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Our standard model-theoretic definition of logical consequence is originally based on Alfred Tarski’s (1936) semantic definition, which, in turn, is based on Rudolf Carnap’s (1934) similar definition. In recent literature, Tarski’s definition is described as a conceptual analysis of the intuitive ‘everyday’ concept of consequence or as an explication of it, but the use of these terms is loose and largely unaccounted for. I argue that the definition is not an analysis but an explication, in the Carnapian sense: the replacement of the inexact everyday concept with an exact one. Some everyday intuitions were thus brought into a precise form, others were ignored and forgotten. How exactly did the concept of logical consequence change in this process? I suggest that we could find some of the forgotten intuitions in Bernard Bolzano’s (1837) definition of ‘deducibility’, which is traditionally viewed as the main precursor of Tarski’s definition from a time before formalized languages. It turns out that Bolzano’s definition is subject to just the kind of natural features—paradoxicality of everyday language, Platonism about propositions, and dependence on the external world—that Tarski sought to tame by constructing an artificial concept for the special needs of mathematical logic. Full article

The meaning of information can be understood as a relationship between information systems. This study presents a brief outline of theoretical tools for the analysis of this relationship. Considering the informational character of reality, it is natural to extend the relationships between signs to include the concept of meaning as another instance of a relation between the informational entities of a sign and its denotation. However, this approach to the semantics of information does not require any specific ontological commitment, as the intention is always directed towards the object presented to us as a structural manifestation of information. Whether there is something that differs from this informational structure and is beyond our capacity to comprehend directly, or whether there are objects that are the result of our own active engagement in their formation, is a matter of ontological position, with respect to which our approach is neutral. The experience of logic tells us about the dangers of self-reference and the problem of the non-definability of the truth, demonstrated by Tarski. To avoid similar problems, we need precise theoretical tools to analyze relationships between information systems and between instances of information involved in semantics. These tools are also necessary for the definition and analysis of levels of informational abstraction that extend beyond the traditional linguistic and logical context. Full article

Organic mathematics is an applied mathematics of philosophical atomism. The order of the chemical elements in the table of elements is the source of order for the logical operations of addition and subtraction of atomic numbers. The inverse square laws of physics are the source of organization of subatomic structures of chemical atoms (atomic and molecular structures). These facts are foundational to the logic of the chemical sciences and are therefore the scientific basis for chemical information theory. The theories and facts of the chemical sciences are so perplex that several forms of symbolic representations are necessary to communicate the broad range of scientific concepts used to inquire into the nature of natural sorts and kinds. The logics proposed by Tarski, Lesniewski and Malatesta are applied to the construction of a numerical “spine” of perplex numbers representing atomic numbers as meta-symbols in meta-languages. The orbital angular momenta of certain collections of electrical particles (also known as “handedness”) are critical components in constructing the logical propositions of the perplex number “spine”. Biological communication channels can function if and only if the natural sorts and kinds are consistent with the matching patterns of the optical isomers. The terms spinners and twisters are introduced to express the electro-mechanical torques necessary for encoding chemical information. This hypothesis can be tested by several categories of experiments, including clinical pharmaco-dynamics and clinical toxico-dynamics of dissymmetric isomers of different sorts and kinds. Full article