quantified modal logic

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The Ontological Argument: Question Begging?

Is the ontological argument question begging? Short answer: some versions yes, others, no. For the long answer, read on:

Many versions of the ontological argument appear to beg the question. The Anselmian version of the argument seems invalid, but there are other formulations of it which avoid its invalidity (cf. Maydole’s chapter on the Ontological Argument in The Blackwell Companion to Natural Theology, eds. Craig + Moreland, 553ff). Then, Alvin Plantinga came along and introduced the modal ontological argument, which relies on S5 modal logic. I have discussed Plantinga’s argument elsewhere, as well as demonstrated its validity.

Therefore, we will focus on a new considerations. Consider the following very simple version of the modal ontological argument:

1) Possibly, God necessarily exists

2) Therefore, God necessarily exists

The argument seems, at first face, to be a very strange argument. However, the argument does not beg the question when viewed through modal logic. 1) does follow from 2) in a non-tautological way, but 1) must be established.

Symbolically, the argument is written as the following (Take “T” to be “God exists”):

1) ◊□T

2) □T

S5 modal logic is based upon this very axiom. Namely, ◊□x⊃□x or  ◊□x iff □x (Hughes & Cressewell, A New Introduction to Modal Logic, 58). For in modal logic:

3) that which is “possible” exists in “some possible world.”

4) That which is necessary exists in all possible worlds

Therefore, if something is possibly necessary, then it must obtain in some possible world (3). however, if it is necessary, then it exists in all possible worlds (4). Therefore, if something is possibly (exists in some possible world) necessary (exists in all possible worlds), then it exists in all possible worlds.

Is this argument question begging? If it is, then it is not obviously so. Alexander Pruss has argued out that the argument is question begging only if it is directed at one who does not understand that 1) entails 2) (Pruss, The Principle of Sufficient Reason, 232). I’m not convinced that this is correct. Soundness of arguments don’t depend upon whether people understand them–they depend on whether they are valid or true.  However, it seems Pruss has an intuitive point here, in that even if this argument isn’t question begging, it appears to be.

How might the theist respond? Well, Pruss argues that if the theist argues for the establishment of S5, then it is no longer question begging (232). Alvin Plantinga does just that in God, Freedom, and Evil, as Pruss points out. We’ve already established elsewhere that Plantinga’s argument doesn’t beg the question regardless (see here), but this symbolic proof is bolstered by providing an argument for S5.

Then, it seems to be the case that if S5 modality is valid, God necessarily exists.

Are there versions of the ontological argument that resist this reduction to the “simplistic” version offered here? Yes, there are.  For example, Stephen Parrish’s ontological argument:

5) The concept of the GPB (Greatest Possible Being) is coherent (and thus broadly logically possible)

6) Necessarily, a being who is the GPB is necessarily existent, and would have (at least) omnipotence, omniscience, and moral perfection essentially.

7) If the concept of the GPB is coherent, then it exists in all possible worlds.

8 ) But if it exists in all possible worlds, then it exists in the actual world.

9) The GPB exists (Parrish, God and Necessity, 82)

This argument may initially seem to be susceptible to the same reduction, but it can avoid this reduction by lengthening it to:

10) the GPB is coherent (and logically possible)

11) the GPB’s coherence entails modal possibility

12) the GPB is necessary

13) modally, if something is possibly necessary, then it is necessary

14) the GPB exists necessarily

The key premise here is 10), because if it is true, then the rest of the argument follows necessarily. What reasons do we have for thinking 10) is true? Such a debate is beyond the scope of this post (good discussions can be found throughout theistic philosophy of religion–see, in particular, Craig and Moreland, Philosophical Foundations for a Christian Worldview and Swinburne, The Coherence of Theism; see also my brief discussion in the post on the argument here); but it seems to me that there is no incoherence in the concept. If that is the case, then I am justified in holding 14).

Therefore, it seems the modal ontological argument is not question begging, particularly if one argues first for the validity of S5 modality. Furthermore, there are other modal arguments which don’t rely on a reduction to a simple modal argument. For example, Parrish’s ontological argument relies instead upon the coherence of the GPB. Such arguments are successful if arguments against the GPB’s coherence are shown to be unsuccessful. In either case, God exists.

Ergo deus est.



Maydole, Robert E. “The Ontological Argument.” The Blackwell Companion to Natural Theology. Edited William Lane Craig and J.P. Moreland. Blackwell, 2009.

Parrish, Stephen E. God and Necessity. University Press of America. 1997.

Pruss, Alexander. The Principle of Sufficient Reason. Cambridge. 2006.


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Book Review: The Nature of Necessity

Alright, I managed to finish another book this weekend. This is one I’ve been working through for months. Literally.

The Nature of Necessity by Alvin Plantinga is quite the philosophy book. In it, Plantinga tackles “The distinction between necessary and contingent truth (1)…” He distinguishes necessity de re and de dicto necessity. De dicto necessity he defines as “a matter of a proposition’s being necessarily true (v)” while de re necessity is “an object’s having a property essentially or necessarily.”

