First-order Logic: Modality and Intensionality

Contemporary use of the term 'intension' derives from the traditional logical Frege-Russell's doctrine that an idea (logic formula) has both an extension and an intension. From the Montague's point of view, the meaning of an idea can be considered as particular extensions in different possible worlds. In this paper we analyze the minimal intensional semantic enrichment of the syntax of the FOL language, by unification of different views: Tarskian extensional semantics of the FOL, modal interpretation of quantifiers, and a derivation of the Tarskian theory of truth from unified semantic theory based on a single meaning relation. We show that not all modal predicate logics are intensional, and that an equivalent modal Kripke's interpretation of logic quantifiers in FOL results in a particular pure extensional modal predicate logic (as is the standard Tarskian semantics of the FOL). This minimal intensional enrichment is obtained by adopting the theory of properties, relations and propositions (PRP) as the universe or domain of the FOL, composed by particulars and universals (or concepts), with the two-step interpretation of the FOL that eliminates the weak points of the Montague's intensional semantics. Differently from the Bealer's intensional FOL, we show that it is not necessary the introduction of the intensional abstraction in order to obtain the full intensional properties of the FOL. Final result of this paper is represented by the commutative homomorphic diagram that holds in each given possible world of this new intensional FOL, from the free algebra of the FOL syntax, toward its intensional algebra of concepts, and, successively, to the new extensional relational algebra (different from Cylindric algebras), and we show that it corresponds to the Tarski's interpretation of the standard extensional FOL in this possible world.Comment: 33 page

Temporal Probabilistic Logic Programs: State and Revision

There are numerous applications where we have to deal with temporal uncertainty associated with events. The Temporal Probabilistic (TP) Logic Programs should provide support for valid-time indeterminacy of events, by proposing the concept of an indeterminate instant, that is, an interval of time-points (event's time-window) with an associated, lower and upper, probability distribution. In particular, we propose the new semantics, for the TP Logic Programs of Dekhtyar and Subrahmanian. Our semantics, based on the possible world semantics is a generalization of the possible world semantics for (non temporal) Probabilistic Logic Programming, and we define the new syntax for PT-programs, with time variable explicitly represented in all atoms, and show how the standard role of Herbrand interpretations used as possible worlds for probability distributions is coherently extended to Temporal Probabilistic Logic Programming.Comment: 10 page

Intensional RDB for Big Data Interoperability

A new family of Intensional RDBs (IRDBs), introduced in [1], extends the traditional RDBs with the Big Data and flexible and 'Open schema' features, able to preserve the user-defined relational database schemas and all preexisting user's applications containing the SQL statements for a deployment of such a relational data. The standard RDB data is parsed into an internal vector key/value relation, so that we obtain a column representation of data used in Big Data applications, covering the key/value and column-based Big Data applications as well, into a unifying RDB framework. Such an IRDB architecture is adequate for the massive migrations from the existing slow RDBMSs into this new family of fast IRDBMSs by offering a Big Data and new flexible schema features as well. Here we present the interoperability features of the IRDBs by permitting the queries also over the internal vector relations created by parsing of each federated database in a given Multidatabase system. We show that the SchemaLog with the second-order syntax and ad hoc Logic Programming and its querying fragment can be embedded into the standard SQL IRDBMSs, so that we obtain a full interoperabilty features of IRDBs by using only the standard relational SQL for querying both data and meta-data.Comment: 30 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1103.0967, arXiv:1403.001

Data Base Mappings and Monads: (Co)Induction

In this paper we presented the semantics of database mappings in the relational DB category based on the power-view monad T and monadic algebras. The objects in this category are the database-instances (a database-instance is a set of n-ary relations, i.e., a set of relational tables as in standard RDBs). The morphisms in DB category are used in order to express the semantics of view-based Global and Local as View (GLAV) mappings between relational databases, for example those used in Data Integration Systems. Such morphisms in this DB category are not functions but have the complex tree structures based on a set of complex query computations between two database-instances. Thus DB category, as a base category for the semantics of databases and mappings between them, is different from the Set category used dominantly for such issues, and needs the full investigation of its properties. In this paper we presented another contributions for an intensive exploration of properties and semantics of this category, based on the power-view monad T and the Kleisli category for databases. Here we stressed some Universal algebra considerations based on monads and relationships between this DB category and the standard Set category. Finally, we investigated the general algebraic and induction properties for databases in this category, and we defined the initial monadic algebras for database instances.Comment: 31 page

Sound and Complete Query Answering in Intensional P2P Data Integration

Contemporary use of the term 'intension' derives from the traditional logical doctrine that an idea has both an extension and an intension. In this paper we introduce an intensional FOL (First-order-logic) for P2P systems by fusing the Bealer's intensional algebraic FOL with the S5 possible-world semantics of the Montague, we define the intensional equivalence relation for this logic and the weak deductive inference for it. The notion of ontology has become widespread in semantic Web. The meaning of concepts and views defined over some database ontology can be considered as intensional objects which have particular extension in some possible world: for instance in the actual world. Thus, non invasive mapping between completely independent peer databases in a P2P systems can be naturally specified by the set of couples of intensionally equivalent views, which have the same meaning (intension), over two different peers. Such a kind of mapping has very different semantics from the standard view-based mappings based on the material implication commonly used for Data Integration. We show how a P2P database system may be embedded into this intensional modal FOL, and how we are able to obtain a weak non-omniscient inference, which can be effectively implemented. For a query answering we consider non omniscient query agents and we define object-oriented class for them which implements as method the query rewriting algorithm. Finally, we show that this query answering algorithm is sound and complete w.r.t. the weak deduction of the P2P intensional logic.Comment: 27 page

