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Three normative theories overly simplified…

Consequentialism: maximize happiness, even if it means compromising your moral integrity.

Deontology: be worthy of happiness, and damn the happiness for yourself and the world around you.

Virtue Ethics: realize that true happiness is found in pursuing worthiness of happiness.

Anselm without Defining God

One of the most common objections that I hear to my ontological arguments is that they use definitions that beg the question.  However, I am careful to note four things that I think block the charge that I “define God into existence”: 1) I require that stipulated definitions be defended as coherent, 2) I specify that I am setting my arguments within the context of free logic, 3) my definitions cannot directly entail, or be semantically equivalent to the conclusion, and 4) I must provided at least one premise that is justified independently from any definition of God, or from the conclusion.

However, I think the concern over the definition is overblown, and we could just derive the conclusion that there is at most one being such that it is not conceivable that there is something greater.  Call it “God” or “Banana Smoothie”.  It really doesn’t matter.  A term like God is emotionally loaded anyways, so maybe there is some rhetorical strategy in abandoning the word “God” altogether.

Here is the argument:

P1) For all x, if x is that than which none greater can be conceived, and there is some other z, which is that than which none greater can be conceived, and x and z are not the same, then  it is conceivable that there is something that can be combined with x as a mereological sum to make a composite whole of it and x as proper parts.

P2) For all x, and all z1, if it is conceivable that there is some mereological sum, which is the whole composed of x and z1 as proper parts, then conceivably there is some thing greater than x (namely the whole, out of which x is a proper part).

P3) All things that are not fictional beings are things that exist in reality.

P4) All fictional beings are things of which a greater can be conceived.

P5) There is something than which none greater can be conceived and either it is fictional or it is not fictional.

C) There is exactly one being in reality such that it is not conceivable that there is something greater.

We have to at least define some predicates, so let,

Fx ≝ x is a fictional being
Rx ≝ x exists in reality
Gxy ≝ x is greater than y
∑xyz ≝ x is the mereological sum of  the proper parts, y and z
©… ≝ it is conceivable that…

