Here, I offer a presentation and explanation of Robert Maydole’s Modal Perfection Argument, an ontological argument for the existence of God. As far as I know, no such comprehensive explanation exists, and so I hope that this will be an all-encompassing guide to the argument. As such, I hope to continually update the page to clarify issues, and the page will be constantly revised according to feedback, particularly where I have been unclear, or failed to adequately explain something. Thus, I would enormously appreciate any feedback and questions regarding the argument – and I also petition that the page is not yet judged as a final product, since there will always be additions to make, at least for the conceivable future. If I have left something out, please alert me, and I will try to modify the page accordingly. Most notably, I have not yet offered an appraisal, or explained the criticisms of the argument – I hope to do this in due course, but if anyone is extra keen to read up on these, Graham Oppy and Thomas Metcalf are the two philosophers to have offered the most well-known objections to the argument. As a disclaimer, I do not personally find the argument to be a convincing one.

This argument was originally put forward in *Philo *in 2003 by Robert E. Maydole. It was reproduced in Maydole’s chapter on the ontological argument in *The Blackwell Companion to Natural Theology* (eds. W.L. Craig and J.P. Moreland) with a very slight change in the logical deduction, though the argument broadly remained the same. I will offer explications of the argument as found in both articles, keeping the symbols constant throughout (thus, slightly modifying the symbols found in the articles, but not changing the argument in substance).

**Contents**

**1. ****The gist of the argument [Complete]**

**2. ****Validity [Complete]
**

- The argument in Philo

- The argument in The Blackwell Companion to Natural Theology

**3. ****Defence of the premises [Forthcoming]**

**4. ****Criticisms [Forthcoming]**

**5. ****Additional notes [Complete]**

**1. ****The gist of the argument [Complete]**

The argument has three premises (M1-M3): the rest follows logically, assuming the validity of 3 axioms (Th 1-Th 3). These premises can be construed thus:

M1: If any property is a perfection, then its negation is not a perfection (i.e. the properties X and not-X cannot both be perfections).

M2: Perfections entail only perfections (i.e. if any property X entails another property Y, and X is a perfection, then Y is also a perfection).

M3: Supremity is a perfection.

A perfection is defined as a property which it is better to have than not to have, and something is supreme (by definition) if there is nothing which is even possibly greater or as great as it.

The argument works, broadly, like this: Suppose it is not possible that there exists a supreme being. In that case, for any x, it is necessarily the case that x is not supreme. And if this is the case then, necessarily, for any x, if x is supreme, then x is not supreme. (since, if “x is not supreme” is true, then any material implication with “x is not supreme” as the consequent is true). Now, suppose supremity is a perfection (M3). And suppose that perfections entail only perfections (M2). If these premises are true, and supremity is not possible, it follows that not being supreme is a perfection (since, if supremity is impossible, then supremity entails non-supremity). But if M1 is accepted (that a property and its negation cannot both be perfections), then not being supreme is not a perfection. This leads to a contradiction: not being supreme is a perfection (by being entailed by a perfection, viz. supremity), and not being supreme is not a perfection (by being the negation of a perfection). So we must reject one of the premises, or we should concede that supremity is possible. If the premises are eminently more plausible than the impossibility of supremity, then we should suppose that supremity is possible – that is, it is possible that there is some being which is supreme.

Maydole then uses the Barcan formula (Th 3), which says that if it is possible that there is some being which has property X, then there is some being which possibly has property X. Thus, it follows that there is some being which, possibly, is supreme. Now, the definition of supremity involves there being no possibility of anything else being greater or as great as the supreme thing. This means that there is a being (call it v) such that it is possible that nothing is possibly greater than v, and that nothing is possibly as great as v. If one accepts the S5 axiom, which implies that, if it is possible that something is impossible, then that thing is impossible (Th 2), then it follows that there exists a being, v, which nothing is possibly greater than, and which nothing is possibly as great as. Thus, v exists, and is the greatest possible being.

I shall now turn to presenting the argument formally, with an explication of the logical deduction. My commentary is in red, and presents a literal translation of the logical symbols, followed by an explanation.

**2. ****Validity [Complete]
**

The argument in *Philo*

P(X) =_{df} it is better to have X than to not. P(X), by definition, means that X is a perfection – that it is better to have X than not to have X.

Gxy =_{df} x is greater than y. Gxy, by definition, means that x is greater than y.