Probabilistic Logic: Many-valuedness and Intensionality

The probability theory is a well-studied branch of mathematics, in order to carry out formal reasoning about probability. Thus, it is important to have a logic, both for computation of probabilities and for reasoning about probabilities, with a well-defined syntax and semantics. Both current approaches, based on Nilsson's probability structures/logics, and on linear inequalities in order to reason about probabilities, have some weak points. In this paper we have presented the complete revision of both approaches. We have shown that the full embedding of Nilsson'probabilistic structure into propositional logic results in a truth-functional many-valued logic, differently from Nilsson's intuition and current considerations about propositional probabilistic logic. Than we have shown that the logic for reasoning about probabilities can be naturally embedded into a 2-valued intensional FOL with intensional abstraction, by avoiding current ad-hoc system composed of two different 2-valued logics: one for the classical propositional logic at lower-level, and a new one at higher-level for probabilistic constraints with probabilistic variables. The obtained theoretical results are applied to Probabilistic Logic Programming.Comment: 15 page

Binary Sequent Calculi for Truth-invariance Entailment of Finite Many-valued Logics

In this paper we consider the class of truth-functional many-valued logics with a finite set of truth-values. The main result of this paper is the development of a new \emph{binary} sequent calculi (each sequent is a pair of formulae) for many valued logic with a finite set of truth values, and of Kripke-like semantics for it that is both sound and complete. We did not use the logic entailment based on matrix with a strict subset of designated truth values, but a different new kind of semantics based on the generalization of the classic 2-valued truth-invariance entailment. In order to define this non-matrix based sequent calculi, we transform many-valued logic into positive 2-valued multi-modal logic with classic conjunction, disjunction and finite set of modal connectives. In this algebraic framework we define an uniquely determined axiom system, by extending the classic 2-valued distributive lattice logic (DLL) by a new set of sequent axioms for many-valued logic connectives. Dually, in an autoreferential Kripke-style framework we obtain a uniquely determined frame, where each possible world is an equivalence class of Lindenbaum algebra for a many-valued logic as well, represented by a truth value.Comment: 21 page

Reduction of Many-valued into Two-valued Modal Logics

In this paper we develop a 2-valued reduction of many-valued logics, into 2-valued multi-modal logics. Such an approach is based on the contextualization of many-valued logics with the introduction of higher-order Herbrand interpretation types, where we explicitly introduce the coexistence of a set of algebraic truth values of original many-valued logic, transformed as parameters (or possible worlds), and the set of classic two logic values. This approach is close to the approach used in annotated logics, but offers the possibility of using the standard semantics based on Herbrand interpretations. Moreover, it uses the properties of the higher-order Herbrand types, as their fundamental nature is based on autoreferential Kripke semantics where the possible worlds are algebraic truth-values of original many-valued logic. This autoreferential Kripke semantics, which has the possibility of flattening higher-order Herbrand interpretations into ordinary 2-valued Herbrand interpretations, gives us a clearer insight into the relationship between many-valued and 2-valued multi-modal logics. This methodology is applied to the class of many-valued Logic Programs, where reduction is done in a structural way, based on the logic structure (logic connectives) of original many-valued logics. Following this, we generalize the reduction to general structural many-valued logics, in an abstract way, based on Suszko's informal non-constructive idea. In all cases, by using developed 2-valued reductions we obtain a kind of non truth-valued modal meta-logics, where two-valued formulae are modal sentences obtained by application of particular modal operators to original many-valued formulae.Comment: 27 page

Data Base Mappings and Theory of Sketches

In this paper we will present the two basic operations for database schemas used in database mapping systems (separation and Data Federation), and we will explain why the functorial semantics for database mappings needed a new base category instead of usual Set category. Successively, it is presented a definition of the graph G for a schema database mapping system, and the definition of its sketch category Sch(G). Based on this framework we presented functorial semantics for database mapping systems with the new base category DB.Comment: 21 page

Intensional RDB Manifesto: a Unifying NewSQL Model for Flexible Big Data

In this paper we present a new family of Intensional RDBs (IRDBs) which extends the traditional RDBs with the Big Data and flexible and 'Open schema' features, able to preserve the user-defined relational database schemas and all preexisting user's applications containing the SQL statements for a deployment of such a relational data. The standard RDB data is parsed into an internal vector key/value relation, so that we obtain a column representation of data used in Big Data applications, covering the key/value and column-based Big Data applications as well, into a unifying RDB framework. We define a query rewriting algorithm, based on the GAV Data Integration methods, so that each user-defined SQL query is rewritten into a SQL query over this vector relation, and hence the user-defined standard RDB schema is maintained as an empty global schema for the RDB schema modeling of data and as the SQL interface to stored vector relation. Such an IRDB architecture is adequate for the massive migrations from the existing slow RDBMSs into this new family of fast IRDBMSs by offering a Big Data and new flexible schema features as well.Comment: 29 pages. arXiv admin note: text overlap with arXiv:1103.0967, arXiv:1103.068