‘Happy’ Reformation Day?

I have been (electronically) wished ‘Happy Reformation Day’ a remarkable number of times this morning. Each time, it has jarred slightly.

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On negotiating Halloween

Yes, actually I have noticed that global injustice is a bigger issue than Halloween. The question, though, is not that simple.

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On the verification of a proof of ‘God’s existence’

Der Spiegel online offered an eye-cathcing headline last week: ‘Computer Scientists “Prove”  God Exists’ (article here). The article referred to a pre-publication report of a paper submitted to arxiv.org last month, entitled ‘Formalization, Mechanization and Automation of Gödel’s Proof of God’s Existence’ (here). Essentially, the paper claims – it is more of an abstract and statement of results – that (a form of) Kurt Gödel’s modal ontological argument had been successfully coded, and that its validity had been demonstrated using – the detail I rather liked – programs running on a MacBook. As my colleague Alan Torrance pointed out on FB, this is not surprising; there has been a fairly general acceptance of Gödel’s logic for a while now. And as Der Spiegel pointed out, the point of the research was not to prove God’s existence but to demonstrate the possibility of computerised verification of logical arguments, using an argument of some fame, and of acknowledged complexity, as a test case. That said, the statement that a proof of God’s existence has been shown to be valid is not uninteresting – and that it is a version of the (in)famous ontological argument is very pleasing to those, like me, who see a restless fascination with the ontological argument as the infallible mark of a truly philosophical mind… Attentive readers will have noted some variation in use of scare quotes in the various headlines: the original piece used none; the newspaper article suggested ‘proof’ was being used wrongly; my title suggests the issue lies with ‘God’s existence’. Obviously, I think my usage is correct, but these differences highlight what is at stake. In defence of the original authors, they were referring to ‘Gödel’s proof of God’s existence’ in the same way they might have referenced ‘Fermat’s last theorem’: it is a well-known logical conundrum of interest to logicians because it appears correct but seeming has proved difficult to demonstrate completely. Der Spiegel‘s headline (at least in translation; I could not find an easy way to navigate to a German original of the article) suggested that the ‘proof’ was dubious; this is precisely wrong; assuming the correctness of the results reported, the heart of the research findings is the fact that the proof is certain, and demonstrated to be so. But what has been proved? Deductive logic always proves the same thing: that, given a certain set of axioms and definitions, a certain set of conclusions follows from a certain set of premises. (In modal logic there is an added complication of which modal logic is in play; if you are interested in the technical details of modal logics, go and get a life the paper claims that KB is sufficient for the proof to work; S5 is not needed.) In serious logic, the axioms and definitions are necessarily seriously abstruse and formal (here, Definition 3 asserts ‘Necessary existence of an individual is the necessary exemplification of all its essences’). The precise proof is that, given five axioms and accepting three definitions, it can be shown that a ‘God-like being’ necessarily exists, where ‘God-like’ means ‘possessing all positive properties’ (this is essentially a formalisation of perfect being theology). So what hesitations might we have before claiming ‘scientists have proved that God exists?’ Two, one more (theologically) significant than the other. First, and less significant, the proof is uninteresting (except as a neat bit of demonstrated logic) if any of the axioms or definitions are dubious. The three required definitions and five stated axioms are as follows (you can look up the symbolic logic in the paper above if you are interested): D1 A ‘God-like’ being possesses all positive properties. D2 An ‘essence’ of an individual is a property possessed by it and necessarily implying any of its properties. D3 ‘Necessary existence’ of an individual is the necessary exemplification of all its essences. A1 Either a property or its negation is positive, but not both. A2 A property necessarily implied by a positive property is positive. A3 The property of being ‘God-like’ is positive. A4 Positive properties are necessarily positive. A5 Necessary existence is a positive property. Now, anyone at all familiar with debate over the ontological argument in its various forms will fairly quickly spot the first point of attack here: is the concept of ‘necessary existence’ coherent/meaningful? (And, concomitantly, is the claim that it is a positive property if coherent justified?) As the authors say in their concluding paragraph, ‘[t]he critical discussion of the underlying concepts, definitions and axioms remains a human responsibility…’ (In reading the proof, I am also struck by the unspecified value-judgement implied in the word ‘positive’; A3, A5, and probably D1 seem to me to smuggle in...

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