Ideas from 'First-order Logic, 2nd-order, Completeness' by Marcus Rossberg [2004], by Theme Structure

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4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / a. Systems of modal logic
With possible worlds, S4 and S5 are sound and complete, but S1-S3 are not even sound
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Henkin semantics has a second domain of predicates and relations (in upper case)
There are at least seven possible systems of semantics for second-order logic
Second-order logic needs the sets, and its consequence has epistemological problems
5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
Logical consequence is intuitively semantic, and captured by model theory
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
Γ|-S says S can be deduced from Γ; Γ|=S says a good model for Γ makes S true
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
In proof-theory, logical form is shown by the logical constants
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
A model is a domain, and an interpretation assigning objects, predicates, relations etc.
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model
5. Theory of Logic / K. Features of Logics / 4. Completeness
Completeness can always be achieved by cunning model-design
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
A deductive system is only incomplete with respect to a formal semantics