Polyvalued vs Standard logic

Under construction Standard logic PC Theorems of the 'ordinary' (two valued, non-modal) Propositional Calculus (PC): (p&q) <-> ~(~pv~q) ; PC1 (pvq) <-> ~(~p&~q) ; PC2 p <-> ~~p ; PC3 (pvq) <-> (qvp) ; PC4 (p&q) <-> (q&p) ; PC5 ((pvq)vr) <-> (pv(qvr)) ; PC6 ((p&q)&r) <-> (p&(q&r)) ; PC7 p <-> (pvp) ; PC8 p <-> (p&p) ; PC9 (p -> q) <-> (~q -> ~p) ; PC10 (p&(qvr)) <-> ((p&r)v(q&r)) ; PC11 (pv(q&r)) <-> ((pvr)&(qvr)) ; PC12 Corresponding theorem of PV1 In polyvalued logic we can prove that all equivalences of PC1-PC12 also are identities that is stronger. As equivalences expresses that that two expressions are true or false together, identities expresses that two expressions always has the same truth value whenever one of them has, i.e. the two expressions expresses identical meaning. (p&q) ~(~pv~q) ; t~& (pvq) ~(~p&~q) ; t~v p ~~p ; t~~ (pvq) (qvp) ; tvqp (p&q) (q&p) ; a3 ((pvq)vr) (pv(qvr)) ; tvv ((p&q)&r) (p&(q&r)) ; t&& p (pvp) ; tvp p (p&p) ; t&p ... about PC10 see below ... (p&(qvr)) ((p&r)v(q&r)) ; t&v& (pv(q&r)) ((pvr)&(qvr)) ; tv&v PC10 is not any identity when the used implication means a conditional implication ('=>') as in everyday logic. The implication of PVL is true conditional. As also the concept of material implication ('->') can be handled in PVL. As material implication is defined by unconditional connectives it expresses only a part of the implication of everyday logic, that delimits '->' to be applicable only to certain contexts. The conditional implication '=>' is applicable to a wide range of everday contexts and it is possible to explicite add more properties for special contexts. PC10 - the law of transposition - will correspond to a whole family of theorem in PVL. These depends on many factors: the choice of implication concept, the context and the certain temporary factual conditions in the context. Transposition - and modus tollens - is not any general rule of PVL. In the conditional version of PC10 with '' (not identical meaning) the two expressions only are equivalent for certain conditions of their domains. Some theorem in PVL that relates/"corresponds" to PC10: (p -> q) (~q -> ~p) ; (p => q) (~q => ~p) ; t=>cp if (p&q) = (~p&~q) then (p => q) = (~q => ~p) ; if (p&~q)=0 and (p&q)>0 and (~p&~q)>0 then (p => q)=1 and (~q => ~p)=1 ; Read more about Modus Tollens and Counter Position. Axiom of PM The best-known axiomatization of PC is Principia Mathematica (PM) by Whitehead and Russell. (pvp) -> p ; A1 q -> (pvq) ; A2 (pvq) -> (qvp) ; A3 (q -> r) -> ((pvq) -> (pvr)) ; A4 Corresponding theorem of PV1 (pvp) p ; ta1pm q => (pvq) ; ta2pm (pvq) (qvp) ; ta3pm (q => r) => ((pvq) => (pvr)) ; ta4pm