4,594 research outputs found
Methods for Solving Necessary Equivalences
Nonmonotonic Logics such as Autoepistemic Logic, Reflective Logic, and Default Logic, are usually
defined in terms of set-theoretic fixed-point equations defined over deductively closed sets of sentences of First
Order Logic. Such systems may also be represented as necessary equivalences in a Modal Logic stronger than
S5 with the added advantage that such representations may be generalized to allow quantified variables crossing
modal scopes resulting in a Quantified Autoepistemic Logic, a Quantified Autoepistemic Kernel, a Quantified
Reflective Logic, and a Quantified Default Logic. Quantifiers in all these generalizations obey all the normal laws
of logic including both the Barcan formula and its converse. Herein, we address the problem of solving some
necessary equivalences containing universal quantifiers over modal scopes. Solutions obtained by these
methods are then compared to related results obtained in the literature by Circumscription in Second Order Logic
since the disjunction of all the solutions of a necessary equivalence containing just normal defaults in these
Quantified Logics, is equivalent to that system
Representing Autoepistemic Logic in Modal Logic
The nonmonotonic logic called Autoepistemic Logic is shown to be representable in a monotonic
Modal Quantificational Logic whose modal laws are stronger than S5. Specifically, it is proven that a set of
sentences of First Order Logic is a fixed-point of the fixed-point equation of Autoepistemic Logic with an initial
set of axioms if and only if the meaning or rather disquotation of that set of sentences is logically equivalent to
a particular modal functor of the meaning of that initial set of sentences. This result is important because the
modal representation allows the use of powerful automatic deduction systems for Modal Logic and unlike the
original Autoepistemic Logic, it is easily generalized to the case where quantified variables may be shared
across the scope of modal expressions thus allowing the derivation of quantified consequences. Furthermore,
this generalization properly treats such quantifiers since both the Barcan formula and its converse hold
Representing Default Logic in Modal Logic
The nonmonotonic logic called Default Logic is shown to be representable in a monotonic Modal
Quantificational Logic whose modal laws are stronger than S5. Specifically, it is proven that a set of
sentences of First Order Logic is a fixed-point of the fixed-point equation of Default Logic with an initial set of
axioms and defaults if and only if the meaning or rather disquotation of that set of sentences is logically
equivalent to a particular modal functor of the meanings of that initial set of sentences and of the sentences in
those defaults. This result is important because the modal representation allows the use of powerful
automatic deduction systems for Modal Logic and because unlike the original Default Logic, it is easily
generalized to the case where quantified variables may be shared across the scope of the components of the
defaults thus allowing such defaults to produce quantified consequences. Furthermore, this generalization
properly treats such quantifiers since both the Barcan Formula and its converse hold
How to Deduce the Other 91 Valid Aristotelian Modal Syllogisms from the Syllogism IAI-3
This paper firstly formalizes Aristotelian modal syllogisms by taking advantage of the trisection structure of (modal) categorical propositions. And then making full use of the truth value definition of (modal) categorical propositions, the transformable relations between an Aristotelian quantifier and its three negative quantifiers, the reasoning rules of classical propositional logic, and the symmetry of the two Aristotelian quantifiers (i.e. some and no), this paper shows that at least 91 valid Aristotelian modal syllogisms can be deduced from IAI-3 on the basis of possible world semantics and set theory. The reason why these valid Aristotelian modal syllogisms can be reduced is that any Aristotelian quantifier can be defined by the other three Aristotelian quantifiers, and the necessary modality ( ) and possible modality ( ) can also be defined mutually. This research method is universal. This innovative study not only provides a unified mathematical research paradigm for the study of generalized modal syllogistic and other kinds of syllogistic, but also makes contributions to knowledge representation and knowledge reasoning in computer science
Representing Reflective Logic in Modal Logic
The nonmonotonic logic called Reflective Logic is shown to be representable in a monotonic Modal
Quantificational Logic whose modal laws are stronger than S5. Specifically, it is proven that a set of
sentences of First Order Logic is a fixed-point of the fixed-point equation of Reflective Logic with an initial set
of axioms and defaults if and only if the meaning of that set of sentences is logically equivalent to a particular
modal functor of the meanings of that initial set of sentences and of the sentences in those defaults. This
result is important because the modal representation allows the use of powerful automatic deduction systems
for Modal Logic and because unlike the original Reflective Logic, it is easily generalized to the case where
quantified variables may be shared across the scope of the components of the defaults thus allowing such
defaults to produce quantified consequences. Furthermore, this generalization properly treats such
quantifiers since all the laws of First Order Logic hold and since both the Barcan Formula and its converse
hold
On the Relationship between Quantified Reflective Logic and Quantified Default Logic
Reflective Logic and Default Logic are both generalized so as to allow universally quantified
variables to cross modal scopes whereby the Barcan formula and its converse hold. This is done by
representing both the fixed-point equation for Reflective Logic and the fixed-point equation for Default both as
necessary equivalences in the Modal Quantificational Logic Z. and then inserting universal quantifiers before
the defaults. The two resulting systems, called Quantified Reflective Logic and Quantified Default Logic, are
then compared by deriving metatheorems of Z that express their relationships. The main result is to show that
every solution to the equivalence for Quantified Default Logic is a strongly grounded solution to the
equivalence for Quantified Reflective Logic. It is further shown that Quantified Reflective Logic and
Quantified Default Logic have exactly the same solutions when no default has an entailment condition
Counting Incompossibles
We often speak as if there are merely possible people—for example, when we make such claims as that most possible people are never going to be born. Yet most metaphysicians deny that anything is both possibly a person and never born. Since our unreflective talk of merely possible people serves to draw non-trivial distinctions, these metaphysicians owe us some paraphrase by which we can draw those distinctions without committing ourselves to there being merely possible people. We show that such paraphrases are unavailable if we limit ourselves to the expressive resources of even highly infinitary first-order modal languages. We then argue that such paraphrases are available in higher-order modal languages only given certain strong assumptions concerning the metaphysics of properties. We then consider alternative paraphrase strategies, and argue that none of them are tenable. If talk of merely possible people cannot be paraphrased, then it must be taken at face value, in which case it is necessary what individuals there are. Therefore, if it is contingent what individuals there are, then the demands of paraphrase place tight constraints on the metaphysics of properties: either (i) it is necessary what properties there are, or (ii) necessarily equivalent properties are identical, and having properties does not entail even possibly being anything at all
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
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