Cantini's "Logical Frameworks"

[Abram Demski]

Hi all,

I've been reading parts of "Logical Frameworks for Truth and Abstraction: an Axiomatic Study." This book develops combinator-style logics inspired by Illative logic and Kripke's theory of truth, which is pretty awesome. Some comments.

- The theories developed are not extensional, so they do not directly support arbitrary substitution of like for like. This is essentially because of the way use/mention is handled; most of the time, objects are manipulated in their "mention" form, so we're handling names of objects. Only when we assert T, the truth-predicate, of a whole sentence do we "use" objects.
- Andrea Cantini likes to talk about specific sets of axioms and inference rules, rather than just semantics, which is good for implementation; however, for some reason Cantini also prefers to keep these rather strictly close to the proof-theoretic power of first-order Peano Arithmetic; somehow stronger systems are seen as a bad thing. This may be an unfortunate feature of the work. On the other hand, since Cantini does give semantic considerations as well, it might not be difficult to add stronger axioms to an extent.
- Most of the book is based on a Kleene valuation scheme, which turns out rather weak for reasons I've brought up before. However, 2 chapters towards the end are devoted to a supervaluation approach, a solution I came to as well... (and Cantini even directs AI people to these 2 chapters in the book's introduction). Supervaluation solves the problem of allowing "2nd-order" stuff; Cantini shows that his supervaluation-based system proves a generalised mathematical induction principle (with a fairly short proof, I might add). As usual, supervaluation also allows proves all the classical tautologies, and allows classical deductions.

--
Abram Demski
http://lo-tho.blogspot.com/
http://groups.google.com/group/one-logic

[YKY (Yan King Yin, 甄景贤)]

I encountered supervaluation when I studied fuzzy logic;  didn't like it (for dealing with fuzziness).  Seems to be just a way to add "dunno" as a truth value.  The value 0.5 can serve the same purpose as it means "neither true nor false".

Secondly, Cantini seems to say in his book (though I haven't read it yet) that his theory does not offer a solution to paradoxes.

KY

[Ernesto Posse]

As with Fuzzy logic and multi-valued logics, supervaluationism
addresses the issue of vagueness, but it should not be confused with
simply adding a "don't know" value. A sentence is "supertrue" iff all
its classical valuations are true. This is not the same as fuzzy truth
value of 0.5.

Supervaluationism was introduced to deal with the sorites paradox and
the paradox of future contingents as far as I know, and at its heart
it distinguishes the principle of bivalence from the law of the
excluded middle, allowing you to deny the first while retaining the
second, thus as the original post mentions, keeping classical
tautologies.

.

2010/11/27 YKY (Yan King Yin, 甄景贤) <generic.in...@gmail.com>:

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Ernesto Posse

Modelling and Analysis in Software Engineering
School of Computing
Queen's University - Kingston, Ontario, Canada

[Abram Demski]

Cantini additionally says that (a version of) the supervaluation approach to formal truth believes in its own consistency, a claim which (at first glace) flies in the face of the 2nd incompleteness theorem! Of course, he means something quite a bit weaker-- just that the system asserts something like T(~[T(x)&F(x)]). Still, if he's claiming this, Cantini does *seem* to be giving the supervaluation approach a rather strong recommendation as a solution to the paradoxes.

My main reason for liking supervaluation, though, is its allowance for an "honest-to-god" induction schema. I've been trying to get this ability in alternative systems of my own creation w/o great success.

Any Kripke-style truth will use a "maybe" sort of value; but what supervaluation does is "sweep this under the carpet" so that we don't have to treat sentences with undefined truth values as special: the system acts as if they have *some* truth value, just a currently undefined one. That's what allows classical deduction. Kleene-based approaches, on the other hand, must prove that a sentence is grounded (and hence has a truth value) before classical logic can be applied to it. This severely restricts the ability to quantify over all predicates, which is why Illative Combinatory Logic can interpret first-order logic but not 2nd-order logic!

I think this is a natural way of doing things: we should not let the existence of strange sentences like the Liar spoil our ability to apply classical logic. The sentences we really care about are the meaningful ones, so we should tailor our inference rules around those... the meaningless sentences just get treated as if they have *some* meaning (who cares what!).

On the other hand, I think I am starting to become a revision theorist... (YKY, I think revision theory *may* be the best to support our idea of probabilistically learned truth predicates...)

--Abram

[YKY (Yan King Yin, 甄景贤)]

On Sat, Nov 27, 2010 at 8:37 PM, Ernesto Posse <epo...@cs.queensu.ca> wrote:

 As with Fuzzy logic and multi-valued logics, supervaluationism
addresses the issue of vagueness, but it should not be confused with
simply adding a "don't know" value. A sentence is "supertrue" iff all
its classical valuations are true. This is not the same as fuzzy truth
value of 0.5.

Supervaluationism was introduced to deal with the sorites paradox and
the paradox of future contingents as far as I know, and at its heart
it distinguishes the principle of bivalence from the law of the
excluded middle, allowing you to deny the first while retaining the
second, thus as the original post mentions, keeping classical
tautologies.

Thanks for explaining =)

My current approach is to build a logic on top of fuzzy-probabilistic semantics a la Bayesian networks (Abram would be very familiar with this).  I wonder what would be the status of LEM in probabilistic logic?  It seems that LEM would be replaced by "P(A) + P(~A) = 1", and the problems it has in classical logic may no longer affect us in the probabilistic setting.

KY