Fuzzy, Monoidal & Higher-Order Logic

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Abstractions about abstract versus concrete representation, fuzzy rules, restricted formulas, t-norm semantics, and second-order quantification.

5 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Abstract and concrete — Distinguish philosophical entities by an explicitly chosen abstract-versus-concrete criterion while preserving paradigm cases, disputed cases, and the fact that no single criterion commands universal agreement.
• Fuzzy rule — Represent a graded IF-THEN relation whose linguistic antecedents and consequent are fuzzy sets, then compute a consequent degree or fuzzy output through declared connective, implication, aggregation, and output conventions.
• Harrop Formula — An intuitionistic formula built so disjunctions and existential quantifiers occur only in negative positions, yielding a stable, computationally disciplined fragment and inspiring hereditary-Harrop logic programming.
• Monoidal t-Norm Logic — The propositional many-valued logic common to all left-continuous t-norms, coupling strong conjunction to residual implication and enforcing prelinearity.
• Second-order logic — Extend first-order languages with quantification over predicates, relations, sets, and functions, while treating full and Henkin semantics as different regimes with different categoricity, completeness, compactness, and axiomatizability behavior.