1. (∀x){~©(∃y)Gyx ∧ (∃z)[~©(∃y)Gyz ∧ ~(z = x)]} ⊃ ©(∃z1)(∃y)∑yxz1} (premise)
2. (∀x)(∀z1){©(∃y)∑yxz1] ⊃ ©(∃y)Gyx}(premise)
3. (∀x)(~Fx ⊃ Rx) (premise)
4. (∀x)(Fx ⊃ ©(∃y)Gyx) (premise)
5. (∃x)[~©(∃y)Gyx ∧ (Fx ∨ ~Fx)] (premise)
6. (∃x){~©(∃y)Gyx ∧ (∃z)[~©(∃y)Gyz ∧ ~(z = x)]} (IP)
7. ~©(∃y)Gyμ ∧ (∃z)[~©(∃y)Gyz ∧ ~(z = μ)] (6 EI)
8. {~©(∃y)Gyμ ∧ (∃z)[~©(∃y)Gyz ∧ ~(z = μ)]} ⊃ ©(∃z1)(∃y)∑yμz1 (1 UI)
9. ©(∃z1)(∃y)∑yμz1 (7,8 MP)
10. ©(∃y)∑yμν (9 EI)
11. (∀z1)[©(∃y)∑yμz1 ⊃ ©(∃y)Gyμ (2 UI)
12. ©(∃y)∑yμν ⊃ ©(∃y)Gyμ (11 UI)
13. ©(∃y)Gyμ (10,12 MP)
14. ~©(∃y)Gyμ (7 Simp)
15. ©(∃y)Gyμ ∧ ~©(∃y)Gyμ (13,14 Conj)
16. ~(∃x){~©(∃y)Gyx ∧ (∃z)[~©(∃y)Gyz ∧ ~(z = x)]} (6-15 IP)
17. (∀x)~{~©(∃y)Gyx ∧ (∃z)[~©(∃y)Gyz ∧ ~(z = x)]} (16 QN)
18. (∀x){~~©(∃y)Gyx ∨ ~(∃z)[~©(∃y)Gyz ∧ ~(z = x)]} (17 DeM)
19. (∀x){©(∃y)Gyx ∨ ~(∃z)[~©(∃y)Gyz ∧ ~(z = x)]} (18 DN)
20. (∀x){©(∃y)Gyx ∨ ~(∃z)~[©(∃y)Gyz ∨ (z = x)]} (19 DeM)
21. (∀x){©(∃y)Gyx ∨ (∀z)[©(∃y)Gyz ∨ (z = x)]} (20 QE)
22. (∀x){©(∃y)Gyx ∨ (∀z)[~~©(∃y)Gyz ∨ (z = x)]} (21 DN)
23. (∀x){©(∃y)Gyx ∨ (∀z)[~©(∃y)Gyz ⊃ (z = x)]} (22 Impl)
24. (∀x)(~©(∃y)Gyx ⊃ Fx) (IP)
25. ~©(∃y)Gyμ ∧ (Fμ ∨ ~Fμ) (5 EI)
26. ~©(∃y)Gyμ ⊃ Fμ (24 UI)
27. Fμ ⊃ ©(∃y)Gyμ (4 UI)
28. ~©(∃y)Gyμ ⊃ ©(∃y)Gyμ (26,27 HS)
29. ~©(∃y)Gyμ (25 Simp)
30. ©(∃y)Gyμ (28,29 MP)
31. ©(∃y)Gyμ ∧ ~©(∃y)Gyμ (29,30 Conj)
32. ~(∀x)(~©(∃y)Gyx ⊃ Fx)(24-31 IP)
33. (∃x)~(~©(∃y)Gyx ⊃ Fx)(32, QN)
34. (∃x)~(~~©(∃y)Gyx ∨ Fx) (33 Impl)
35. (∃x)~(©(∃y)Gyx ∨ Fx)(34 DN)
36. (∃x)(~©(∃y)Gyx ∧ ~Fx) (35 DeM)
37. ~©(∃y)Gyμ ∧ ~Fμ (36 EI)
38. ~Fμ ⊃ Rμ (3 UI)
39. ~Fμ (37 Simp)
40. Rμ (38,39 MP)
41. ~©(∃y)Gyμ (37 Simp)
42. ©(∃y)Gyμ ∨ (∀z)[~©(∃y)Gyz ⊃ (z = μ)] (23 UI)
43. (∀z)[~©(∃y)Gyz ⊃ (z = μ)] (41,42 DS)
44. ~©(∃y)Gyμ ∧ (∀z)[~©(∃y)Gyz ⊃ (z = μ)] (41,43 Conj)
45. ~©(∃y)Gyμ ∧ (∀z)[~©(∃y)Gyz ⊃ (z = μ)] ∧ Rμ (40,44 Conj)
46. (∃x){~©(∃y)Gyx ∧ (∀z)[~©(∃y)Gyz ⊃ (z = x)] ∧ Rx} (45 EG)

But now 46 say that there is exactly one being than which none greater exists and it exists in reality.

QED

Suppose we add to our lexicon:

g ≝ (ɿx)~©(∃y)Gyx

Then we could easily reach:

47. Rg (46 theory of descriptions given the def of “g”)

And this is precisely the conclusion I reach here.  So using a definite description at the outset saves space, but requires additional defenses for the premise that are made explicit here.

Vecchio’s Variation on Anselm’s Ontological Argument (VVAOA)

P1) All things that are not fictional are things that exist in reality.
P2) All things that are fictional are such that it is conceivable that there is something greater.
C) God, i.e. the being of which it is not conceivable that there is a greater, exists in reality.

Defense of P1: Fictional and non-fictional are complementary classes, and non-fictional is definitionally synonymous with real. So, if something is not fictional, it is non-fictional, and, thus, real.