Sx =_{df} (¬◊(∃y)Gyx & ¬◊(∃y)(x≠y & ¬Gxy)). Sx, by definition, means that x is supreme – that it is not possible that there exists some y such that y is greater than x, and that it is not possible that there exists some y such that (x is not identical to y, and x is not greater than y). This is a long-winded way of saying that, if x is supreme, then nothing is possibly greater than x, and nothing else is possibly as great as x.

[aF] =_{df} the property of being an F.

[aF]x =_{df} x has the property of being an F.

M1: (X)(P[aX]) → ¬P([a¬X])). For any property X, if the property of being X is a perfection, then it is not the case that the property of being not-X is a perfection. In other words, being X and being not-X cannot both be perfections.

M2: (Y)(P(Y) → (Z)(□(x)(Yx → Zx) → P(Z)). For any property Y, if Y is a perfection, then for any property Z, if it is necessarily the case that (for any x, if x has property Y then x has property Z), then Z is a perfection). Again, this is a long-winded way of saying that if a property Y is a perfection, and if having property Y entails having property Z, then property Z is a perfection. In other words, perfections entail only perfections.

M3: P([aS]). The property of being supreme is a perfection.

Th 1: ◊(p & q) → (◊p & ◊q). If, possibly, (p and q), then possibly, p and possibly, q.

Th 2: ◊¬◊p → ¬◊p. If it is possible that it is not possible that p, then it is not possible that p. This is an instance of the S5 axiom of modal logic.

Th 3: ◊(∃x)Fx → (∃x)◊Fx. If it is possible that there exists some x such that x is F, then there exists some x such that it is possible that x is F.

1. ¬◊(∃x)Sx (AIP). It is not possible that there exists some x such that x is supreme. This is an assumption for an indirect proof, i.e. “Suppose it is not possible that there exists some x such that x is supreme”.

2. □¬(∃x)Sx (1, ME, DN). Necessarily, it is not the case that there exists some x such that x is supreme. This follows from premise 1 by modal equivalence: if it is not possible (i.e. impossible) that there is a supreme being, then *necessarily* there is not a supreme being.

3. □(x)¬Sx (2, QN, DN). Necessarily, for any x, x is not supreme. This follows from premise 2 by quantifier negation: if it is necessarily not the case that there exists a supreme being, then it is necessarily the case that any x is not supreme.

4. (x)¬Sx (3, NE, CP). For any x, x is not supreme. This follows from premise 3 by necessity elimination: all that we have done is removed the “necessarily” qualifier from the start.

5. ¬Sa (4, UI). a is not supreme. This follows from premise 4 by universal instantiation: if, for any x, x is not supreme, then a is not supreme.

6. ¬Sa v ¬Sa (5, Add). a is not supreme or a is not supreme. This follows from premise 5 by addition: if a is not supreme, then obviously it is the case that either a is not supreme or a is not supreme.

7. Sa → ¬Sa (6, Impl). If a is supreme, then a is not supreme. Given premise 6, this is true merely by the truth conditions of material implications: if the consequent of a material implication is true (in this case, ¬Sa), then the material implication is true.

8. (x)([aS]x ≡ Sx) (Abs). For any x, x has the property of being supreme if and only if x is supreme.

9. (x)([a¬S]x ≡ ¬Sx) (Abs). For any x, x has the property of being not supreme if and only if x is not supreme.

10. ([aS]a ≡ Sa) (8, UI). a has the property of being supreme if and only if a is supreme. This follows from a universal instantiation of premise 8.

11. ([a¬S]a ≡ ¬Sa) (9, UI). a has the property of being not supreme if and only if a is not supreme. This follows from a universal instantiation of premise 9.

12. [aS]a → [a¬S]a (7, 10, 11, Equiv, Simp, HS). If a has the property of being supreme, then a has the property of being not supreme. This follows from substituting the identity relations in premises 10 and 11 into premise 7.

13. (x)([aS]x → [a¬S]x) (12, UG). For any x, if x has the property of being supreme, then x has the property of being not supreme. This follows from a universal generalisation of premise 12.

14. (x)¬Sx → (x)([aS]x → [a¬S]x) (4-13, CP). For any x, if x is not supreme, then (for any x, if x has the property of being supreme, then x has the property of being not supreme). This is just a summary of the derivation of premise 13 from premise 4.

15. □[(x)¬Sx → (x)([aS]x → [a¬S]x)] (14, NI). Necessarily, for any x, if x is not supreme, then (for any x, if x has the property of being supreme, then x has the property of being not supreme). This follows from 14 by necessity introduction: it just recognises that the derivation of premise 13 from premise 4 is logically valid, and so it is a necessary truth that premise 13 follows from premise 4.

