A Dempster-Shafer theory inspired logic.

Issues of formalising and interpreting epistemic uncertainty have always played a prominent role in Artificial Intelligence. The Dempster-Shafer (DS) theory of partial beliefs is one of the most-well known formalisms to address the partial knowledge. Similarly to the DS theory, which is a generalisation of the classical probability theory, fuzzy logic provides an alternative reasoning apparatus as compared to Boolean logic. Both theories are featured prominently within the Artificial Intelligence domain, but the unified framework accounting for all the aspects of imprecise knowledge is yet to be developed. Fuzzy logic apparatus is often used for reasoning based on vague information, and the beliefs are often processed with the aid of Boolean logic. The situation clearly calls for the development of a logic formalism targeted specifically for the needs of the theory of beliefs. Several frameworks exist based on interpreting epistemic uncertainty through an appropriately defined modal operator. There is an epistemic problem with this kind of frameworks: while addressing uncertain information, they also allow for non-constructive proofs, and in this sense the number of true statements within these frameworks is too large. In this work, it is argued that an inferential apparatus for the theory of beliefs should follow premises of Brouwer's intuitionism. A logic refuting tertium non daturìs constructed by defining a correspondence between the support functions representing beliefs in the DS theory and semantic models based on intuitionistic Kripke models with weighted nodes. Without addional constraints on the semantic models and without modal operators, the constructed logic is equivalent to the minimal intuitionistic logic. A number of possible constraints is considered resulting in additional axioms and making the proposed logic intermediate. Further analysis of the properties of the created framework shows that the approach preserves the Dempster-Shafer belief assignments and thus expresses modality through the belief assignments of the formulae within the developed logic

A Dempster-Shafer theory inspired logic

Issues of formalising and interpreting epistemic uncertainty have always played a prominent role in Artificial Intelligence. The Dempster-Shafer (DS) theory of partial beliefs is one of the most-well known formalisms to address the partial knowledge. Similarly to the DS theory, which is a generalisation of the classical probability theory, fuzzy logic provides an alternative reasoning apparatus as compared to Boolean logic. Both theories are featured prominently within the Artificial Intelligence domain, but the unified framework accounting for all the aspects of imprecise knowledge is yet to be developed. Fuzzy logic apparatus is often used for reasoning based on vague information, and the beliefs are often processed with the aid of Boolean logic. The situation clearly calls for the development of a logic formalism targeted specifically for the needs of the theory of beliefs. Several frameworks exist based on interpreting epistemic uncertainty through an appropriately defined modal operator. There is an epistemic problem with this kind of frameworks: while addressing uncertain information, they also allow for non-constructive proofs, and in this sense the number of true statements within these frameworks is too large. In this work, it is argued that an inferential apparatus for the theory of beliefs should follow premises of Brouwer's intuitionism. A logic refuting tertium non daturìs constructed by defining a correspondence between the support functions representing beliefs in the DS theory and semantic models based on intuitionistic Kripke models with weighted nodes. Without addional constraints on the semantic models and without modal operators, the constructed logic is equivalent to the minimal intuitionistic logic. A number of possible constraints is considered resulting in additional axioms and making the proposed logic intermediate. Further analysis of the properties of the created framework shows that the approach preserves the Dempster-Shafer belief assignments and thus expresses modality through the belief assignments of the formulae within the developed logic.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

A Survey of Probabilistic Reasoning in Justification Logic

Σε αυτήν τη διπλωματική εργασία μελετούμε την έννοια της επιχειρηματολογίας (justification), αναπαριστάμενη σε ένα λογικό φορμαλισμό. Μελετούμε την επιστημική / δοξαστική αναπαράσταση της justification logic, μίας επέκτασης της κλασικής λογικής (classical logic) με φόρμουλες της μορφής t:F, που μεταφράζονται ως "Το t είναι επιχείρημα που υποδεικνύει την αλήθεια της θέσης (ή την πίστη στη θέση) F.". Παρουσιάζουμε τις βασικές σημασιολογίες της justification logic, συνοδευόμενες από τα αντίστοιχα θεωρήματα ορθότητας και πληρότητας και αναλύουμε πώς εκλαμβάνει η κάθε μία την έννοια της επιχειρηματολογίας. Επίσης, αναλύουμε την έννοια τις επιχειρηματολογίας συνυφασμένη με την έννοια της αβεβαιότητας, παρουσιάζοντας τις θεμελιώδεις probabilistic justification logics. Διατυπώνουμε τις αντίστοιχες σημασιολογίες, μαζί με τα αντίστοιχα θεωρήματα ορθότητας και πληρότητας και εξετάζουμε πώς η κάθε μία λογική αντιλαμβάνεται την αβεβαιότητα στο πλαίσιο της επιχειρηματολογίας. Τέλος, μελετούμε μία νέα σημασιολογία που προτάθηκε και μελετήθηκε εκ των E. Lehmann και T. Studer τα τελευταία τρία χρόνια, ονόματι subset models. Ελέγχουμε πώς τα subset models θα μπορούσαν να συνδυαστούν με τη θεωρία πιθανοτήτων, στην προσπάθεια κατασκευής μίας πιθανοτικής λογικής που διαχωρίζει μεταξύ της αβεβαιότητας υπό το πρίσμα της πειστικότητας του επιχειρήματος, της αβεβαιότητας υπό το πρίσμα της αποδεικτικότητας της θέσης εκ του επιχειρήματος και τις αβεβαιότητας ισχύς της θέσης.In this thesis, we study the notion of justification, interpreted in a logical formalism. Specifically, we study the epistemic/doxastic interpretation of justification logic; i.e., an expansion of classical logic with formulae of the form t:F, which translate as "t is an evidence of the truth of F.". We present the basic semantics for justification logic, along with the corresponding theorems of soundness and completeness, and analyze how each one of them perceives the notion of justification. Moreover, we examine the notion of justification in relation to the notion of uncertainty, by presenting the fundamental probabilistic justification logics. We present the corresponding semantics, accompanied with the corresponding soundness and (sort of) completeness and we investigate how each one of these perceives the uncertainty in the context of justification. Last but not least, we define the subset models, a recent semantics for justification logic proposed and studied by E. Lehmann and T. Studer. We analyze the ontology of justification, as it is expressed in this framework, and we examine how subset models could probably combine with the notion of uncertainty, in a way that distinguishes between the suasiveness of the evidence t, the conclusiveness of evidence t over assertion F, and the certainty of F