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'.
|
14296
|
Dispositions are physical states of mechanism; when known, these replace the old disposition term [Quine]
|
|
Full Idea:
Each disposition, in my view, is a physical state or mechanism. ...In some cases nowadays we understand the physical details and set them forth explicitly in terms of the arrangement and interaction of small bodies. This replaces the old disposition.
|
|
From:
Willard Quine (The Roots of Reference [1990], p.11), quoted by Stephen Mumford - Dispositions 01.3
|
|
A reaction:
A challenge to the dispositions and powers view of nature, one which rests on the 'categorical' structural properties, rather than the 'hypothetical' dispositions. But can we define a mechanism without mentioning its powers?
|
17400
|
Mendeleev focused on abstract elements, not simple substances, so he got to their essence [Mendeleev, by Scerri]
|
|
Full Idea:
Because he was attempting to classify abstract elements, not simple substances, Mendeleev was not misled by nonessential chemical properties.
|
|
From:
report of Dmitri Mendeleev (The Principles of Chemistry [1870]) by Eric R. Scerri - The Periodic Table 04 'Making'
|
|
A reaction:
I'm not fully clear about this, but I take it that Mendeleev stood back from the messy observations, and tried to see the underlying simpler principles. 'Simple substances' were ones that had not so far been decomposed.
|