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'.
|
2427
|
Maybe understanding doesn't need consciousness, despite what Searle seems to think [Searle, by Chalmers]
|
|
Full Idea:
Searle originally directed the Chinese Room against machine intentionality rather than consciousness, arguing that it is "understanding" that the room lacks,….but on Searle's view intentionality requires consciousness.
|
|
From:
report of John Searle (Minds, Brains and Science [1984]) by David J.Chalmers - The Conscious Mind 4.9.4
|
|
A reaction:
I doubt whether 'understanding' is a sufficiently clear and distinct concept to support Searle's claim. Understanding comes in degrees, and we often think and act with minimal understanding.
|
7390
|
If bigger and bigger brain parts can't understand, how can a whole brain? [Dennett on Searle]
|
|
Full Idea:
The argument that begins "this little bit of brain activity doesn't understand Chinese, and neither does this bigger bit..." is headed for the unwanted conclusion that even the activity of the whole brain won't account for understanding Chinese.
|
|
From:
comment on John Searle (Minds, Brains and Science [1984]) by Daniel C. Dennett - Consciousness Explained 14.1
|
|
A reaction:
In other words, Searle is guilty of a fallacy of composition (in negative form - parts don't have it, so whole can't have it). Dennett is right. The whole shebang of the full brain will obviously do wonderful (and commonplace) things brain bits can't.
|