An Extension of Propositional Logic by Arthur Shevenyonov 1 Propositional Logic An Extension Nary Anthropic
The proposed extension of propositional logic appears to bridge gaps across areas as diverse as, inductive strength and deductive validity, morphisms and Russellian attempts at formal axiomatization, anthropic alternates, and generalized games ultimately pointing to gradiency and orduality rationales. Russell's Dream Come True (Almost) The conventional logic of propositions may not do any better job than set theory does, as they both stumble into the issue of whether sets can inherently be related (without invoking explicit groupor functional theoretic mappings) Along similar lines, the Euler VennBallentine diagrams do not show exactly how the sets in question are related beyond naive implication or conjunction and disjunction. The logic of predicates might improve upon the exposition while eking out on the semantic as well as pragmatic detail; but the very extent of interim detail, or Azimuthality (Shevenyonov, 2016a) might prove prohibitive when capturing the least informative utterances, let alone while connoting the finer shades of meaning. For that matter, predicates appear to emulate the nature of functions, in which light their ability to substantiate it would have to be taken with a grain of salt. Higher order predicates or logic (HOL) could further generalize so far as to capture functional analytic cognates (e.g. 'functions of lines, functional compositions as an extension of matrix product, and path integrals) but that would be just an unnecessarily involved and still 'cardinal' proxy for Orduality or relational premises (Shevenyonov, 2016c) What could be the more parsimonious way of extending the simplest possible, propositional logic while providing bridges between deductive validity, inductive strength, functionality, and outright remote areas such as generalized game theory? In what ways could the latter equivalence be deployed in rationalizing the more arcane views of the 'anthropic' principle and the 'multiverse' approach to endogenizing world constants or parameters underpinning the key regularities? 1 This tentative and incomplete effort is a dedication to Zoya Kosmodemyanskaya, Sandino, Patrice Lumumba, Che, and mayhap to Fidel alongside Leonardo Peltier and Joan of Arc. No deference to leftism per se, but rather reverence for the many hues of sanctity as a response to the perfidious and cherry picking ways of mediocrity. An Extension of Propositional Logic by Arthur Shevenyonov 2 To begin with, consider a conventional implication of the form, A B, which equivalently can be rendered as In predicate terms, this could build upon forms like x A(x) B(x) iff x (x) (x) AND x (x) (x) OR x (x) B(x) Thus far, there appears little that cannot be demonstrated with the magic of conventional diagrams even though one would have difficulty to either rigorously distinguish between the universal specification (existence) operators or conceptualize the conjugate or dual predicate stretching forms. In other words, induction appears near meaningless in general, unless extra specifications and modes are phased in. Needless to say, one would have liked for any such modifier operators to strike an optimum balance of completeness and simplicity (which invariably refers to Gradiency and Orduality) A good starting point might be to seek generalizations of the form, 1.1 (pA qB) ?B ?A) This would simultaneously act to expand upon the previous while accounting for any interim operator or relational combinations. In the meantime, it could be instrumental to define a general existence operator as follows: px p(x) lim p 0 px ~NOT~ lim p 1 px (x)~x lim p q 1 pqx (pA qB) (A B) lim p q 0 pqx (pA qB) (B A) Incidentally, this is meant to be a prior illustration, so further extensions along the (1.1) will not hinge upon its validity or refined conventions. Moreover, the p and q operators could be used as a variety of predicates (to represent an existential mode as opposed to an existential measure, probabilistic or otherwise qualitative) In a sense, these very notions could be endogenized or 'solved for' as structural conventions, with operator 'values' referring to a structural status in a functional or relational space. For now, the convention will be so simple as to overlook any particular (while capturing all) narrowings along the aforementioned lines. [Somewhat mnemonically, or as a matter of meta formalization or automorphism, that could imply: pp pp( p) Again, this has little to do with the way the meta induction proceeds from now on. An Extension of Propositional Logic by Arthur Shevenyonov 3 To begin with, one may refer to the standard conventions for set operations (Appendix) to arrive at the following: 