The first few chapters compose Plantinga’s argument for de re necessity by showing that such things can be shown in de dicto terms. It is quite an interesting section, and one in which I think Plantinga makes a strong case. Chapter IV I found particularly interesting, as Plantinga’s discussion here starts to turn to modal logic. The discussion of how we should define worlds, books (which I take to mean a list of propositions that belong to a given world), actuality, possibility, and the like. This discussion is extremely interesting and leads to some conclusions like “A possible world… is a possible state of affairs in the broadly logical sense… a state of affairs S is complete or maximal if for every state of affairs S’, S includes S’ or S precludes S’. And a possible world is simply a possible state of affairs that is maximal (45).”

But then, “Equally obviously [as that one state of affairs obtains], at most one obtains; for suppose two worlds W and W* are distinct worlds, there will be some state of affairs S such that W includes S and W* precludes S. But then if both W and W* are actual, S both obtains and does not obtain; and this, as they say, is repugnant to the intellect (45).”

These kinds of conclusions are to be found throughout The Nature of Necessity and it is this that makes the book such a good read. Plantinga’s descriptions of various states of affairs never fails to entertain, as his sense of humor is present at places throughout the book, while he still maintains his extremely rigorous treatment of the items at hand.

In later chapters (notably VI), Plantinga defends his version of necessity from alternatives, and in the process raises some objections to these alternatives. I’d write more of these out, but the fact of the matter is that the Nature of Necessity has that problem that I believe most books that are so heavy on analytic philosophy have: if one doesn’t read the whole, it is hard to understand any one part. This isn’t really a strike against the book, but is rather simply of note. The way that Plantinga builds on each previous point throughout the book helps give his case clarity.

Once Plantinga has gotten the bulk of the text out of the way, having made a strong case for necessity, he gets into how it is that these concepts apply to God, particularly the theistic God. The chapters about God and evil and God and necessity are largely recycled (or perhaps reworked) versions of his book God, Freedom, and Evil. As such, I didn’t spend a lot of time on these chapters, but in them Plantinga presents his case for the “Victorious Modal” version of the Ontological Argument. It is a version that I think has great potential, and has been the subject of some debate (see Graham Oppy, for instance… or the Blackwell Companion to Natural Theology‘s treatment). Plantinga’s defense of theism from the problem of evil is equally impressive. I have written about it in the past, so I will not do so again, but I do think that Plantinga’s various arguments carry some weight and certainly merit discussion among apologists.

The Nature of Necessity closes with an appendix of some length (30 pages) about modal logic. Specifically, it is directed at answering objections (and more specifically, Quine’s objections) to modal logic. I believe that Plantinga does a fine job of taking on this task, and making a valid case for modal logic, but readers must be absolutely warned that a deep understanding of symbolic logic is required to even begin to approach the appendix. I recommend tackling some texts on logic before reading the appendix. As it was, I had to write out a list of the symbols being used in order to understand this section. It took me several days to get through the appendix alone. But once one does so, they will find that it is yet another rewarding section in an overall wonderful book.

The book weighs in at about 251 pages (including the appendix, but not including the preface), but the content that it includes is immense. Not only that, but the text itself is so heavy that a single page can often take minutes to work through and ponder. I recommend The Nature of Necessity to those who wish to explore, well, necessity and what it means further. It is by no means an easy work to read, and will require quite a bit of pondering and rereading, but in the end the payoff is worth it.

Scores (5 is truly average):

Quality of Arguments (if it applies): 10

Overall Content: 8

Difficulty: 9

Clarity: 9

Theology/Doctrine: N/A- other than theism, this doesn’t really have enough to judge the work based on Doctrinal or Theological stances

Value (price): 8- There’s a lot here, and I think it warrants the purchase price.

Relevance: 6- I don’t know how much use I’ll get out of the book as an apologetic work. Plantinga’s God, Freedom, and Evil may be more useful in this regard, but that doesn’t really take away from how much I liked the book

Review Criteria:

The Quality of arguments is just what it says. Obviously this is subjective. Do I think the arguments presented in the book (if there are any) are valid and/or useful?

Overall Content is a general judge of how good I felt the book is.

Difficulty is the amount of work it takes to get through the work. Higher values don’t necessarily mean the book is better, just more difficult to read.

Clarity simply outlines how clear I believe the author was.

Theology/Doctrine is my judgment, clearly based on my presuppositions, of how good I felt the author’s theological or doctrinal content was (if there is any).

Value is a determination of whether I believe the book is worth the asking price.

Relevance outlines whether I think the book has real-life applications. A low score in this doesn’t necessarily mean the book is bad, just that I believe there may not be much to use. In other words, a book could score low on this criterium, but I might still find it quite good.

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