namely ‘something than which nothing greater can be conceived’,he understands what he hears,and what he understands is in his understanding, even if hedoes not understand that it exists [in reality].For it is something else that a thing exists in theunderstanding than to understand that a thing exists [inreality].

Proslogion II

Even the fool, then, is forced to agree that something thanwhich nothing greater can be conceived exists in theunderstanding, since he understands this when he hears it, andwhatever is understood is in the understanding.

And, surely, that than which a greater cannot be conceived

cannot exist in the understanding alone.

For if it exists solely in the understanding, it canbe conceived to exist in reality also, which isgreater.If, then, that than which a greater cannot beconceived exists in the understanding alone, thissame than which a greater cannot be conceived is[something] than which a greater can be conceived.But surely this cannot be.Something than which a greater cannot be conceivedtherefore exists without doubt, both in theunderstanding and in reality.

Two Examples of Formal Reconstructions ofProslogion II

Morscher 1991: “Was sind und was sollen die Gottesbeweise?”

(1) ¬A id quo does not exist in reality(2) ¬A→ B if id quo does not exist in reality

then something greater can be conceived(3) B → C if something greater than id quo can be conceived

then: Contradiction(4) C Contradiction

(nothing greater can be conceived than id quoand something greater can be conceived than id quo)

(5) A id quo exists in reality

Two Examples of Formal Reconstructions ofProslogion II

Lewis 1970: “Anselm and Actuality”

Premisses:

(1) ∀x(Ux → ∃w(Ww ∧ Exw))

(2) ∀x∀w∀v((Ux ∧Ww ∧Wv ∧ Exw ∧ ¬Exv)→ G(xw)(xv))

(3A) ∃x(Ux ∧¬∃w∃y(Ww ∧G(yw)(xa))) for the actual world a

(4) Wa

Conclusion: ∃x(Exa ∧ ¬∃w∃y(Ww ∧G(yw)(xa)))

Criteria for Formal Reconstructions

(I) An ideal formal reconstruction of an informal argument should

(1) be locally conform to the piece of text to be reconstructed(2) be globally conform to what the author of the text said elsewhere(3) capture the fundamental structure of the argument(4) have, from the author’s standpoint, plausible premisses

(II) A general maxim:

(5) conformity trumps cogency!

(1) be locally conform to the piece of text to be reconstructed

(2) be globally conform to what the author of the text said elsewhere(3) capture the fundamental structure of the argument(4) have, from the author’s standpoint, plausible premisses

(1) be locally conform to the piece of text to be reconstructed(2) be globally conform to what the author of the text said elsewhere

(3) capture the fundamental structure of the argument(4) have, from the author’s standpoint, plausible premisses

(1) be locally conform to the piece of text to be reconstructed(2) be globally conform to what the author of the text said elsewhere(3) capture the fundamental structure of the argument

(4) have, from the author’s standpoint, plausible premisses

(III) An ideal formal reconstruction of an informal ontologicalargument should

(6) have premisses that are analytical truths

(IV) In an ideal formal reconstruction of Anselm’s informalontological argument

(7) the conclusion of the argument should (if valid) be hidden in thepremisses

(III) An ideal formal reconstruction of an informal ontologicalargument should

(6) have premisses that are analytical truths

(IV) In an ideal formal reconstruction of Anselm’s informalontological argument

(7) the conclusion of the argument should (if valid) be hidden in thepremisses

Central Concepts in Anselm’s Argument

(1) Existence

(2) God

(3) Greaterness

(4) Conceivability

Central Concepts in Anselm’s Argument: Existence

Existence in the understanding (‘esse in intellectu’):

basic mode of existencebest modelled in terms of quantifiers (without existential import)‘∃xϕ’ means ‘There is something in the understanding such that ϕ’

Existence in reality (‘esse in re’):

substantial property an object can have or lackbest modelled in terms of a simple predicate E!

Necessary existence (‘esse quod non possit cogitari non esse’):

substantial property an object can have or lackbest modelled in terms of a complex predicate �E!

Central Concepts in Anselm’s Argument: God

In the critical passage of Anselm’s argument in Ch. II, Anselmdoes not refer to ‘God’, but only to

something which nothing greater can be conceived (‘aliquid quomaius nihil cogitari possit’)

that than which nothing greater can be conceived (‘id quo maiusnihil cogitari possit’)

What is proved in chapter II is that aliquid quo exists in reality

id quo plays the role of an auxiliary name that is used to provesomething from an existence assumption (not a definitedescription)

Central Concepts in Anselm’s Argument: Greaterness

Anselm’s use of the term ‘greater’ is unique to his ontologicalargument and his definition of God

‘Being greater than’ is understood as a relation > amongindividuals, which defines some kind of ordering amongindividuals

It’s certain properties that determine this ordering !

