Abstract
An alternative title of this chapter could be ‘On Vetoer in Language, Logic and Mathematics’ because this text can be considered as an example comparing definitions of a veto player in a natural language, i.e. in real world, in first-order logic and, generally, in mathematics. It is intuitively quite simple to understand how the vetoer can be defined by means of a natural language as a person with absolute negative power, i.e. a person who can impede any social initiative. Here we show how in first-order logic the existence of a vetoer is acknowledged by a first-order formula which is inconsistent with some well-known axioms of traditional Social Choice Theory. In addition, we present how the exact power of a vetoer can be mathematically calculated and transparently defined by means of a weighted voting system.
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References
Anderson, A. R., Belnap, Jr., N. D. (1975). Entailment: The Logic of Relevance and Necessity. Vol. I, Princeton University Press, Princeton.
Arrow, K. (1963). Social Choice and Individual Values. John Wiley, New York.
Arrow, K. J., Sen, A. (2002). Handbook of Social Choice and Welfare. Vol 1, Gulf Professional Publishing, Houston.
Arrow, K. J., Sen, A., Suzumura, K. (2010). Handbook of Social Choice and Welfare. Vol 2, Elsevier, Amsterdam.
Artemidiadis, N. (2004). History of Mathematics. American Mathematical Society, Providence.
Bass, S. M., Kwakernaak, H. (1977). Rating and Ranking of Multiple–Aspect Alternatives Using Fuzzy Sets. Automatica, 18, 47–58.
Blau, J. H., Deb, R. (1977). Social Decision Functions and the Veto. Econometrica, 45 (4), 871–879.
Boričić, B. (1988a). On certain normalizable natural deduction formulations of some propositional intermediate logics. Notre Dame Journal of Formal Logic 29, 563–568.
Boričić, B. (1988b). A note on sequent calculi intermediate between LJ and LK. Studia Logica, 47, 151–157.
Boričić, B., Konjikušić, S. (2004a). Preference logic on rough and fuzzy sets. Economic Annals, XLIV 160, 131–146.
Boričić, B. (2004b). Nobel prize in economics—Amartya K. Sen. Economic Annals, XLIV 160, 215–220.
Boričić, B. (2007). Logical and historical determination of impossibility theorems by Arrow and Sen. Economic Annals, LI 172, 7–20.
Boričić, B. (2009). Dictatorship, liberalism and the Pareto rule: possible and impossible. Economic Annals, LIV 181, 45–54.
Boričić, B. (2011). Logic and Proof. Faculty of Economics, University of Belgrade, Belgrade, 2011. (Zbl 1214.03043; MR 2011k:03001)
Boričić, B. (2014a). Multiple von Wright’s preference logic and social choice theory. The Bulletin of Symbolic Logic, 20, 224–225.
Boričić, B. (2014b). Impossibility theorems in multiple von Wright’s preference logic. Economic Annals, LIX 201, 69–84.
Boričić, B. (2018). Model, proving and refuting, in Quantitative Models in Economics (Ed. J. Kočović et all), Faculty of Economics, University of Belgrade, Belgrade, 3–19.
Boričić, B. (2023). A note on dictatorship, liberalism and the Pareto rule, Economic Annals (to appear)
Boyer, C. B. (1968). A History of Mathematics. J. Wiley and Sons, New York.
Bufardi, A. (1999). On the fuzzification of the classical definition of preference structure. Fuzzy Sets and Systems, 104, 323–332.
Chichilnisky, G. (1982). The topological equivalence of the Pareto condition and the existence of a dictator. Journal of Mathematical Economics, 9, 223–233.
da Costa, N. C. A., Krause, D., Bueno, O. (2007). Paraconsistent logics and paraconsistency. Handbook of the Philosophy of Science, Elsevier, 791–911.
De Baets, B., Van de Walle, B., Kerre, E. E. (1998a). A plea for the use of Lukasiewicz triplets in the definition of fuzzy preference structures, Part I: General argumentation. Fuzzy Sets and Systems, 97, 1998, 349–359.
