Mattila, Jorma K.

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KybernetikaJorma K. MattilaModified modus ponens and modal logicKybernetika, Vol. 28 (1992), No. Suppl, 28--32Persistent URL: http://dml.cz/dmlcz/124190Terms of use:© Institute of Information Theory and Automation AS CR, 1992Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This paper has been digitized, optimized for electronic delivery and stamped withdigital signature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czS U P P L E M E N T TO K Y B E R N E T I K A VOLUME 28(1992), P A G E S 2 8 - 3 2 MODIFIED MODUS PONENS AND MODAL LOGIC JORMA K. MATTILA This paper discusses an inference rule called by modified modus ponens, which is used in the logical system LPC + Ch which is first order (or lower) predicate calculus equipped with additional axiomati-zation of modifier operators. This basic forms a system like generalized modal system with several pairs of modal operators. The main properties of the system LPC + Ch necessary for introducing this topic are considered. It suffices well a propositional system PC + Ch for these purposes. The modal version of modified modus ponens is proved to hold in standard modal systems. 1. I N T R O D U C T I O N First we give a short description of the Ch-extension of classical propositional logic PC . As an alphabet of our C/i-language we adopt the alphabet of classical propositional calculus choosing the connectives -i standing for negation, —• standing for implication, as primitives, and connectives V standing for disjunction, A standing for conjunction, and «-> s tanding for equivalence are derived from those in the known way. We adopt the set of proposition letters PR = {p,; | i = 0 , 1 , . . . , n,...} straight from PC. So, for PC + Ch we get the logical alphabet from PC. We further need some added characters for formalizing a set of characteristic operators. The symbolic alphabet consists of a set of modifier symbols 0 = { $ , T\,T~2, • • •} where the operators T\, T2, • • • are substantiating (abbreviated by 5 ~ T) and 3 is an identity operator. We can denote these modifier operators by metavariables 7i, T, V,.. • (with or without numerical subscripts). For any modifier T € 0, we can form its dual modifier T* — ~^T^, and the set of duals we symbolize by 0*. For the identity operator 9 it holds 9* = 5 . Modifiers belonging to 0* are called weakening operators (abbreviated by T* ~ S ) . The formation of well-formed formulae (wffs) is similar to tha t of PC. We give the definition of the set W of wffs of PC + Ch as follows: D e f i n i t i o n 1 . 1 . W is the set of wffs of PC + Ch if (1°) the set W of wffs of PC is a subset of W ; (2°) i f o e W and TeO then T(a) € W ; (3°) if a <= W then - a g W ; (4°) if a, 0 <= W then (a -» 0) € W ; Modified Modus Ponens and Modal Logic 2 9 (5°) All the wffs are generated by the steps (1°) - (4°). The formal semantics of PC + Ch is given in Mattila [7], and we do not consider it here. Instead, we go straight to the axiomatization. In addition to the axiomatization of PC we need in our proof-theoretical system a characteristic axiom schemata governing the logical properties of the modifier operators. Our axiomatization for our system PC + Ch are as follows: Axiom schemata of Ch. (i) All the tautologies of PC are axioms. (ii) If n, T <E 0 U 0*, and W ~ T (n is at most as strong as T), then for all a € W T(a)-*n(a) . (AxCh) is an axiom, (iii) For all wffs a € W and for the identity operator S € 0 3(a) •-> a (Axld) is an axiom. We also adopt the following inference rules: Modus ponens: a-+0, a\-f), (MP) Modified modus ponens: a -> /t, T(a)\- T(f3) (MMP) where T £ 0 is an arbitrary operator. Rule of Substantiation. For wffs o S W and all substantiating operators T € 0 f- a => h T(a) (RS) So, a Ch-system is any non-empty set X, such that the tautologies of PC, (AxCh) and (Axld) are included in X, and X is closed under (MP), (MMP), and (RS). In the sequel we need the following properties of PC + Ch, which are proved in Mattila [6], [7]. For any P £ W, \-T(P)-+P (9f ~ /•) (1.1) h P ^ H ( P ) (W ~ 3?) (1.2) Then we consider situations in which operators are associated with connected wffs. We have the following result: If T is a substantiating operator and W'= ->T~i, and P, Q € W, then T(P-> Q) \- T(P)->T(Q). (1.3) 30 J.K. MATTILA 2. SOM E SYNTACTICAL SIMILARITIES B E T W E E N Ch- AND MODAL SYS-T E M S We consider first the system T giving its axioms and other rules and definitions we need (for details, see [1]). One useful way for axiomatizing modal systems is to built the system over PC, as usually is done. Thus the system T consists of the axioms for propositional logic, and of the axioms basing on necessity, D P - > P (TA1) a(P^Q) -> (ap - DQ) (TA2) T contains also inference rule MP and so-called Rule of Necessitation: | - p = 4 . j - ap (N) T h e modal concept of possibility, is defined by the condition O P = d f - D - . P (Def. O) for any wff P, i .e . possibility is the dual of necessity. If we interpret the operators T and H to be the modal operators D standing for necessity, and O standing for possibility of aletic modal logic, respectively, and the system has only this dual pair of operators, we. get a modal system which contains the modal system T . In the modal interpretation of modifier operators the identity operator 3 corresponds to modal operator 'actuality' (abbreviated often by O ) - The formal evidence for tha t is e .g. the equation Q(P) = -> O (~'I>) f ° r aH P € W . T h e modal counterparts of (AxCh) in the system T is L- ap -» OP (AxCh") from which it follows the reflexivity laws by means of the actuality operator . \- D P -4 P (2.1) |_ O P - O P ' (2.2) Thus in this modal interpretation (TA1) is equivalent to (1.1) operator D being sub-s tant ia t ing. (TA2) follows directly from the modal counterpart of (1.3). Because M P belongs to the both systems and in PC + Ch substantiat ing operators have the same formal property than D in T , namely (N), we have showed tha t T belongs to the modal version PC + Ch. Clearly (2.1) implies both (2.2), and (AxCh") , and also (2.2) implies both (2.1) and (AxCh") . Modified Modus Ponens and Modal Logic 31 3. MODAL VERSION OF MODIFIED MODUS PONENS There are also other standard modal systems like S I , S 2, S3, S 4, and S5, which are the most usual ones. The subsystem relations between these are S 1 C S 2 C S 3 C S 4 c S 5 a n d S l c S 2 c T c S 4 c S 5 , Thus S 3 is in a way alternative to the system T. It suffices to restrict our considerations to S 1. The rule of Necessitation does not hold without restrictions in standard systems S I , S2 and S 3 (see [1], p. 225, 230, and 235). The restricted form is h p c o => hsi D a (3.1) For any wff a of PC. Axiom (TA2) of T is a theorem in S 1. It is proved in Hughes and Cresswell [1] p. 225 and numbered by TS1.21. We need this result below. We now prove the following Proposit ion 3.1. The modal version of the rule MMP holds in S 1, i.e. h P - Q, h M(P) =* h M(Q) (3.2) where M is a modal operator of S 1. Proof . Suppose \- P ^> Q, and h M(P) hold. It is remarkable that especially hpc P —* Q- For M = a vie have the deduction. 1. P -*Q given 2. op given 3. D ( P - Q ) appl. (3.1) to 1 4. D ( P - + Q ) - (DP-+DQ) TSl.21 inH.&C. 5. OP -> DQ MP,3,4 6. °Q MP,2,5 M = O we have the deduction 1. P-+Q given 2. OP given 3. -OQ premise 4. D-Q OQ = - D - Q 5. ( P - » Q ) -> (^Q^-P) LAЗ 6. - Q - - P MP,1,5 7. D(-Q - - P ) appl. (3.1) to 6 8. D ( - Q ^ - P ) - (D-Q-> D-P) TS2.21 inH.&C. 9. D-G -+ D-.F MP,7,8 10. D - P MP,4,9 11. - O P OP = - D - P 12. - O P л OP A,2,ll 13. OQ PC,3,12. 32 J.K. MATTILA D From this result and from the subsystem relations it follows directly P r o p o s i t i o n 3 . 2 . The modal version of the rule MMP holds in T , i .e . h P - Q, h M(P) => |- M(Q) (3.3) Where M is a modal operator of T . This can be also proved very easily without the knowledge of Proposition 3.1. Because T is also a subsystem of Brouwerian system, the modal version of M M P holds also in it. R E F E R E N C E S  [1] G.E. Hughers and M.J. Cress well: An Introduction to Modal Logic. Methuen and Co., 1985. [2] S.A. Kripke: Semantical considerations on modal logic. Acta Phil. Fenn. XVI, 1963. [3] C. Lakoff: Hedges: A study in meaning criteria and the logic of fuzzy concepts. J. Philos. Logic 2 1973. [4] E.J. Lemmon: An Introduction to Modal Logic. American Philosophical Quarterly, Monograph No. 11 ( K. Segerberg, ed.), Oxford 1977. [5] J.K. Mattila: Proof-theoretical aspects of fuzzy logic. In: Recent Developments in the Theory and Applications of Fuzzy Sets (Bandler and Kandel, eds.), Proceedings of North American Fuzzy Information Processing Society NAFIPS'86 Conference, New Orleans, Lousiana, June 2 - 4 , 1986, pp. 386 - 398. [6] J.K. Mattila: Calculus of Modifier Operators in Fuzzy Logic. Publications of the Institute for Applied Mathematics No. 14, University of Turku, Turku, Finland 1989. [7] J. K. Mattila: The construction of a propositional modifier logic. Lappeenranta University of Tech-nology, Department of Information Technology, Research Report No. 21, Lappeenranta, Finland 1990. Acting professor, Docent Jorma K. Matiila, PhD, Department of Information Technology, Lappeen-ranla University of Technology, P. O. Box 20, SF-53851 Lappeenranta. Finland. 

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Modified modus ponens and modal logic

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