16. P([aS]) → (Z)[(□(x)([aS]x → Zx)] → P(Z) (M2, UI). If the property of being supreme is a perfection, then for any property Z, if it is necessarily the case that (for any x, if x has the property of being supreme then x has property Z), then Z is a perfection). This is a universal instantiation of premise M2, and just says that if the property of being supreme is a perfection, and if having the property of being supreme entails any other property Z, then Z is also a perfection.

17. P([aS]) (M3). The property of being supreme is a perfection. This is just premise M3.

18. (Z)[(□(x)([aS]x → Zx)] → P(Z) (16, 17, MP). For any property Z, if it is necessarily the case that (for any x, if x has the property of being supreme then x has property Z), then Z is a perfection). This follows from premises 16 and 17 by modus ponens.

19. (x)[([aS]x → [a¬S]x) → P([a¬S])] (18, UI). For any x, if (if x has the property of being supreme, then x has the property of being not supreme), then the property of being not supreme is a perfection. This follows from a universal instantiation of premise 18, simply substituting [a¬S] for Z.

20. □(x)([aS]x → [a¬S]x) (3, 15, MMP). Necessarily, for any x, has the property of being supreme, then x has the property of being not supreme. This follows from premises 3 and 15 by modal modus ponens.

21. P([a¬S]) (19, 20, MP). The property of being not supreme is a perfection. This follows from premises 19 and 20 via modus ponens.

22. P([aS]) → ¬P([a¬S]) (M1, UI). If the property of being supreme is a perfection, then it is not the case that the property of being not supreme is a perfection. This is a universal instantiation of premise M1, which is just to say that the properties being supreme and being not supreme cannot both be perfections.

23. ¬P([a¬S]) (17, 22, MP). It is not the case that the property of being not supreme is a perfection. This follows from premise 17 and 22 by modus ponens.

24. ◊(∃x)Sx (21, 23, IP). It is possible that there exists some x such that x is supreme. This follows from an indirect proof: note that premises 21 and 23 are contradictory – and so, given the premises, the original assumption in premise 1 must be false. And, of course, if premise 1 is false, then it is possible that a supreme being exists.

25. ◊(∃x)Sx → (∃x)◊Sx (Th 3). If it is possible that there exists some x such that x is supreme, then there exists some x such that it is possible that x is supreme. This is an instance of theorem 3, the controversial Barcan Formula.

26. (∃x)◊Sx (24, 25, MP). There exists some x such that it is possible that x is supreme. This follows from premises 24 and 25 by modus ponens.

27. ◊Sv (26, EI). It is possible that v is supreme. This is an existential instantiation of premise 26: it is just to say that v, some existing thing (whatever it is), is possibly supreme.

28. ◊[¬◊(∃y)Gyv & ¬◊(∃y)(v≠y & ~Gvy)] (27, def. “S”). Possibly, (it is not possible that there exists some y such that y is greater than v, and it is not possible that there exists some y such that (v is not identical to y, and v is not greater than y)). This is just a restatement of premise 27, with the definition of supremity substituted in.

29. ◊¬◊(∃y)Gyv & ◊¬◊(∃y)(v≠y & ~Gvy)] (28, Th 1, MP). It is possible that it is not possible that there exists some y such that y is greater than v, and it is possible that it is not possible that there exists some y such that (v is not identical to y, and v is not greater than y). This follows from theorem 1 and premise 28 via modus ponens.

30. ¬◊(∃y)Gyv & ¬◊(∃y)(v≠y & ~Gvy) (29, Simp, Com, Th 2, MP, Conj). It is not possible that there exists some y such that y is greater than v, and it is not possible that there exists some y such that (v is not identical to y, and v is not greater than y). This follows from theorem 2 (axiom S5) and premise 29 by modus ponens.

31. Sv (30, def. “S”). v is supreme. This follows from premise 30, since premise 30 just says that v is supreme (with the full definition of supremity).

32. (∃x)Sx (31, EG). There exists some x such that x is supreme. This follows from premise 31: since v is an existing thing and it is supreme, it follows trivially that there exists something, viz. v, which is supreme.

The argument in The Blackwell Companion to Natural Theology

The first section argues for the possibility of a supreme being, corresponding to the argument up to premise 24 in the *Philo* paper – he uses the structure of Godel’s argument for this possibility premise, but with Maydole’s own premises instead of Godel’s axioms. Godel’s argument is found in the Appendix of Maydole’s chapter – I have substituted in Maydole’s premises.