De Baets, B., Van de Walle, B., Kerre, E. E. (1998b). Part II: The identity case. Fuzzy Sets and Systems, 99, 303–310.
Debreu, G. (1959). The Theory of Value: an Axiomatic Analysis of Economic Equilibrium. Wiley, New York.
de Swart, H. (2018). Philosophical and Mathematical Logic. Springer Undergraduate Texts in Philosophy, Springer, Berlin.
Gabbay, D. M. (1981). Semantical Investigations in Heyting’s Intuitionistic Logic. D. Reidel Publ. Comp., Dordrecht.
Fishburn, P. C. (1973). The Theory of Social Choice. Princeton University Press, Princeton.
Hofstadter, D. R. (1979). Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books, New York.
Hosoi, T. (1967). On intermediate logics I. J. Fac. Sci. Univ. Tokyo Sect. I, 14, 293–312.
Hosoi, T. (1969). On intermediate logics II. J. Fac. Sci. Univ. Tokyo Sect. I, 16, 1–12.
Ilić, M. (2016). An alternative Gentzenization of \(RW_+^o\). Mathematical Logic Quarterly, 62(6), 465–480.
Ilić, M. (2017). A natural deduction and its corresponding sequent calculus for positive contractionless relevant logics. Reports on Mathematical logic, 52, 93–124.
Ilić, M. (2021a). A note on an alternative Gentzenization of \(R_+^o\). Mathematical Logic Quarterly, 67(2), 186–192.
Ilić, M. (2021b). A cut–elimination proof in positive relevant logic with necessity, Studia Logica 109(3), 2021, pp. 607–638.
Ilić, M., Boričić, B. (2014). A cut–free sequent calculus for relevant logic \(RW^\star \). Logic Journal of IGPL, 22(4), 673–695.
Ilić, M., Boričić, B. (2021). A note on the system GRW with the intensional contraction rule. Logic Journal of IGPL, 29(3), 333–339.
Kang, M. S. (2010). Voting as veto. Michigan Law Review, 108, 1221–1281.
Kelly, J. S. (1978). Arrow Impossibility Theorems. Academic Press, London.
Li, H. X., Yen, V. C. (1995). Fuzzy Sets and Fuzzy Decision-Making. CRC Press, New York.
Mas–Colell, A., Sonnenschein, H. (1972). General possibility theorems for group decisions. The Review of Economic Studies, 39, 185–192.
Moulin, H. (1981). The Proportional Veto Principle. The Review of Economic Studies, 48, 407–416.
Moulin, H. (1982). Voting with Proportional Veto Power. Econometrica, 50, 145–162.
Murakami, Y. (1968). Logic and Social Choice. Routledge, London.
Routhley, R. (1979). Repairing proofs of Arrow’s general impossibility theorem and enlarging the scope of the theorem. Notre Dame Journal of Formal Logic, 20, 879–890.
Schofield, N. J. (1985). Social Choice and Democracy. Springer, Berlin.
Sen, A. K. (1969). Quasi-Transitivity, Rational Choice and Collective Decisions. The Review of Economic Studies, 36 (3), 381–393.
Sen, A. K. (1970a). The impossibility of a Paretian liberal. Journal of Political Economy, 78 (1), 152–157.
Sen, A. K. (1970b). Collective Choice and Social Welfare. Holden–Day, San Francisco (The fourth edition: Elsevier, Amsterdam, 1995).
Skala, H. J. (1975). Non-Archimedean Utility Theory. Springer, Berlin.
Srećković, M. (2017). Measuring the veto power. Ekonomske ideje i praksa, 25, 89–96.
Takeuti, G. (1975). Proof Theory. North–Holland, Amsterdam (Second edition: Dover Publications, 2013).
Taylor, A. D. (1995). Mathematics and Politics—Strategy, Voting, Power and Proof. Springer, Berlin.
van Dalen, D. (1980). Logic and Structure. Springer, Berlin (Fifth edition 2013).
