[FOM] A minor issue in modal logic

Keith Brian Johnson joyfuloctopus at yahoo.com
Sat Jul 3 13:53:31 EDT 2010

All and sundry:  I have encountered an issue in modal logic that I haven't seen and don't know where to look for a resolution of, and my local philosophy department's members haven't been much help.  Perhaps FOMers can help.  

The fundamental question, I think--I'll give its origin in a moment--is this:  For any proposition p, where "Np" means "It is necessarily true that p" and "Np[a]" means "p is necessarily true in the actual world," are Np and Np[a] equivalent--or, if not, does Np[a] at least entail Np?  

In analyzing Charles Hartshorne's version of St. Anselm's ontological argument for God's existence, I found him using modal modus tollens--i.e., N(p-->q)-->(N~q-->N~p)--to go from his premiss N(g-->Ng) ("g"="God exists") to N~Ng-->N~g, as part of getting to Ng v N~g (i.e., Pg-->Ng, where "Pg" means "It is possible that g").  Now, I'm willing to grant him his Ng v N~g (i.e., Pg-->Ng) as a definition or characterization of God, and then criticize the rest of his argument; but when I do look at this part of his argument, and in particular at his use of modal modus tollens, I note that if "g-->Ng" is supposed to mean "g[a]-->Ng," then modal modus tollens should get him N~Ng-->N~g[a], not N~Ng-->Ng.  (If one takes an expression like "Aig[i]" to mean "g is true in every possible world w[i]," then modal modus tollens would get him his desired conclusion if N(g-->Ng) were interpreted not as Ai(g[a]-->Aig[i]) but rather as Ai(g[i]-->Aig[i]).  But if Hartshorne
 had that, he wouldn't be starting off with N(g[a]-->Ng), as he seems to think he is.)  

It would seem natural to say that N~g[a] just means N~g, since if it is necessarily true that g is false in the actual world, that seems to just mean that g is false in every possible world, i.e., N~g.  But someone like Alvin Plantinga (Hartshorne doesn't use possible-worlds terminology) seems to want to talk about what is true of one possible world from the standpoint of another, suggesting a distinction between "Necessarily, g is false" and "Necessarily, g is false in the actual world."  They sound alike without possible-worlds terminology, but with it, they sound different:  "In all possible worlds, g is false" vs. "In all possible worlds, 'g is true in the actual world' is false."   (At least, they sound different if the term "actual world" is always taken to refer to this world, no matter which world it is used in, rather than as referring to the world in which it is uttered.)  Hence the question, Are Np and Np[a] equivalent--or, since
 entailment of Np by Np[a] would be enough to save the Hartshornean derivation, does Np[a] entail Np?

Does anyone know of anywhere these sorts of questions are addressed?

Keith Brian Johnson


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