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Stephen Read - One of the best experts on this subject based on the ideXlab platform.

  • General-Elimination Harmony and the Meaning of the Logical Constants
    Journal of Philosophical Logic, 2010
    Co-Authors: Stephen Read
    Abstract:

    Inferentialism claims that expressions are meaningful by virtue of Rules governing their use. In particular, logical expressions are autonomous if given meaning by their introduction-Rules, Rules specifying the grounds for assertion of propositions containing them. If the Elimination-Rules do no more, and no less, than is justified by the introduction-Rules, the Rules satisfy what Prawitz, following Lorenzen, called an inversion principle. This connection between Rules leads to a general form of Elimination-Rule, and when the Rules have this form, they may be said to exhibit "general-Elimination" harmony. Ge-harmony ensures that the meaning of a logical expression is clearly visible in its I-Rule, and that the I- and E-Rules are coherent, in encapsulating the same meaning. However, it does not ensure that the resulting logical system is normalizable, nor that it satisfies the conservative extension property, nor that it is consistent. Thus harmony should not be identified with any of these notions.

  • Harmony and Autonomy in Classical Logic
    Journal of Philosophical Logic, 2000
    Co-Authors: Stephen Read
    Abstract:

    Michael Dummett and Dag Prawitz have argued that a constructivist theory of meaning depends on explicating the meaning of logical constants in terms of the theory of valid inference, imposing a constraint of harmony on acceptable connectives. They argue further that classical logic, in particular, classical negation, breaks these constraints, so that classical negation, if a cogent notion at all, has a meaning going beyond what can be exhibited in its inferential use. I argue that Dummett gives a mistaken elaboration of the notion of harmony, an idea stemming from a remark of Gerhard Gentzen"s. The introduction-Rules are autonomous if they are taken fully to specify the meaning of the logical constants, and the Rules are harmonious if the Elimination-Rule draws its conclusion from just the grounds stated in the introduction-Rule. The key to harmony in classical logic then lies in strengthening the theory of the conditional so that the positive logic contains the full classical theory of the conditional. This is achieved by allowing parametric formulae in the natural deduction proofs, a form of multiple-conclusion logic.

C Riba - One of the best experts on this subject based on the ideXlab platform.

  • strong normalization as safe interaction
    Logic in Computer Science, 2007
    Co-Authors: C Riba
    Abstract:

    When enriching the lambda-calculus with rewriting, union types may be needed to type all strongly normalizing terms. However, with rewriting, the Elimination Rule (orE) of union types may also allow to type non normalizing terms (in which case we say that (orE) is unsafe). This occurs in particular with non-determinism, but also with some confluent systems. It appears that studying the safety of (orE) amounts to the characterization, in a term, of safe interactions between some of its subterms. In this paper, we study the safety of (orE) for an extension of the lambda-calculus with simple rewrite Rules. We prove that the union and intersection type discipline without (orE) is complete w.r.t. strong normalization. This allows to show that (orE) is safe if and only if an interpretation of types based on biorthogonals is sound for it. We also discuss two sufficient conditions for the safety of (orE), and study an alternative biorthogonality relation, based on the observation of the least reducibility candidate.

  • LICS - Strong Normalization as Safe Interaction
    22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007), 2007
    Co-Authors: C Riba
    Abstract:

    When enriching the lambda-calculus with rewriting, union types may be needed to type all strongly normalizing terms. However, with rewriting, the Elimination Rule (orE) of union types may also allow to type non normalizing terms (in which case we say that (orE) is unsafe). This occurs in particular with non-determinism, but also with some confluent systems. It appears that studying the safety of (orE) amounts to the characterization, in a term, of safe interactions between some of its subterms. In this paper, we study the safety of (orE) for an extension of the lambda-calculus with simple rewrite Rules. We prove that the union and intersection type discipline without (orE) is complete w.r.t. strong normalization. This allows to show that (orE) is safe if and only if an interpretation of types based on biorthogonals is sound for it. We also discuss two sufficient conditions for the safety of (orE), and study an alternative biorthogonality relation, based on the observation of the least reducibility candidate.

Hannu Nurmi - One of the best experts on this subject based on the ideXlab platform.

  • The No-Show Paradox Under a Restricted Domain
    Homo Oeconomicus, 2019
    Co-Authors: Dan S Felsenthal, Hannu Nurmi
    Abstract:

    The no-show paradox occurs whenever a group of identically-minded voters is better off abstaining than by voting according to its preferences. Moulin’s (J Econ Theory 45:53–64, 1988 ) result states that if one wants to exclude the possibility of the no-show paradox, one has to resort to procedures that do not necessarily elect the Condorcet winner when one exists. This paper examines ten Condorcet-consistent and six Condorcet-non-consistent procedures in a restricted domain, viz., one where there exists a Condorcet winner who is elected in the original profile and the profile is subsequently modified by removing a group of voters with identical preferences. The question asked is whether the no-show paradox can occur in these settings. It is found that only two of the ten Condorcet-consistent procedures investigated (Maximin and Schwartz’s procedure) are not vulnerable to the no-show paradox, whereas only two of the six non-Condorcet-consistent ranked procedures investigated (Coombs’ and the Negative Plurality Elimination Rule procedures) are vulnerable to this paradox in the restricted domain. In other words, for a no-show paradox to occur when using Condorcet-consistent procedures it is not, in general, necessary that a top Condorcet cycle exists in the original profile, while for this paradox to occur when using (ranked) non-Condorcet-consistent procedures it is, almost always, necessary that the original profile has a top cycle.