Defense of P2: For anything fictional, one can conceive of a concrete correlate. For instance, Platonism is a conceivable metaphysical system in which entities that non-Platonists might conceive of as abstracta are conceived as concrete extra-mental realities that have causal powers. So it is for anyone who conceives of God as a fictional object of thought. One can also conceive God to be a concrete extra-mental reality that has causal powers. Given that one can conceive of a concrete correlate of a fictional being, these correlates will be similar in description except that the latter would be conceived to have causal powers, while the former, as an abstract fiction, would not. Ceteris paribus, if one thing lacks causal powers while the latter has causal powers, the latter is greater than the former, in as much as by ‘greatness’ one should understand ‘greater in capacity, ability, or power’.

The following formal proof shows the above to be valid.
Let,

Fx ≝ x is a fictional being
Rx ≝ x exists in reality
Gxy ≝ x is greater than y
©… ≝ it is conceivable that…
g ≝ (ɿx)~©(∃y)Gyx

1. (∀x)(~Fx ⊃ Rx) (premise)
2. (∀x)(Fx ⊃ ©(∃y)Gyx) (premise)
3. Fg (Assumption Indirect Proof)
4. Fg ⊃ ©(∃y)Gyg (2 by Universal Instantiation)
5. ©(∃y)Gyg (3,4 by Modus Ponens)
6. (∃x){[~©(∃y)Gyx ∧ (∀z)[~©(∃y)Gyz ⊃ (z = x)] ∧ ©(∃y)Gyx} (5 by theory of descriptions)
7. [~©(∃y)Gyμ ∧ (∀z)[~©(∃y)Gyz ⊃ (z = μ)] ∧ ©(∃y)Gyμ (6 by Existential Instantiation)
8. [(∀z)[~©(∃y)Gyz ⊃ (z = μ)] ∧ ~©(∃y)Gyμ] ∧ ©(∃y)Gyμ (7 by Commutation)
9. (∀z)[~©(∃y)Gyz ⊃ (z = μ)] ∧ [~©(∃y)Gyμ ∧ ©(∃y)Gyμ] (8 by Association)
10. ~©(∃y)Gyμ ∧ ©(∃y)Gyμ (9 by Simplification)
11. ~Fg (3-10 by Indirect Proof)
12. ~Fg ⊃ Rg (1 by Universal Instantiation)
13. Rg (11,12 by Modus Ponens)
QED

An Ontological Argument Using Aristotelian Logic

The following argument should receive an Aristotelian interpretation for existential import, but neutral on the question of whether one is discussing fictional or non-fictional existence. This is in-line with the Anselmian point that the question isn’t whether God exists, but the mode of God’s existence, i..e in reality or in the understanding alone:

1) All fictional beings are things of which a greater can be conceived (premise).

2) No being that is identical to the being than which none greater can be conceived is a thing of which a greater can be conceived (premise).

3) No being that is identical to the being than which none greater can be conceived is a fictional being (from 1,2 by Modus Camestres).

4) Some beings that are identical to the being than which none greater can be conceived are not fiction beings (from 3 by Sub-Alternation).

5) Some beings that are identical to the being than which none greater can be conceived are non-fictional beings (from 4 by Obversion).

6) There is some x, such that x is identical to the being than which none greater can be conceived and x is non-fictional (from 5 by Semantic Equivalence).

QED

The Aptness of the Ontological Argument

There is a kind of abductive argument from aptness, or fittingness, that some philosophers and theologians have employed in the past.  For example, Bl. Duns Scotus develops an argument for the Immaculate Conception from its fittingness.

What is aptness?  It seems to be an explanatory feature like parsimony, or conservativeness.  It is something that, were we to discover its truth, we would not be surprised, given what we presently understand of the topic.  Moreover, in contemplating the aptness of a hypothesis, one has a sense that such a truth, though unsurprising, is nonetheless illuminating.

