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Could there be a 1-ary connective # two successive applications ofwhich amounted to a single application of (say) classicalnegation?[] If we work in - an austereversion of this question amounts to asking about the fate of #subjected, alongside the standard structural rules, to the conditionthat for all φ:

For conditions other than those corresponding to (closure undercertain) rules we can also ask whether a connective satisfying thecondition in question can safely—i.e. conservatively—bepostulated to exist. One such condition is the following, hereexpressed as a condition on a consequence relation ⊢ with abinary connective # in its language, all variables understood asuniversally quantified:

For this section the focus will be on rules governing the variousconnectives, beginning with a deferred topic from Section: the transfer ofCarnap-inspired questions of the range of valuations on which thesequents provable in this or that logic hold to question about therange of valuations on which the sequent-to-sequent rules preserve theproperty of holding. We will also revisit the sequent-calculusoperational rules from early inSection and look at some oftheir properties, as well as at some interesting variations on thestructural rules which work alongside them. One topic—or pair oftopics—raised by such investigations is the issue of theexistence and uniqueness of connectives exhibiting prescribed logicalbehaviour (such as might be enshrined in various rules), an issuebearing on all approaches to logic and not just the sequent calculusapproach. Finally we look at a pair of concepts applying toconnectives as they behave in particular logics: the concept of auniversally representative connective and the concept of a‘special’ connective.

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John Seely Brown presents an interesting notion that the internet leverages the small efforts of many with the large efforts of few. The central premise is that connections created with unusual nodes supports and intensifies existing large effort activities. Brown provides the example of a Maricopa County Community College system project that links senior citizens with elementary school students in a mentor program. The children “listen to these “grandparents” better than they do their own parents, the mentoring really helps the teachers…the small efforts of the many- the seniors – complement the large efforts of the few – the teachers.” (2002). This amplification of learning, knowledge and understanding through the extension of a personal network is the epitome of connectivism.

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Connectivism presents a model of learning that acknowledges the tectonic shifts in society where learning is no longer an internal, individualistic activity. How people work and function is altered when new tools are utilized. The field of education has been slow to recognize both the impact of new learning tools and the environmental changes in what it means to learn. Connectivism provides insight into learning skills and tasks needed for learners to flourish in a digital era.

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Section 1 introduces the idea of formal sentential languages as(absolutely free) algebras, their primitive connectives having thestatus of the fundamental operations of these algebras, and explainsthe notion of a consequence relation on such a language. At a moregeneral level, Galois connections are also introduced, and a simpleobservation made concerning what we call binary relationalconnections, to be applied in later sections. Section 2 introduces thefundamental idea of a valuation for a language (a bivalent truth-valueassignment to its formulas). Truth-functionality and a generalizationthereof called pseudo-truth-functionality (of a connective withrespect to a class of valuations) are explained, as well as the ideaof sequents and sequent-to-sequent rules. We then move to a moregeneral conception of sequents as originally conceived, allowing a setof formulas to play the conclusion role rather than just a singleformula, as well as to the associated idea of generalized consequencerelations, emphasizing Carnap's motivation for making thisgeneralization. That motivation can be summarised with reference toone connective. We take negation as our example. Consider theconsequence relation determined by the class of all valuations onwhich the usual negation truth-function connective is associated withthe negation connective. This can be regarded as the classical logicof negation, at least if logics are identified with consequencerelations. Now, there are additional valuations consistent with thisconsequence relation on which the negation truth-functionis not associated with the negation connective. (Theterminology of a valuation's being consistent with a consequencerelation is adapted from Dana Scott. A precise definition appears inSection 2.) But when we consider instead generalized consequencerelations, the analogous discrepancy does not arise. Section 2includes a statement of a result of D. Gabbay which isolates preciselythe truth-functions for which such discrepancies do not arise(roughly: generalizations of the conjunction truth-function andvarious pro jection functions), as well as of a result ofW. Rautenberg showing that despite this expressive weakness in theapparatus of consequence relations, they are sufficientlydiscriminating—without the need to pass to generalizedconsequence relations—to distinguish the logical properties ofany two truth-functions (of the same arity).

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Connectivism is the integration of principles explored by chaos, network, and complexity and self-organization theories. Learning is a process that occurs within nebulous environments of shifting core elements – not entirely under the control of the individual. Learning (defined as actionable knowledge) can reside outside of ourselves (within an organization or a database), is focused on connecting specialized information sets, and the connections that enable us to learn more are more important than our current state of knowing.

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Section 3 picks up the theme of sequent-to-sequent rules,concentrating on semantic analysis in the style of J. W. Garson ofsuch rules, and also on the existence and uniqueness of connectivessatisfying given rules (a topic introduced by H. Hiż). These lasttwo are seen to be connected with what we call cut-inductivity andid-inductivity of specifically sequent-calculus rules, via, in theformer case, a proposal identifying the existence of a connective(with prescribed logical behaviour, given the perspective provided bya logic not explicitly providing such a connective) with conservativeextension (of the given logic by rules for the new connectivedetailing that behaviour). This proposal was made by N. D. Belnap;some problems for conservative extension as a sufficient condition forgranting the existence or intelligibility of a new connective withsuch-and-such logical properties are noted. Section 4 collects severalcases of interest in which one might wonder about the existence ofconnectives with various such properties. For instance (this beingExample 3 out of a total of six) one might ask if it makes sense tohave a 1-ary connective with the property that two applications ofthis connective are equivalent to a single application of (classical)negation. Section 5 assembles notes and references.