1. ¬◊(∃x)Sx (AIP). It is not possible that there exists some x such that x is supreme. This is an assumption for an indirect proof – the idea is that if we suppose this premise, along with the other premises of the argument, there will be a contradiction – so we ought to reject this premise and agree that a supreme being is possible.

2. ¬◊(∃x)Sx → □(x)(Sx → ¬Sx) (theorem). If it is not possible that there exists some x such that x is supreme then, necessarily, for any x, if x is supreme then x is not supreme. This is a theorem: if some property is impossible, then the fact that any given x does not have that property is a necessary truth, and so is entailed (or necessarily implied) by any proposition – in this case Sx.

3. □(x)(Sx → ¬Sx) (1, 2, MP). Necessarily, for any x, if x is supreme then x is not supreme. This follows from premises 1 and 2 by modus ponens.

4. □(x)([a¬S]x ≡ ¬Sx) (Abs, NI). Necessarily, for any x, x has the property of being not supreme if and only if x is not supreme.

5. (□(x)(Sx → ¬Sx) & □(x)([a¬S]x ≡ ¬Sx)) → □(x)(Sx → [a¬S]x) (theorem). If (necessarily, for any x, if x is supreme then x is not supreme) and (necessarily, for any x, x has the property of being not supreme if and only if x is not supreme) then, necessarily, for any x, if x is supreme then x has the property of being not supreme. This theorem just expresses the fact that if the abstraction in premise 4 is correct (as it surely is), then [a¬S]x can be substituted for ¬Sx in premise 3.

6. □(x)(Sx → [a¬S]x) (3, 4, 5, Conj, MP). Necessarily, for any x, if x is supreme then x has the property of being not supreme. This is just the consequent of premise 5 by modus ponens, since the conjunction of premises 3 and 4 fulfill the antecedent.

7. P(S) (M3). Supremity is a perfection. This is premise M3 of the argument.

8. (P(S) & □(x)(Sx → [a¬S]x)) → P[a¬S]) (M2, UI). If (supremity is a perfection and necessarily, for any x, if x is supreme then x has the property of being not supreme) then the property of being not supreme is a perfection. This is a universal instantiation of premise M2, with supremity and the property of being not supreme being the relevant properties.

9. P[a¬S] (6, 7, 8, Conj, MP). The property of being not supreme is a perfection. This is the consequent of premise 8, which follows by modus ponens given the antecedent of premise 8 (which is the conjunction of premises 6 and 7).

10. P(S) → ¬P[a¬S] (M1, UI, Equiv, Simp). If supremity is a perfection, then the property of being not supreme is not a perfection. This is a universal instantiation of premise M1, which just says that a property and its negation cannot both be perfections – so, if supremity is a perfection, not being supreme cannot also be a perfection.

11. ¬P[a¬S] (9, 10, MP). The property of being not supreme is not a perfection. This follows from premises 9 and 10 by modus ponens.

12. ◊(∃x)Sx (1-11, IP). Possibly, there exists some x such that x is supreme. This follows by the indirect proof: premises 9 and 11 follow from premise 1 (and M1-M3), and are contradictory. Thus, premise 1 should be rejected (given M1-M3), and the possibility of a supreme being accepted.

Then, in the main body of the chapter (section 5), Maydole offers the deduction from the possibility premise to the conclusion that a supreme being exists:

1. ◊(∃x)Sx (from previous argument). Possibly, there exists some x such that x is supreme.

2. ◊(∃x)Sx → (∃x)◊Sx (Th 3). If it is possible that there exists some x such that x is supreme, then there exists some x such that possibly, x is supreme. This is an instance of the Barcan Formula.

3. (∃x)◊Sx (1, 2, MP). There exists some x such that possibly, x is supreme. This follows from premises 1 and 2 by modus ponens.

4. ◊Sv (3, EI). Possibly, v (some existing thing) is supreme. This is an existential instantiation of premise 3.

5. ◊(¬◊(∃y)Gyv & ¬◊(∃y)(v≠y & ¬Gvy)) (4, def. “S”). Possibly, (it is not possible that there exists some y such that y is greater than v, and it is not possible that there exists some y such that (v is not identical to y, and v is not greater than y)). This simply substitutes in the definition of supremity to premise 4.