Now, the aptness of a hypothesis, insofar as it seems to be an abductive explanatory feature, does not appear to be the intuition of merely an analytical or tautological truth, even, say, within counterfactual contemplation.  For example, I wouldn’t really say that it is apt that, should there be a sound proof that the Goldbach conjecture be true, that the proof would be mathematical in nature.  For, to say that is just really to state the implicit tautology that if there is a sound mathematical proof for x, then there is a sound mathematical proof for x, and tautologies like that are not, in any way, illuminating, which is at least part of what we mean by “apt.”

I have explained aptness through a kind of subjunctive conditional, i.e. ‘if x were true of y, it would be apt that x is true of y.’  That alone might be some reason to think it is probable that x is true of y.  However, if there is also evidence that is consistent with the claim of aptness, it would be reasonable, all the more, to increase the likelihood that x is true of y.

So, what of the ontological argument?  Or more precisely, what of a priori arguments that purport to establish the existence of God, i.e. a maximal great, supreme, or perfect being.  It seems apt that if any concrete object should be established solely through a priori considerations, then God would be a candidate.  Nay, given the supposed chasm between creature and creator, it is apt that, among concreta, God alone should be proved to exist solely by a priori considerations.

Put in terms of our formula, we could say, “If God alone, among concreta , could be proved to exist solely by a priori considerations, then it would be apt that God alone, among concreta, can be proved to exist solely by a priori considerations.”  And this might be some reason to think a version of the ontological argument is plausible.  Now consider the evidence in support of the hypothesis — i.e. the lack of ontological arguments for any other concrete objects (leaving aside the possibility of ontological arguments for objects in an abstract realm), and the plethora of candidate ontological arguments for a supreme being.  These facts of philosophical history, that ontological arguments seem only to be suited to establish the existence of God, I would contend, a good abductive reason to think it is plausible that there be a sound ontological argument.

Perhaps, though, you are not moved that the aptness of the ontological argument should makes us think that such an argument is probably sound.  Aptness may still be sufficient to establish the soundness of an ontological argument.  How so?  Well, it seems to me that if we should think that a feature increases the probability of a hypothesis this entails not that it is broadly logically possible, but that we should think it is broadly logically possible.  That is, if our considered judgment is that we think there is evidence for a hypothesis, which increases the likelihood of that hypothesis, then we are committed to thinking the prior probability of the hypothesis is not 0.  This does not establish that a sound ontological argument is, in fact, possible, but that one who is committed to their being evidence for a sound ontological argument is, in fact, committed to the real possibility for it.  But then, if one thinks such an argument is possible, one should also think such an argument, in fact, exists.

We might reason as follows:

P1. If one should think there is good evidence to support the claim that it is apt that God alone, among concreta, can be proved to exist solely by a priori considerations, then one should think that the probability of the hypothesis ‘God alone, among concreta, can be proved to exist solely by a priori considerations’ has increased.
P2. If one should think that the probability of the hypothesis ‘God alone, among concreta, can be proved to exist solely by a priori considerations’ has increased, then one should think that it is broadly logically possible that God alone, among concreta, can be proved exist solely by a priori considerations.
P3. If one should think that it is broadly logical possible that God alone, among concreta, can be proved exist solely by a priori considerations, then one should think that, in fact, God alone, among concreta, can be proved exist solely by a priori considerations.
P4. One should think there is good evidence to support the claim that it is apt that God’s existence alone, among concreta, can be proved to exist solely by a priori considerations.
C.  So, one should think that, in fact, God alone, among concreta, can be proved to exist solely by a priori considerations.

Defense of the premises:

In defense of P1, one can say that to have good evidence to support the explanation for a hypothesis just is to make that hypothesis more likely than it otherwise would be.  We can simply stipulate that this is what we mean by good evidence, i.e. it is sufficient to imply the plausibility of the hypothesis in question.

