Ideas from 'First-order Logic, 2nd-order, Completeness' by Marcus Rossberg [2004], by Theme Structure
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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
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Henkin semantics has a second domain of predicates and relations (in upper case)
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There are at least seven possible systems of semantics for second-order logic
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Second-order logic needs the sets, and its consequence has epistemological problems
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5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
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Logical consequence is intuitively semantic, and captured by model theory
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5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
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Γ |- S says S can be deduced from Γ; Γ |= S says a good model for Γ makes S true
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5. Theory of Logic / E. Structures of Logic / 1. Logical Form
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In proof-theory, logical form is shown by the logical constants
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5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
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A model is a domain, and an interpretation assigning objects, predicates, relations etc.
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5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
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If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model
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5. Theory of Logic / K. Features of Logics / 4. Completeness
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Completeness can always be achieved by cunning model-design
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5. Theory of Logic / K. Features of Logics / 5. Incompleteness
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A deductive system is only incomplete with respect to a formal semantics
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