6. ◊(¬◊(∃y)Gyv & ¬◊(∃y)(v≠y & ¬Gvy)) → (◊¬◊(∃y)Gyv & ◊¬◊(∃y)(v≠y & ¬Gvy)) (Th 1). If, possibly, (it is not possible that there exists some y such that y is greater than v, and it is not possible that there exists some y such that (v is not identical to y, and v is not greater than y)), then (possibly, it is not possible that there exists some y such that y is greater than v, and possibly, it is not possible that there exists some y such that v is not identical to y and v is not greater than y). This is just what Theorem 1 says, with ¬◊(∃y)Gyv representing ‘p’ and ¬◊(∃y)(v≠y & ¬Gvy) representing ‘q’.

7. (◊¬◊(∃y)Gyv & ◊¬◊(∃y)(v≠y & ¬Gvy)) (5, 6, MP). Possibly, it is not possible that there exists some y such that y is greater than v, and possibly, it is not possible that there exists some y such that (v is not identical to y, and v is not greater than y). This follows from premises 5 and 6 by modus ponens.

8. ◊¬◊(∃y)Gyv (7, Simp). Possibly, it is not possible that there exists some y such that y is greater than v. This is the first half of premise 7.

9. ◊¬◊(∃y)(v≠y & ¬Gvy) (7, Com, Simp). Possibly, it is not possible that there exists some y such that (v is not identical to y, and v is not greater than y). This is the second half of premise 7.

10. ◊¬◊(∃y)Gyv → ¬◊(∃y)Gyv (Th 2). If it is possible that it is not possible that there exists some y such that y is greater than v, then it is not possible that y is greater than v. This is an instantiation of theorem 2 (axiom S5).

11. ◊¬◊(∃y)(v≠y & ¬Gvy) → ¬◊(∃y)(v≠y & ¬Gvy) (Th 2). If it is possible that it is not possible that there exists some y such that (v is not identical to y, and v is not greater than y), then it is not possible that there exists some y such that (v is not identical to y, and v is not greater than y). This is another instantiation of theorem 2.

12. ¬◊(∃y)Gyv (8, 10, MP). It is not possible that there exists some y such that y is greater than v. This follows from premises 8 and 10 by modus ponens.

13. ¬◊(∃y)(v≠y & ¬Gvy) (9, 11, MP). It is not possible that there exists some y such that (v is not identical to y, and v is not greater than y). This follows from premises 9 and 11 by modus ponens.

14. ¬◊(∃y)Gyv & ¬◊(∃y)(v≠y & ¬Gvy) (12, 13, Conj). It is not possible that there exists some y such that y is greater than v, and it is not possible that there exists some y such that (v is not identical to y, and v is not greater than y). This is just a conjunction of premises 12 and 13.

15. Sv (14, def. “S”). v (some existing thing) is supreme – this just recognises that premise 14 states the definition of supremity with respect to v.

16. (∃x)Sx (15, EG). There exists some x such that x is supreme. This is an existential generalisation of premise 15.

**3. ****Defence of the premises [Forthcoming]**

**4. ****Criticisms [Forthcoming]**

**5. ****Additional notes [Complete]**

The argument uses second order quantificational modal logic, with the following abbreviations used for the inferences in this system:

Conj – Conjunction

Add – Addition

Simp – Simplification

DS – Disjunctive Syllogism

E-MI – Excluded Middle Introduction

MP – Modus Ponens

MT – Modus Tollens

HS – Hypothetical Syllogism

CD – Constructive Dilemma

Dist – Distribution

Assoc – Association

DN – Double Negation

DeM – DeMorgan

Trans – Transposition

Exp – Exportation

Equiv – Equivalence

EI – Existential Instantiation

EG – Existential Generalisation

UI – Universal Instantiation

UG – Universal Generalisation

QN – Quantifier Negation

II – Identity Introduction

IE – Identity Elimination

CP – Conditional Proof

IP – Indirect Proof

Additionally, the system uses 5 Modal Inference Rules:

NE – Necessity Elimination

MMP – Modal Modus Ponens

NI – Necessity Introduction

ME – Modal Equivalence

PN – Possibility Necessity

The system also uses the Principle of Abstraction (Abs): (x)([aF]x ≡ Fx) where [aF]x =_{df} x has the property of being an F. And finally, it includes three pertinent theorems:

Th 1: ◊(p & q) → (◊p & ◊q).

Th 2: ◊¬◊p → ¬◊p.

Th 3: ◊(∃x)Fx → (∃x)◊Fx.

Th 1: ◊(p & q) → (◊p & ◊q). If, necessarily, (p and q), then necessarily, p and necessarily, q.