1.2 (pA qB) (1 pA)(qB) (pA)(qB) 1.3 ?B ?A) (1 ?B)?A) ?B)?A) Based on the (1.1) identity or definition, it obtains that: 1.4 (pA)(qB) ?B)?A) 1.5 (1 pA)(qB) (1 ?B)?A) The dual operators can be solved for based on the knowledge of the direct ones (p, q) subject to the particular nature of the objects making up the basis(A, B) Rather than providing closed form solutions of the A A(B) sort (which might be less than meaningful outside intended functional maps) based on the weighted average type reductions as per the special case of p q, ?A pB p AB(p it should be enough as a starting point to further restrict this to the A B instance, to arrive at B B(p One should be able to appreciate that even the very special cases could prove fairly involved while acting to generalize the convention of B B (as in the Appendix) Even more importantly, in the event the A B restriction is applied alone, it follows that 1.6 p, q Additional special cases may have to be studied prior to reverting to the crossdisciplinary implications. For instance, the syllogisms of the kind X (valid for some X) versus X (invalid for all X) could be studied in line with restricting either p or q to a zero value (irrespective of how a minimal existential status is defined for practical purposes) In the former case, it follows that, qB (1 ?B)?A) In the latter, either 0 or A~ or ?B 1. In other words, even the degenerate cases can be formalized rigorously as well as consistently. By contrast, should either one be set to unity (which captures induction without collapsing to the naive A B instance) it would obtain that, either B(pA) ?A( p B) or, (qB) ?A)?B) An Extension of Propositional Logic by Arthur Shevenyonov 4 Once both these restrictions apply, equivalence obtains, which identically reduces to the naive case. This can be deployed to infer some extra relationships: 1.7 B? ?A)?B) ?A( B) It would appear that the linear operator or homogeneity case could be one solution, or operator concept (albeit not necessarily applicable outside the corner conditions) 1.8 B? ?A? This appears to be a non commutative generalization of the naive case equivalence (Appendix) B(1 A) A(1 B) OR (A B) (B A) Alternatively, (1.7) could more neutrally be reduced to, 1.9 B? ?A)?B) By directly juxtaposing the terms pairwise, it can be obtained that, ?A if and only if B ?B) This appears to be valid insofar as it refers to the identity definitions as implied from the outset. Rather than making any use of the homogeneity assumption, it embarks on the symmetry of [double] negation, albeit without assuming any general operator, i.e. The more 'rigorous' approach would be to embark on the initial conventions (Appendix) (1 A)B (1 ?B)?A) OR (A B) ?B ?A) Since this merely refers to the original equivalence (1.1) yet to be reduced, this checks the correctness of (1.7) through (1.9) without providing any further reduced or closed forms. This might reiterate the multitude of meta solutions or operator concepts as per the initial conjecture, with the linearity or homogeneity case (1.8) still yielding a tentative yet parsimonious non commutativity of the form: 1.10 (A? 1 (B? In passing, please note that an alternate juxtaposition as per (1.9) could be: ?B iff B ?A Combined, this might suggest the reduced forms: 1.11 ?A) B ?B) Should linearity or homogeneity be maintained in this case, the composite or product could indeed amount to the universal negation (inversion) operator or acting as commutator An Extension of Propositional Logic by Arthur Shevenyonov 5 consistent with (1.8) and (1.10) One should bear in mind that this pertains to the special case still offering some nontrivial implications. Haunting Overlaps: Generalized Games, Anthropic Principle Relational Premises In actuality, the dual operators as in (1.8) delineate the existence domain or specification modality for the applicable subset of X. Better yet, as long as the primal operator couple secures the actual direct mapping, the conjugate counterparts point to the inverse mapping, thus providing for a bridge in between propositional logic and functional or relational representations: (p,q)(A, B) (A (p,q) B) (pA qB) 1~{f: A B} (p,(A, B) (B (p, A) ?B ?A) 1~{f: B A} Among other things, the above may refer to the A p<1 B Gradiency rationale as in Shevenyonov (2016a) or the Orduality premises (A, B) as in Shevenyonov (2016c) Moreover, if one were to expand (1.5) directly as, 2.1 (qB pqAB) ?A ?AB) a striking equivalence is revealed between this general propositional logic (GPL) and the generalized game (GG) featuring a consistent treatment of cooperative and non cooperative games alike (in fact, an entire continuum of interior cases) as in Shevenyonov (2016o) 2.2 )1s? )2s )1 s? )2 The parenthetical terms (or their ratios) r