Winter, E. (1996). Voting and Vetoing. The American Political Science Review, 90, 813–823.
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Appendix: Glossary of Notation and Definitions
Appendix: Glossary of Notation and Definitions
Axiomatic System
The context that enables accurate argumentation based on given statements (axioms) and rules. After Euclid, the axiomatic approach is used in many sciences to avoid doubts and ambiguities. Basic formal properties of a deduction relation are induced by nature of logic on which axiomatic system is build up.
Classical Logic
An axiomatic system defining logical rules of reasoning based on the Aristotle’s two-valued principle. The law of excluded middle, that each statement or its negation is provable, is characteristic for classical logic.
Deduction Relation
Relationship between a set of hypotheses \(\varGamma \) and its conclusion A, denoted by \(\varGamma \vdash A\), with meaning that A can be derived from \(\varGamma \) in a corresponding axiomatic system.
Dictatorship
A form of social organization in which one person possesses absolute power without effective limits. Particularly, in social choice theory, dictator is defined as an individual who can impose her/his personal preferences to the whole society.
Impossibility
In this book, this is a synonym for inconsistency.
Inconsistent Statements
Two or more statements are inconsistent if they always enable inferring untruth. In this book, the fact that X and Y are inconsistent is denoted as \(X,Y\vdash \), by means of a deduction relation.
Logical Connectives
The usual starting point of a logical formalism are propositional connectives, such as: and, or, not, if ... then, if and only if, and quantifiers: for all and for some (or there exists). The list of basic logical symbols is the following one: \(\wedge \)—conjunction (for and), \(\vee \)—disjunction (for or), \(\neg \)—negation (for not), \(\to \)—implication (for if ... then), \(\leftrightarrow \)—equivalence (for if and only if), \(\forall \)—universal quantifier (for for all), and \(\exists \)—existential quantifier (for there exists).
Non-classical Logic
An axiomatic system defining a logic denying the Aristotle’s two-valued principle. The most popular non-classical logics are classified as relevant, intuitionistic, linear, substructural or paraconsistent.
Pareto Rule
In social choice theory, Pareto rule means that if each member of a society prefers the same state, then the society prefers this state.
Preference Logic
An axiomatic system, based on rationality choice axioms, which enables formal reasoning about individual or social preferences.
Preference Relation
A binary relation expressing relationship between two alternatives. For instance, ’xPy’ expresses that ‘the alternative x is preferred to the alternative y’.
Rationality Choice Axioms
Asymmetric, i.e., \(\forall x,y(xPy\to \neg yPx)\), and transitive, i.e., \(\forall x,y,z\in X(xPy\wedge yPz\to xPz)\), binary relation P is called the strict preference relation, and we say that it satisfies rationality choice axioms. Each strict preference relation P generates an indifference relation I, as follows: xIy iff \(\neg xPy\wedge \neg yPx\), and a weak preference relation R, by definition: xRy iff \(xPy\vee xIy\).
Social Choice Theory
Also known as theory of social decision-making or theory of public choice, is treated in this book as a mathematically founded study of collective decision-making procedures. It is focused on the problem of how to define a social state derived from the personal states, and when it is possible.
Vetoer
A person participating in the process of decision-making with veto power, i.e., a voter having the possibility to stop any action or block any decision-making process or its result.
Voting System
Set of rules and procedures defining process of decision-making by means of voting. This is a part of social choice theory.
Weighted Voting System
A more general form of a voting system which enables expressing various powers of voters including the possibility that some voters possess more than one vote.
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Boričić, B., Srećković, M. (2023). Vetoing: Social, Logical and Mathematical Aspects. In: Hounkonnou, M.N., Martinovic, D., Mitrović, M., Pattison, P. (eds) Mathematics for Social Sciences and Arts. Mathematics in Mind. Springer, Cham. https://doi.org/10.1007/978-3-031-37792-1_6
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