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This paper introduces a comprehensive logical framework to reason about threshold-driven diffusion and threshold-driven link change in social networks. It considers both monotonic dynamics, where agents can only adopt new features and create new connections, and non-monotonic dynamics, where agents may also abandon features or cut ties. Three types of operators are combined: one capturing diffusion only, one capturing link change only, and one capturing both at the same time. We first characterise the models on which diffusion of a unique feature and link change stabilise, whilst discussing salient properties of stable models with multiple spreading features. Second, we show that our operators (and any combination of them) are irreplaceable, in the sense that the sequences of model updates expressed by a combination of operators cannot always be expressed using any other operators. Finally, we analyse classes of models on which some operators can be replaced.

Lattice-ordered abelian groups, or abelian $$\ell $$ ℓ -groups in what follows, are categorically equivalent to two classes of 0-bounded hoops that are relevant in the realm of the equivalent algebraic semantics of many-valued logics: liftings of cancellative hoops and perfect MV-algebras. The former generate the variety of product algebras, and the latter the subvariety of MV-algebras generated by perfect MV-algebras, that we shall call $$\textsf{DLMV}$$ DLMV . In this work we focus on these two varieties and their relation to the structures obtained by forgetting the falsum constant 0, i.e., product hoops and DLW-hoops. As main results, we first show a characterization of the free algebras in these two varieties as particular weak Boolean products; then, we show a construction that freely generates a product algebra from a product hoop and a DLMV-algebra from a DLW-hoop. In other words, we exhibit the free functor from the two algebraic categories of hoops to the corresponding categories of 0-bounded algebras. Finally, we use the results obtained to study projective algebras and unification problems in the two varieties (and the corresponding logics); both varieties are shown to have (strong) unitary unification type, and as a consequence they are structurally and universally complete.

We show how to obtain a probabilistic semantics and calculus for a logic presented by a valuation specification. By identifying general forms of valuation constraints we are able to accommodate a wide class of propositional based logics encompassing multi-valued logics like Łukasiewicz 3-valued logic and the Belnap–Dunn four-valued logic as well as paraconsistent logics like $${\textsf{mbC}}$$ mbC and $${\textsf{LFI1}}$$ LFI 1 . The probabilistic calculus is automatically generated from the valuation specification. Although not having explicit probability constructors in the language, the rules of the calculus reflect the valuation constraints in a probabilistic way. Indeed the probability of the premises of each rule coincides with the probability of the conclusions. Moreover, a failed exhaustive attempt of proving a formula in this calculus means non-derivability. Nevertheless when the non-derived formula is consistent then it is possible to extract a satisfying valuation from the failed exhaustive attempt. Soundness and completeness of the calculi are established with respect to the probabilistic semantics consisting of probability spaces also induced by the valuation specification. Furthermore we prove the equivalence between the probabilistic and the valuation semantics.

A standard sequent system (the system $$\mathcal{L}\mathcal{J}$$ L J ), a standard natural deduction system (the system $$\mathcal{N}\mathcal{J}$$ N J ) and an extended natural deduction system (the system $$\mathcal{N}\mathcal{E}$$ N E ) will be considered. The $$\mathcal{L}\mathcal{J}$$ L J -images of $$\mathcal{N}\mathcal{J}$$ N J -derivations and $$\mathcal{N}\mathcal{E}$$ N E -derivations ( $$\mathcal {GLJ}$$ GLJ -derivations and $$\mathcal {PLJ}$$ PLJ -derivations) and the $$\mathcal{N}\mathcal{E}$$ N E -images of $$\mathcal{N}\mathcal{J}$$ N J -derivations ( $$\mathcal {ENE}$$ ENE -derivations) will be presented. It will be shown that $$\mathcal {PLJ}$$ PLJ -derivations and $$\mathcal {GLJ}$$ GLJ -derivations have special cuts, nde-cuts and nd-cuts, respectively. Nd-cuts corresponding to maximum segments of $$\mathcal{N}\mathcal{J}$$ N J -derivations (ndam-cuts) and nde-cuts corresponding to maximum segments of $$\mathcal{N}\mathcal{E}$$ N E -derivations (ndeam-cuts) will be studied. It will be shown that (A) an $$\mathcal{N}\mathcal{J}$$ N J -derivation is normal iff its $$\mathcal{N}\mathcal{E}$$ N E -image is normal iff its $$\mathcal{L}\mathcal{J}$$ L J -image has no ndam-cuts; (B) an $$\mathcal {GLJ}$$ GLJ -derivation has no ndam-cuts iff its $$\mathcal{N}\mathcal{J}$$ N J -image is normal iff its $$\mathcal{N}\mathcal{E}$$ N E -image is normal; and (C) an $$\mathcal {ENE}$$ ENE -derivation is normal iff its $$\mathcal{N}\mathcal{J}$$ N J -image is normal iff its $$\mathcal{L}\mathcal{J}$$ L J -image has no ndeam-cuts.

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