10751
|
Second-order logic needs the sets, and its consequence has epistemological problems [Rossberg]
|
|
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.
|
10753
|
Logical consequence is intuitively semantic, and captured by model theory [Rossberg]
|
|
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.
|
10752
|
Γ |- S says S can be deduced from Γ; Γ |= S says a good model for Γ makes S true [Rossberg]
|
|
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.
|
10756
|
A model is a domain, and an interpretation assigning objects, predicates, relations etc. [Rossberg]
|
|
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.
|
10758
|
If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model [Rossberg]
|
|
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'.
|
21091
|
It would be absurd if even a free constitution did not impose restraints, for the public good [Hume]
|
|
Full Idea:
A republican and free form of government would be an obvious absurdity, if the particular checks and controls, provided by the constitution, had really no influence, and made it not the interest, even of bad men, to act for the public good.
|
|
From:
David Hume (That Politics may be reduced to a Science [1750], p.14)
|
|
A reaction:
Presumably if you attain absolute power you can write any old constitution you like (Clause 1: the presidency is for life). But there does seem much point in doing it - unless it is to facilitate the use of the law for persecutions.
|
21092
|
Nobility either share in the power of the whole, or they compose the power of the whole [Hume]
|
|
Full Idea:
A nobility may possess power in two different ways. Either every nobleman shares the power as part of the whole body, or the whole body enjoys the power as composed of parts, which each have a distinct power and authority.
|
|
From:
David Hume (That Politics may be reduced to a Science [1750], p.15)
|
|
A reaction:
He says the first type is found in Venice, and is preferable to the second type, which is found in Poland. Presumably in the shared version there is some restraint on depraved nobles. The danger is each noble being an autocrat.
|