Unless you’ve made a mistake in the symbolization, you should change “necessarily” to “possibly.”

Also, the first two theorems are *not* controversial.

Thanks for the correction, Czar. The mistake was in my translation, not the symbols. Now corrected.

May be worth talking about S5, as accessibility is nontrivial, and plausibly atheist may consider restoring reflective equilibrium with this argument by just rejecting S5. (Although, for my part, I’m sufficiently sceptical about modal epistemology that M1 and 2 don’t exert any great intuitive pull).

I don’t know a lot about formal logic but it seems to be a rule of logic that impossible properties entail their negations?

Doesn’t M2 run counter to that? (if the definitions are taken into account)

Is that allowed, and what consequences would that have?

It seems to me to be the logical equivalent of dividing by zero.

I believe there is a typo in line 9/10 of the sixth paragraph under the contents (the long paragraph) where it says that M1 says that not being supreme is a perfection, a missing “not” I would think.

Eric, impossibilities entail *anything* – so they can entail a proposition AND its negation. So I’m not sure that M2 contradicts anything in logic. Re: the typo: M1 doesn’t directly say that not being supreme is a perfection – rather, when added to the other premises and the assumption that it is not possible that there exists a supreme being, it jointly implies that not being supreme is a perfection. But this contradicts the premise that being supreme is a perfection. So the whole point is that this contradiction leads us to either reject one of the premises, or reject the assumption. The idea is that the premises are much more plausibly true than the assumption, so we should reject the assumption and conclude instead that it *is* possible that there exists a supreme being. Hope that explains.

Re:typo, as it stands you demonstrate “not being supreme is a perfection” twice, on lines 7 and 9 and “not being supreme is not a perfection” pops up in line 10 with no explicit explanation (though it’s not hard to figure out) also the “But” on line 8 seems malplaced if you have the same conclusion preceding it as following it.

If impossibilities entail their own negation isn’t defining something as not entailing its negation even if impossible defining it as exempt from the rule that impossibilities entail their own negation? How could thast be justified?

Couldn’t, say, the nonexistance of a supreme being be similarly “immunized” against disproof? How can we ever disprove anything if this is valid?

I see the issue now. Yes, there was a typo – thanks for noting – changed now! While I don’t think the argument is successful, I don’t think it defines anything as not entailing its negation, so I’m unsure about your latter query.

Sorry about not being clearer about the typo from the beginning.

M1 and M2 may not be part of a definition but they quite clearly add up to the claim that a Perfection can’t entail the negation of any perfection (including itself). However I think I’ve been unclear on the process of “doing logic”, if I can just say “I don’t think M1 and M2 are resonable premises therefore I don’t accept the conclusion.” then thats good enough for me and I don’t need them to be “illegal” or anything.

As you mention, the Barcan Formula (BF) is controversial. Maydole defends it in another article and not in the Blackwell article. I found this disappointing, since his entire argument hangs on it. Are you familiar with Maydole’s defense of this formula? If so, I think it would be very helpful to add it to your account of the argument. That way, we have everything relevant to the argument covered. Just a thought.

Hi, I received this objection to the premise M2:

“You have this phrase M2:

M2: (Y)(P(Y) → (Z)(□(x)(Yx → Zx) → P(Z))

You have this entailment in the consequent:

(Z)(□(x)(Yx → Zx) → P(Z))

You have this in the premise:

□(x)(Yx → Zx)

Which is this:

{m} □(x)(Zx ∨ ¬Yx)

If this:

□(x)(¬Yx)

Then this premise is true. Then you have:

(Y)(P(Y) → (Z)(⊤ → P(Z))

And since this is:

(Y)(P(Y) → (Z)( P(Z) ∨ ¬⊤)

…then:

(Y)(P(Y) → (Z)( P(Z) ∨ ⊥)

…so:

(Y)(P(Y) → (Z)( P(Z) )

…and M2 should not do this. M2 should only

assert P(Z). The purpose of coincidental ¬Yx’s

in {m} above is to select for cases where Yx

so that Zx can be asserted, but only where Yx.

If such cases are an empty set, there’s no selection

going on; true wins by default. But you let this

imply P(Z), hence the problem with M2.

Philosophically, it’s a problem because you should

only accept axioms that entail properties that meet

definitions, because if the definitions apply different

properties when a thing exists versus when it doesn’t,

then you can no longer sensibly talk about that thing’s

existence or non-existence”

I’d really appreciate an evaluation on this.