What’s a negative property?

In discussing the Gödelian ontological argument recently articulated by Alexander Pruss[1] (here and here) there was a need to define what we mean by “positive property”. In the first post, we defined a positive property (in a very Anselmian way) as a property that is better or greater to have than not. In his second paper, Pruss suggests a different route: define “negative property” first, and then define “positive property” from that. That’s what we’re going to do here. If people don’t like the Anselmian intuitions behind our first definition, this should be a more acceptable route. Note what our goal is: we seek a coherent and non-gerrymandered definition of “negative” such that the axioms in the argument are true. I’m not here seeking to give an analysis of negative properties as if they’re an already established concept. Rather, I’m giving a definition.

Limitation as negativeness

We’ll try to flesh out the suggestion by Pruss[2]:

We might, however, proceed differently, by taking as our primitive the notion of a negative property, which is actually more natural than the Gödelian notion of a positive property. We can think of a negative property as one that limits a being in some way.

So, we’ll define a property of some being to be “negative” if and only if it is limiting to that being in some way. Perhaps this is too rough. After all, properties can be limiting in some circumstances but not others. We can be more specific, then, and say that a property of a being is negative if in every circumstance, it is limiting to that being in some way. What do we mean by “limiting to a being”? It seems natural to take this to mean something along the following lines: in exemplifying the property, the being has less control or is less capable in some sense than if it wasn’t exemplifying the property. That is, the property leads to more restrictions or hindrances to the being than its negation. Note that the property is limiting to the being. Self-existence, for example, is the ultimate form of ontological independence, and as such is limiting to a being’s dependence upon other beings. However, it is certainly not limiting to the being itself. As such, it isn’t a negative property.

We’ll say a property is “positive”, then, if and only if its negation is negative.

How about some examples? Sure: not knowing that 1+1=2 is a negative property, for no matter what circumstance a being (that exemplifies this property) finds itself in, it’s cognitive abilities will always be limited by not knowing that 1+1=2. Being trapped is a negative property since it limits the being’s freedom. Moving to more of the relevant properties, omnipotence is positive, since omnipotence is plausibly just perfect freedom of will and perfect efficacy of will[3], both of which are clearly positive properties. Since not being self-existent is limiting to one’s control, freedom and self-sufficiency, it follows that it’s negative. And so self-existence, which is the ultimate property of self-sufficiency and independence, would be positive. Omniscience would also be a positive property. Perfect goodness would also be a positive property, since being evil is limiting to a being’s goodness. We might be able to push this a bit further: surely not being the paradigm of moral goodness is limiting in the sense that there is a standard above one to which one must submit? In this case, being the paradigm of moral goodness (which is stronger than being merely perfect good) would be a positive property.

Back to the ontological argument

Ok, that seems like enough examples. Let’s turn back to the ontological argument from earlier. The following two axioms follow quite naturally from how we’ve defined negative properties:

F1*. If a property P is negative, then ~P is not negative.

F2*. If a property P is negative, and a property Q entails P, then Q is negative.

Awesomely, (F1*) and (F2*) entail (F1) and (F2) from our earlier post. Furthermore, since self-existence is positive and it entails necessary existence, it follows from (F2) that necessary existence is positive, giving us (N1).[4]

We could go through the whole argument again, this time maybe even talking in terms of negative properties rather than positive ones. But that’s too much effort, so once again, I leave it as an exercise for the reader 😛


  1. He’s formulated and defended this argument in two papers: “A Gödelian Ontological Argument Improved” in Religious Studies (2009) and “A Gödelian Ontological Improved Even More” in M. Szatkowski (ed.), Ontological Proofs Today (2012).
  2. This appears in the second of the two papers mentioned in the previous note.
  3. See “Understanding omnipotence” by Kenneth Pearce and Alexander Pruss, Religious Studies 48 (2012): 403-414. Even if we don’t accept their analysis of it, any good analysis of omnipotence should be that of having unlimited power, which is sufficient for it to be a positive property.
  4. We can also get their via the property of being the paradigm of moral goodness like before.

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