  • the in vulnerability of 20 voting procedures to the no show paradox in a restricted domain
    2019
    Co-Authors: Dan S Felsenthal, Hannu Nurmi
    Abstract:

    The No-Show paradox occurs whenever a group of identically-minded voters is better off abstaining than by voting according to its preferences. Moulin’s (Journal of Economic Theory 45:53–64, 1988) result states that if one wants to exclude the possibility of the No-Show paradox, one has to resort to procedures that do not necessarily elect the Condorcet winner when one exists. This paper examines 10 Condorcet-consistent and 10 Condorcet-non-consistent procedures in a restricted domain, viz., one where there exists a Condorcet winner who is elected in the original profile and the profile is subsequently modified by removing a group of voters with identical preferences. The question asked is whether the No-Show paradox can occur in these settings. It is found that only 2 of the 10 Condorcet-consistent procedures investigated (Minimax and Schwartz’s procedure) are invulnerable to the No-Show paradox, whereas only 3 of the 10 non-Condorcet-consistent ranked procedures investigated (Coombs’s, the Negative Plurality Elimination Rule, and the Majority Judgment procedures) are vulnerable to this paradox in the restricted domain. In other words, for a No-Show paradox to occur when using Condorcet-consistent procedures it is not, in general, necessary that a top Condorcet cycle exists in the original profile, while for this paradox to occur when using (ranked) non-Condorcet-consistent procedures it is, almost always, necessary that the original profile has a top cycle.

  • Monotonicity Violations by Borda’s Elimination and Nanson’s Rules: A Comparison
    Group Decision and Negotiation, 2018
    Co-Authors: Dan S Felsenthal, Hannu Nurmi
    Abstract:

    Abstract This paper compares the vulnerability of Borda Elimination Rule (BER) and of Nanson Elimination Rule (NER) to monotonicity paradoxes under both fixed and variable electorates. It is shown that while NER is totally immune and BER is vulnerable to monotonicity failure in 3-candidate elections, neither of these two Rules dominates the other in n-candidate elections (n > 3) when no Condorcet Winner exists. When the number of competing alternatives is larger than three and no Condorcet Winner exists, we find profiles where NER violates monotonicity while BER does not, profiles where BER violates monotonicity while NER does not, as well as profiles where both NER and BER violate monotonicity. These findings extend to both fixed and variable electorates, as well as to situations where the initial winners under both Rules are the same, as well as to situations where the initial winners under both Rules are different. So, which of the two Rules should be preferred in terms of monotonicity in n-candidate elections (n > 3) where no Condorcet Winner exists, depends on the kind of profiles one can expect to encounter in practice most often. Nevertheless, in view of the results of 3-candidate elections under other scoring Elimination Rules, we conjecture that inasmuch as BER and NER exhibit monotonicity failures, it is more likely to occur in closely contested elections.

Daniele Francesco Santamaria - One of the best experts on this subject based on the ideXlab platform.

  • an optimized ke tableau based system for reasoning in the description logic shdlssx
    2018
    Co-Authors: Domenico Cantone, Marianna Nicolosi Asmundo, Daniele Francesco Santamaria
    Abstract:

    We present a \ke-based procedure for the main TBox and ABox reasoning tasks for the description logic $\dlssx$, in short $\shdlssx$. The logic $\shdlssx$, representable in the decidable multi-sorted quantified set-theoretic fragment $\flqsr$, combines the high scalability and efficiency of Rule languages such as the Semantic Web Rule Language (SWRL) with the expressivity of description logics. %In fact it supports, among other features, Boolean operations on concepts and roles, role constructs such as the product of concepts and role chains on the left hand side of inclusion axioms, and role properties such as transitivity, symmetry, reflexivity, and irreflexivity. Our algorithm is based on a variant of the \ke\space system for sets of universally quantified clauses, where the KE-Elimination Rule is generalized in such a way as to incorporate the $\gamma$-Rule. The novel system, called \keg, turns out to be an improvement of the system introduced in \cite{RR2017} and of standard First-Order \ke x \cite{dagostino94}. Suitable benchmark test sets executed on C++ implementations of the three systems show that in several cases the performances of the \keg-based reasoner are up to about 400\% better than the ones of the other systems.

  • an optimized ke tableau based system for reasoning in the description logic shdlssx extended version
    arXiv: Logic in Computer Science, 2018
    Co-Authors: Domenico Cantone, Marianna Nicolosiasmundo, Daniele Francesco Santamaria
    Abstract:

    We present a \ke-based procedure for the main TBox and ABox reasoning tasks for the description logic $\dlssx$, in short $\shdlssx$. The logic $\shdlssx$, representable in the decidable multi-sorted quantified set-theoretic fragment $\flqsr$, combines the high scalability and efficiency of Rule languages such as the Semantic Web Rule Language (SWRL) with the expressivity of description logics. %In fact it supports, among other features, Boolean operations on concepts and roles, role constructs such as the product of concepts and role chains on the left hand side of inclusion axioms, and role properties such as transitivity, symmetry, reflexivity, and irreflexivity. Our algorithm is based on a variant of the \ke\space system for sets of universally quantified clauses, where the KE-Elimination Rule is generalized in such a way as to incorporate the $\gamma$-Rule. The novel system, called \keg, turns out to be an improvement of the system introduced in \cite{RR2017} and of standard first-order \ke x \cite{dagostino94}. Suitable benchmark test sets executed on C++ implementations of the three mentioned systems show that the performances of the \keg-based reasoner are often up to about 400\% better than the ones of the other two systems. This a first step towards the construction of efficient reasoners for expressive OWL ontologies based on fragments of computable set-theory.

Hatem Hadda - One of the best experts on this subject based on the ideXlab platform.