Which properties exactly make something greater depends onwhich textual evidence one takes into account

‘Being greater than’ is a relation among individuals that is determinedby the properties in a certain (consistent) class P of great-makingproperties and E! is in P.

Central Concepts in Anselm’s Argument: ‘cogitare’and ‘esse in intellectu’

Two main lines of understanding the role of Anselm’s notion notionof conceivability

(1) Predicative reading: ‘conceivable’ as a predicate true of objects

(2) Operator reading: ‘conceivable’ as an operator attached tosentences

Accordingly, two readings of the central locution ‘x is somethingthan which nothing greater can be conceived’

(1) there is no conceivable being that is greater than x

(2) it is not conceivable that there is a being greater than x

On the predicative reading, the point of ‘conceivability is tointroduce the reader into non-factual discourse about ‘conceivableobjects’ as an alternative to discourse about objects that have realexistence

On the operator reading, the point of ‘conceivability’ is to introducethe reader into non-factual discourse about ‘conceivablecircumstances’ and the objects that exist in conceivablecircumstances are the objects that exist in the understanding

From a logical point of view, the distinction between both readingsdetermines the basic structure of any reconstruction; modal ornon-modal!

A modal-free reconstruction

Definition. Gx :↔ ¬∃y(y > x)

Premisses:

(1) ExUnd: ∃xGx

(2) Ax.GR: x > y ↔ ∀PF (Fy → Fx) ∧ ∃PF (Fx ∧ ¬Fy)

(3) Realization: ∀PF∃x(∀F (F(F )↔ F (x)))

to compare a thing with itself

(4) Quasi-Id: x ≡D y :↔ ∀PF (¬D(F )→ (Fx ↔ Fy))

to be proved:

God!: ∃x(Gx ∧ E!x)

Background Modal Logic

Propositional part of background logic satisfies usual laws of anormal modal logic and scheme (T)

(K) �(α→ β)→ (�α→ �β)

(T) α→ ♦α

Quantificational part of background logic consists of classicalquantification theory plus the Barcan- and Converse Barcanformulas:

(BF) ∀x�α(x)→ �∀xα(x)

(CBF) �∀xα(x)→ ∀x�α(x)

Actual existence is expressed by a simple predicate E!

(K) �(α→ β)→ (�α→ �β)

(T) α→ ♦α

(BF) ∀x�α(x)→ �∀xα(x)

(K) �(α→ β)→ (�α→ �β)

(T) α→ ♦α

(BF) ∀x�α(x)→ �∀xα(x)

A first Approximation

Definition: Gx :↔ ¬♦∃y(y > x)

Premisses:

(1) ExUnd: ∃xGx

(2) PossEx: ∀x♦E!x

(3) Great: ∀x(¬E!x ∧ ♦E!x → ∃z♦(z > x))

⇒ good approximation!

⇒ argument is valid, i.e. ∃x(Gx ∧ E!x) follows from the premisses (1),(2) and (3).

Premisses:

(1) ExUnd: ∃xGx

(3) Great: ∀x(¬E!x ∧ ♦E!x → ∃z♦(z > x))

Premisses:

(1) ExUnd: ∃xGx

(3) Great: ∀x(¬E!x ∧ ♦E!x → ∃z♦(z > x))

Problem and (a) Solution

We do not merely want to say that if id quo does not actuallyexist, then id quo’s greatness would be exceeded by something

We want to be able to say that if id quo does not actually exist,then id quo would be exceeded in greatness by itself

We need a means to compare id quo’s greatness with its owngreatness in conceivable circumstances!

The solution is to introduce(1) a primitive function g that assigns ‘greatnesses’ to individuals(2) a relation � (‘exceeds’) between ‘greatnesses’

The basic idiom ‘x is greater than y ’ now translates to g(x) � g(y)

A Two Possibilities

Premisses:

(1) ExUnd: ∃xGx

(3) Great’: ∀x(¬E!x ∧ ♦E!x → ∃z(z = g(x) ∧ ♦(g(x) � z))

Definition 1: Gx :↔ ¬♦∃y(g(y) � g(x))

Definition 2: Gx :↔ ∃z(z = g(x) ∧ ¬♦∃y(g(y) � z)

⇒ both versions are plausible candidates.

⇒ argument is valid with definition 2, but invalid with definition 1.

A Two Possibilities

Premisses:

(1) ExUnd: ∃xGx

A Two Possibilities

Premisses:

(1) ExUnd: ∃xGx

What did we learn?

Due to the ambiguitiy of the notion of ‘conceivability’, there isnothing in Anselm’s argument in chapter II that could force us toadopt a particular background logic (modal or classical)

What makes a formal reconstruction of this seemingly simpleargument so intricate is that there is no simple way to comparesomething with itself ‘conceived of’ with different properties

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