In defense of P2, we are not actually, as some might fear, shifting from epistemic possibility to broad logical possibility, strictly speaking.  We are couching this implication within what one should think, given one’s epistemic duties. Whether or not something is, in fact, broadly logically possible, if one thinks something is not, a priori impossible, one cannot, rationally, at the same time remain agnostic to its broad logical possibility.  To think that a hypothesis might become more likely, given the evidence, entails that one, thinking appropriately, also thinks the hypothesis is inherently possible.  Otherwise, no evidence would improve the probability.  Hence, the rational person who thinks there is evidence that is suggestive some sound ontological argument, that person ought to think that such an argument is really possible, in a robust sense.

For P3, I would simply note that, given the fact that a priori ontological arguments derive modally necessary conclusions, from a priori necessary truths.  Such an argument would, in effect, be sound across possible worlds.  Indeed, the very counterfactual I have contemplated in this post “If God alone, among concreta , can be proved to exist solely by a priori considerations, then it would be apt that God alone, among concreta, can be proved to exist solely by a priori considerations” could be assessed as true, like other subjunctive conditionals, in terms of possible worlds, e.g. in the nearest possible world where God alone, among concreta, can be proved to exist solely by a priori considerations, it is explanatorily fitting and apt that such is the case, and that is just to affirm P3.

Finally, P4 is based on the above considerations.  I think it is basically intuitive that it should be apt that the ontological argument should work only for God, and for no other concrete object.  The fact that there have been dozens of formulations of the ontological argument that are, at the very least, plausibly sound, and no ontological argument for any other concrete thing only goes to support this aptness, and so make this aptness not only likely, but provide abductive support for one to embrace the possibility of some sound ontological argument for God.  Now, one might say that there are evidential matters to consider, but I am not compelled to think so.  If Plantinga’s own version of the argument is correct, then the possibility of a maximally great being is at least reasonable to believe on its own.  Moreover, there appears to be substantive responses to arguments for the incoherence of theism, and I take that to be the primary counter-evidence to the ontological argument.  Given that, I think the aptness of the ontological argument, and the evidential support for it, is sufficient to make it plausible that there is sound ontological argument.

From this, it follows that we should think God exists.

QED

De Ente and the Falsity of Naturalism

Thomas Aquinas writes:

…[E]verything that is in a genus has a quiddity beyond its existence, since the quiddity or nature of the genus or species is not in the order of nature distinguished in the things of which it is the genus or species, but the existence is diverse in diverse things (De Ente V.).

Given some basic modal theorems and axioms, and the above considerations, the following argument occurred to me:

P1. If naturalism is true, everything is in the genus “nature”.
P2. If everything is in the genus “nature”, then everything has a quiddity beyond its existence.
P3. Necessarily, if there is some x such that its quiddity is nothing other than its existence, then necessarily there is some x such that its quiddity is nothing other than its existence.
P4. If there is something x such that its quiddity is nothing other than its existence, then not everything has its quiddity beyond its existence.
P5. Possibly, there is some x such that its quiddity is nothing other than its existence.
C. It is not the case that naturalism is true.

Defense of P1: Naturalism just is the thesis that everything that exists is natural, and so belongs to the generic class “nature”.

Defense of P2: According to Aquinas, if the quiddity, or essence, of a thing is in a genus, then its quiddity cannot be its existence, since a genus admits of more than one instance, and whatever has its existence as its quiddity cannot admit of more than one instance.

Defense of P3: If something has its existence as its quiddity, then it has existence per se and so necessarily so.  This is necessarily implied, since it is analytically true.

Defense of P4: This would be based on the notion that if a quiddity is the same as its existence, then its quiddity would not also be beyond its existence, for then the quiddity and existence could not be the same.

Defense of P5: This is just to say that it is at least metaphysically possible that something’s quiddity and facticty are the same.  There does not appear to be anything impossible about such a notion, at least prima facie.

