[FOM] A minor issue in modal logic
Steven Ericsson Zenith
steven at semeiosis.org
Sun Jul 4 13:59:12 EDT 2010
The following assumes that I have guessed correctly what you are referring to by Np[a]. It really isn't clear what it could mean to have "a proposition that is necessarily true in the actual world." It suggests a metaphysics that exists in "the actual world" independent from propositions that are propositions by logicians and mathematicians.
The question of whether you can access "the actual world" under any circumstances occupied all Positivist logicians, at least. The question is the core theme of Peirce's semeiotics and underlies any axiomatic proof theory.
It is at least dangerous to assert that Np[a] entails Np given the inaccessibility of Np[a]. Under these circumstances the assertion provides justification for the truth of any proposition what-so-ever.
If you wish to make the case that Np[a] entails Np you must first establish the manner of distinguishing Np[a] from Np.
Epistemically the best we can do is speak about distinguishing Np from p. From my point of view there is no such thing as an Np[a] and so it makes no sense to say Np and Np[a] are equivalent.
Dr. Steven Ericsson-Zenith
Institute for Advanced Science & Engineering
On Jul 3, 2010, at 10:53 AM, Keith Brian Johnson wrote:
> 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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