Propositional Logic

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What would happen if we tried to read the thinking and some writings of Machiavelli using the propositional logic? This short paper moves from this aim: clarify the arguments of the Florentine thinker, making explicit the inferences just through the propositional logic, its rules and its theorems. In particular, the logical patterns, that we try to find inside the Machiavelli's text, are the reductio ad absurdum (negation Introduction), the case analysis and a particular version of the constructive dilemma in some its proofs: the conditional proof. The paradigmatic parts of the of the Machiavelli's writings, that we will use for extract some important topics of his arguments, are the twenty-fifth chapter of the Il Principe and the ninth chapter of the Discorsi sopra la prima deca di Tito Livio's third book.

Che cosa succederebbe se provassimo ad interpretare il pensiero e alcuni scritti di Machiavelli tramite la logica proposizionale? Questo short paper muove da questo intento: chiarire le argomentazioni del pensatore fiorentino, esplicitandone le inferenze tramite la logica proposizionale, le sue regole e i suoi teoremi. Nello specifico, gli schemi logici utilizzati, che si provano a rintracciare all'interno del dettato machiavelliano, sono quelli della riduzione all'assurdo e del dilemma costruttivo in alcune sue varianti. Le parti paradigmatiche degli scritti del fiorentino, che vengono utilizzate per estrapolarne alcuni luoghi del suo pensiero, sono il Capitolo XXV de Il Principe e il Libro III, Capitolo 9 dei Discorsi sopra la prima decade di Tito Livio.

An introduction to the formal approach that underlies university mathematics. Topics covered are: Sets and Logic, Relations and Functions, Number Systems, Infinity. The book has arisen out of a first year course taught to Mathematics majors at Shiv Nadar University, with the aim of helping students bridge the different requirements of rigour at school and university.

Equilibrium logic, introduced in [20], is a conservative extension of answer set semantics for logic programs to the full language of propositional logic. In this paper we initiate the study of first-order variants of equilibrium logic. In particular, we focus on a quantified version QN 5 of the propositional many-valued logic N 5 of here-and-there with strong negation, and define the condition of equilibrium via a minimal model construction. We verify Skolem forms and Herbrand theorems for QN 5 and show that, like its propositional counterpart, the ...

From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categories of syntax and semantics, mediated by homming into a common dualizing object, in this case 2. In the present work, we generalize the entire arrangement from propositional to first-order logic. Boolean algebras are replaced by Boolean categories presented by theories in first-order logic, and spaces of models are replaced by topological groupoids of models and their isomorphisms. A duality between the resulting categories of syntax and semantics, expressed first in the form of a contravariant adjunction, is established by homming into a common dualizing object, now $\Sets$, regarded once as a boolean category, and once as a groupoid equipped with an intrinsic topology. The overall framework of our investigation is provided by topos theory. Direct proofs of the main results are given, but the specialist will recognize toposophical ideas in the background. Indeed, the duality between syntax and semantics is really a manifestation of that between algebra and geometry in the two directions of the geometric morphisms that lurk behind our formal theory. Along the way, we construct the classifying topos of a decidable coherent theory out of its groupoid of models via a simplified covering theorem for coherent toposes.

Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen as the logic of subsets of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic.

In Data Mining, to discover closed item’s relation among the items, association rule can be used. I propose a method to find closed and dependent items using propositional logic property Biimplication. Generating association rules, finding confidence of rules and apply propositional logic with consideration of minimum confidence we can find closed items, thereby we can consider those items as closed and dependable items one on each other.

Lindenbaum method is named after the Polish logician Adolf Lindenbaum who prematurely and without a clear trace disappeared in the turmoil of the Second World War at the age of about 37. The method is based on the symbolic nature of formalized languages of deductive systems and opens a gate for applications of algebra to logic and, thereby, to Abstract algebraic logic.

Effectively Propositional Logic (EPR), also known as the Bernays-Schoenfinkel class, allows encoding problems that are propositional in nature, but EPR encodings can be exponentially more succinct than purely propositional logic encodings. We recently developed a DPLL-based decision procedure that builds on top of efficient SAT solving techniques to handle the propositional case efficiently while maintaining the succinctness offered by the

Propositional bounded model checking has been applied successfully to verify embedded software but is limited by the increasing propositional formula size and the loss of structure during the translation. These limitations can be reduced by encoding word-level information in theories richer than propositional logic and using SMT solvers for the generated verification conditions. Here, we investigate the application of different SMT solvers to the verification of embedded software written in ANSI-C. We have extended the encodings from previous SMT-based bounded model checkers to provide more accurate support for finite variables, bit-vector operations, arrays, structures, unions and pointers. We have integrated the CVC3, Boolector, and Z3 solvers with the CBMC front-end and evaluated them using both standard software model checking benchmarks and typical embedded applications from telecommunications, control systems and medical devices. The experiments show that our approach can analyze larger problems and substantially reduce the verification time.

