Modal operator

About: Modal operator is a research topic. Over the lifetime, 1151 publications have been published within this topic receiving 22865 citations. The topic is also known as: modal connective.

Alethic-Deontic Logic and the Alethic-Deontic Octagon

Abstract: This paper will introduce and explore a set of alethic-deontic systems. Alethic-deontic logic is a form of logic that combines ordinary (alethic) modal logic, which deals with modal concepts such as necessity, possibility and impossibility, and deontic logic, which investigates normative expressions such as “ought”, “right” and “wrong”. I describe all the systems axiomatically. I say something about their properties and prove some theorems in and about them. We will be especially interested in how the different deontic and modal concepts are related to each other in various systems. We will map these relationships in an alethic-deontic octagon, a figure similar to the classical so-called square of opposition.

From logic programming semantics to the consistency of syntactical treatments of knowledge and belief

Abstract: This paper concerns formal theories for reasoning about the knowledge and belief of agents. It has seemed attractive to researchers in artificial intelligence to formalise these propositional attitudes as predicates of first-order predicate logic. This allows the agents to express stronger introspective beliefs and engage in stronger meta-reasoning than in the classical modal operator approach. Results by Montague [1963] and Thomason [1980] show, however, that the predicate approach is prone to inconsistency. More recent results by des Rivieres & Levesque [1988] and Morreau & Kraus [1998] show that we can maintain the predicate approach if we make suitable restrictions to our set of epistemic axioms. Their results are proved by careful translations from corresponding modal formalisms. In the present paper we show that their results fit nicely into the framework of logic programming semantics, in that we show their results to be corollaries of well-known results in this field. This does not only allow us to demonstrate a close connection between consistency problems in the syntactic treatment of propositional attitudes and problems in semantics for logic programs, but it also allows us to strengthen the results of des Rivieres & Levesque [1988] and Morreau & Kraus [1998].

Alethic-Deontic Logic : Some Theorems

Abstract: The purpose of this paper is to prove some theorems in alethic-deontic logic. Alethic-deontic logic is a kind of bimodal logic that combines ordinary alethic (modal) logic and deontic logic. Ordinary alethic logic is a branch of logic that deals with modal concepts, such as necessity and possibility, modal sentences, arguments and systems. Deontic logic is the logic of norms. It is about normative words, such as “ought”, “right” and “wrong”, normative sentences, arguments and systems. Alethic-deontic logic contains both modal and normative concepts and can be used to study how these interact. This paper contains some interesting theorems that can be proved in alethic-deontic logic. I will show that all primitive deontic operators are redundant when prefixed to the alethic operators in some systems. I will prove that necessarily equivalent sentences have the same deontic status in many systems. I will establish that the set of sentences in some alethic-deontic systems can be partitioned into five, mutually exclusive, exhaustive subsets. Finally, I will show that there are exactly ten distinct modalities in some alethic-deontic systems.

On a theory of weak implications

Abstract: The purpose of this paper is to study logical implications which are much weaker than the implication of intuitionistic logic.In §1 we define the system SI (system of Simple Implication) which is obtained from intuitionistic logic by restricting the inference rules of intuitionistic implication. The implication of the system SI is called the “simple implication” and denoted by ⊃, where the simple implication ⊃ has the following properties:(1) The simple implication ⊃ is much weaker than the usual intuitionistic implication.(2) The simple implication ⊃ can be interpreted by the notion of provability, i.e., we have a very simple semantics for SI so that a sentence A ⊃ B is interpreted as “there exists a proof of B from A”.(3) The full-strength intuitionistic implication ⇒ is definable in a weak second order extension of SI; in other words, it is definable by help of a variant of the weak comprehension schema and the simple implication ⊃. Therefore, though SI is much weaker than the intuitionistic logic, the second order extension of SI is equivalent to the second order extension of the intuitionistic logic.(4) The simple implication is definable in a weak modal logic MI by the use of the modal operator and the intuitionistic implication ⇒ with full strength. More precisely, A ⊃ B is defined as the strict implication of the form ◽(A ⇒ B).In §1, we show (3) and (4). (2) is shown in §2 in a more general setting.Semantics by introduction rules of logical connectives has been studied from various points of view by many authors (e.g. Gentzen [4], Lorentzen [5], Dummett [1], [2], Prawitz [8]. Martin-Lof [7], Maehara [6]). Among them Gentzen (in §§10 and 11 of [4]) introduced such a semantics in order to justify logical inferences and the mathematical induction rule. He observed that all of the inference rules of intuitionistic arithmetic, except for those on implication and negation, are justified by means of his semantics, but justification of the inference rules on implication and negation contains a circular argument for the interpretation by introduction rules, where the natural interpretation of A ⊃ B by ⊃-introduction rule is “there exists a proof of B from A ” (cf. §11 of Gentzen [4]).

The Square of Opposition in Orthomodular Logic

Abstract: In Aristotelian logic, categorical propositions are divided in Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative. Possible relations between two of the mentioned type of propositions are encoded in the square of opposition. The square expresses the essential properties of monadic first order quantification which, in an algebraic approach, may be represented taking into account monadic Boolean algebras. More precisely, quantifiers are considered as modal operators acting on a Boolean algebra and the square of opposition is represented by relations between certain terms of the language in which the algebraic structure is formulated. This representation is sometimes called the modal square of opposition. Several generalizations of the monadic first order logic can be obtained by changing the underlying Boolean structure by another one giving rise to new possible interpretations of the square.