HOL (Interpretation)

Current License: CC BY-SA 2.5

9 events

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Nov 11, 2010 at 20:24	comment	added	Bubba88		I want to thank everyone for the answers, cause everything is clear to me now.
Nov 11, 2010 at 20:19	vote	accept	Bubba88
Nov 11, 2010 at 15:59	comment	added	Andrés E. Caicedo		There is certainly a viewpoint that even what is sometimes called "full" second order logic (of models of set theory) is essentially set theory. There is a famous quote by Quinne, saying that second order logic is "set theory in sheep's clothing" (this is in his "Philosophy of logic", 2nd ed., Harvard U. Press, 1986).
Nov 11, 2010 at 14:45	comment	added	Carl Mummert		@Andrej: Thanks for pointing out that gaping omission in my comment. Certainly there are also many mathematical logicians who wouldn't favor viewing HOL as just a jargon for set theory; I was not intending to leave them out.
Nov 11, 2010 at 14:18	comment	added	Andrej Bauer		I would be one of those logicians who disagree with set-theory-reductionist viewpoint. Higher-order logic can have many other interpretations (toposes come to mind).
Nov 11, 2010 at 12:58	history	edited	Joel David Hamkins	CC BY-SA 2.5	corrected spelling
Nov 11, 2010 at 12:33	comment	added	Carl Mummert		Also, although I (as a mathematical logician) am fine with simply viewing full higher-order semantics as being defined relative to a model of set theory, and thus really a sort of first-order semantics that incorporates set theory, that is only one way of looking at it. A logician of a less mathematical nature might simply view the semantics as being disquotational, for example. The distinction here is somewhat philosophical, but it's worth pointing out that not everyone would agree with our set-theory-reductionist viewpoint.
Nov 11, 2010 at 12:27	comment	added	Carl Mummert		The key point is that, when discussing higher order logic, we have to be very explicit about the semantics that we want to use. There is nothing syntactic about higher-order logic that cannot be done with first-order logic, but full higher-order semantics are much stronger than first-order (Henkin) semantics.
Nov 11, 2010 at 12:01	history	answered	Joel David Hamkins	CC BY-SA 2.5