Ideas of Marcus Rossberg, by Theme

[German, fl. 2008, Professor at the University of Connecticut.]

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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic

10751	Second-order logic needs the sets, and its consequence has epistemological problems
	Full Idea: Second-order logic raises doubts because of its ontological commitment to the set-theoretic hierarchy, and the allegedly problematic epistemic status of the second-order consequence relation.
	From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §1)
	A reaction: The 'epistemic' problem is whether you can know the truths, given that the logic is incomplete, and so they cannot all be proved. Rossberg defends second-order logic against the second problem. A third problem is that it may be mathematics.

10757	Henkin semantics has a second domain of predicates and relations (in upper case)
	Full Idea: Henkin semantics (for second-order logic) specifies a second domain of predicates and relations for the upper case constants and variables.
	From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3)
	A reaction: This second domain is restricted to predicates and relations which are actually instantiated in the model. Second-order logic is complete with this semantics. Cf. Idea 10756.

10759	There are at least seven possible systems of semantics for second-order logic
	Full Idea: In addition to standard and Henkin semantics for second-order logic, one might also employ substitutional or game-theoretical or topological semantics, or Boolos's plural interpretation, or even a semantics inspired by Lesniewski.
	From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3)
	A reaction: This is helpful in seeing the full picture of what is going on in these logical systems.

5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence

10753	Logical consequence is intuitively semantic, and captured by model theory
	Full Idea: Logical consequence is intuitively taken to be a semantic notion, ...and it is therefore the formal semantics, i.e. the model theory, that captures logical consequence.
	From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2)
	A reaction: If you come at the issue from normal speech, this seems right, but if you start thinking about the necessity of logical consequence, that formal rules and proof-theory seem to be the foundation.

5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-

10752	Γ |- S says S can be deduced from Γ; Γ |= S says a good model for Γ makes S true
	Full Idea: Deductive consequence, written Γ|-S, is loosely read as 'the sentence S can be deduced from the sentences Γ', and semantic consequence Γ|=S says 'all models that make Γ true make S true as well'.
	From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2)
	A reaction: We might read |= as 'true in the same model as'. What is the relation, though, between the LHS and the RHS? They seem to be mutually related to some model, but not directly to one another.

5. Theory of Logic / E. Structures of Logic / 1. Logical Form

10754	In proof-theory, logical form is shown by the logical constants
	Full Idea: A proof-theorist could insist that the logical form of a sentence is exhibited by the logical constants that it contains.
	From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2)
	A reaction: You have to first get to the formal logical constants, rather than the natural language ones. E.g. what is the truth table for 'but'? There is also the matter of the quantifiers and the domain, and distinguishing real objects and predicates from bogus.

5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models

10756	A model is a domain, and an interpretation assigning objects, predicates, relations etc.
	Full Idea: A standard model is a set of objects called the 'domain', and an interpretation function, assigning objects in the domain to names, subsets to predicate letters, subsets of the Cartesian product of the domain with itself to binary relation symbols etc.
	From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3)
	A reaction: The model actually specifies which objects have which predicates, and which objects are in which relations. Tarski's account of truth in terms of 'satisfaction' seems to be just a description of those pre-decided facts.

5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms

10758	If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model
	Full Idea: A mathematical theory is 'categorical' if, and only if, all of its models are isomorphic. Such a theory then essentially has just one model, the standard one.
	From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3)
	A reaction: So the term 'categorical' is gradually replacing the much-used phrase 'up to isomorphism'.

5. Theory of Logic / K. Features of Logics / 4. Completeness

10761	Completeness can always be achieved by cunning model-design
	Full Idea: All that should be required to get a semantics relative to which a given deductive system is complete is a sufficiently cunning model-theorist.
	From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §5)

5. Theory of Logic / K. Features of Logics / 5. Incompleteness

10755	A deductive system is only incomplete with respect to a formal semantics
	Full Idea: No deductive system is semantically incomplete in and of itself; rather a deductive system is incomplete with respect to a specified formal semantics.
	From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3)
	A reaction: This important point indicates that a system might be complete with one semantics and incomplete with another. E.g. second-order logic can be made complete by employing a 'Henkin semantics'.