We give a logical analysis of quantum measurements as forms of information update. We enumerate some of the “lessons” that Logic can learn from Quantum Mechanics: (1) the importance of logical dynamics; (2) the fact that quantum physics does not require any modification of the classical laws of “static” propositional logic, but only a non-classical dynamics of information flow; (3) the fact that all information-gathering actions have ontic side-effects; (4) the fact that this ontic impact might in its turn affect the flow of information, leading to non-classical epistemic effects and to states of “objectively imperfect information”.

In 1933 Godel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that Godel's provability calculus is nothing but the forgetful projection of LP. This also achieves Godel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which resisted formalization since ...

In this paper, we study the verification of dense time properties by discrete time analysis. Interval Duration Logic, (IDL), is a highly expressive dense time logic for specifying properties of real-time systems. Validity checking of IDL formulae is in general undecidable. A corresponding discrete-time logic QDDC has decidable validity. In this paper, we consider a reduction of IDL validity question to QDDC validity using notions of digitization. A new notion of Strong Closure under Inverse Digitization, SCID, is proposed. For all SCID formulae, the dense and the discrete-time validity coincide. Moreover, SCID has good algebraic properties which can be used to conveniently prove that many IDL formulae are SCID. We also give some approximation techniques to strengthen/weaken formulae to SCID form. For SCID formulae, the validity of dense-time IDL formulae can be checked using the validity checker for discrete-time logic QDDC.

We describe a formal specification and verification in PVS for the general theory of SRT division, and for the hardware design of a specific implementation. The specification demonstrates how attributes of the PVS language (in particular, predicate subtypes) allow the general theory to be developed in a readable manner that is similar to textbook presentations, while the PVS table construct allows direct specification of the implementation's quotient look-up table. Verification of the derivations in the SRT theory and for the data path and look-up table of the implementation are highly automated and performed for arbitrary, but finite precision; in addition, the theory is verified for general radix, while the implementation is specialized to radix 4. The effectiveness of the automation derives from PVS's tight integration of rewriting with decision procedures for equality, linear arithmetic over integers and rationals, and propositional logic. This example demonstrates that the resources of an expressive specification language and of a general-purpose theorem prover are not inimical to highly automated verification in this domain, and can contribute to clarity, generality, and reuse.

The Marlo diagram can represent and solve, at least in theory, logic problems with an unlimited number of terms. In López Aznar (2014), (2018b), it was confirmed that the diagram far exceeded the requirements which, according to Moktefi & Shin (2013), are expected from any diagrammatic system. Diagrams are created when trying to add the ability of expressing implicit possibilities to the Venn diagram. This idea was developed in 2013 when studying Johnson Laird's theory of mental models, in González Labra (1998). Diagrams prioritise the concept of "being associated to" instead of the concept of belonging and inclusion. This has allowed the recombining of their DNA with the connective network models, described in Crespo (2006). In López Aznar (2017), (2018a), the Marlo diagram is translated into tree diagrams that show the logical connec-tives as the simple elimination of different possibilities. This idea is already present in the classic truth tables, but logical trees make it possible to simulate how truth and falsity propagate through the networks of a cognitive system. The diagram has been in development since 2013, in high school classes, with the help of students and other teachers who are experts in mathematics.

This paper is about sentences of form "To be human is to be an animal", "To live is to fight", etc. I call them 'infinitive sentences'. I define an augmented propositional language able to express them and give a matrix-based semantics for it. I also give a tableau proof system, called IL for Infinitive Logic. I prove soundness, completeness and a few basic theorems.

The purpose of this master thesis is to investigate the basic concepts of propositional logic for knowledge representation and formalization of reasoning in Artificial Intelligence. The different properties of logical propositions are discussed. The basic and derived logical connectives are used to establish the compound statements, and the truth tables are constructed to investigate the properties of logical connectives. Such propositions as tautology, satisfiability, contradiction, contingency, logical entailment and logical equivalence are analyzed. Three algebraic normal forms - negation normal form, disjunctive normal form and conjunctive normal form are studied. Horn clauses are implemented. Two forms of valid inferences as modus ponens and modus tollens are considered. Some examples are provided to better understand the main properties of propositional logic. Keywords: Propositional Logic, Logical Connectives, Normal Forms, Horn Clauses, Modus Ponens, Modus Tollens.