Formal Proof:

Let,

N ≝ Naturalism is true
Gxy ≝ x is in the genus y
Q(F,x) ≝ F is ths quiddity of x
B(F,G,x) ≝ F is beyond G for x
E! ≝ existence
n ≝ nature
Theorem of K: ☐(p → q) → (♢p → ♢q)
Theorem of S5: ♢☐p → ☐p
Axiom M: ☐p → p

1. N → (∀x)Gxn (premise)
2. (∀x)Gxn → (∀x)(∀F)[(Q(F,x) → B(F,E!,x)] (premise)
3. ☐{(∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)} → ☐(∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)]}} (premise)
4. (∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)} → ~(∀x)(∀F)[(Q(F,x) → B(F,E!,x)] (premise)
5. ♢(∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)} (premise)
6. N → (∀x)(∀F)[(Q(F,x) → B(F,E!,x)] (1,2 HS)
7. ☐{(∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)] → ☐(∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)]}} → {♢(∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)]} → ♢☐(∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)]}} (Theorem of K)
8. ♢(∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)]} → ♢☐(∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)]} (3,7 MP)
9. ♢☐(∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)]} (5,8 MP)
10. ♢☐(∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)]} → ☐(∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)]} (Theorem of S5)
11. ☐(∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)]}(9,10 MP)
12. ☐(∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)]} → (∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)]} (Axiom M)
13. (∃x){[Q(E!,x)∧(∀F)~(F = E!)]→ ~Q(F,x)]}(11,12 MP)
14. ~(∀x)(∀F)[(Q(F,x) → B(F,E!,x)] (4,13 MP)
15. ~N (6,14 MT)

QED

Physicalism v. Hylomorphism

3jqqon.jpg

Well, he didn’t

Let

Fx ≝ x is a financier
Rxy ≝ x ran a sex trafficking ring out of y
Kxy ≝ x killed y
j ≝ (ɿx)(Fx ∧ Rxl)
l ≝ Little St. James Island

1. ~(∃x)[(Fx ∧ Rxl) ∧ Kxx](premise)
2. Kjj (Assumption for Indirect Proof)
3. (∃x){[(Fx ∧ Rxl) ∧ (∀y)[(Fy ∧ Ryl)→ (y = x)] ∧ Kxx} (2 theory of descriptions)
4. [(Fμ ∧ Rμl) ∧ (∀y)[(Fy ∧ Ryl)→ (y = μ)] ∧ Kμμ (3 EI)
5. (∀x)~[(Fx ∧ Rxl) ∧ Kxx] (1 QN)
6. ~[(Fμ ∧ Rμl) ∧ Kμμ] (5 UI)
7. [(Fμ ∧ Rμl) ∧ (∀y)[(Fy ∧ Ryl)→ (y = μ)] (4 Simp)
8. Fμ ∧ Rμl (7 Simp)
9. Kμμ (4 Simp)
10. (Fμ ∧ Rμl)∧ Kμμ (8,9 Conj)
11. [(Fμ ∧ Rμl)∧ Kμμ] ∧ ~[(Fμ ∧ Rμl) ∧ Kμμ] (6,10 Conj)
12. ~Kjj (2-11 Indirect Proof)

QED

Hempel’s Raven Paradox and God’s Existence

elsa
Hempel’s Raven Paradox: a red apple is evidence that “all ravens are black things”, since it counts as evidence for the logically equivalent proposition “all non-black things are non-ravens”.
 
Likewise, the fictional character, Elsa of Arendelle, is evidence that God is non-fictional, since “All things identical to God are non-fictional beings” is logically equivalent to “All fictional beings are non-identical to God”.
 
If it is evidence, it isn’t very good evidence, I would admit. But, even if it isn’t very good evidence, there is a potential infinity of fictional characters that are non-identical to God. How would you block that from generating a cumulative case for God, simply by increasingly inventing fictional characters that are not God?
 
This is an honest question I have. I am not trying to provide some bizarre cumulative case for God’s existence, as I perceive that this would prove too much.
 
I suppose this just is Hempel’s paradox, but I don’t think I have seen it applied to existential claims about singular terms before.

The Paradox of the Inconsistent Triad

  1. This is an inconsistent triad.
  2. If, at most, two of the propositions in this triad are true, then this is an inconsistent triad.
  3. At most, two of the propositions in this triad are true.