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10751
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Second-order logic needs the sets, and its consequence has epistemological problems [Rossberg]
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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.
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From:
Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §1)
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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.
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10753
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Logical consequence is intuitively semantic, and captured by model theory [Rossberg]
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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.
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From:
Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2)
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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.
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10752
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Γ |- S says S can be deduced from Γ; Γ |= S says a good model for Γ makes S true [Rossberg]
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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'.
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From:
Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2)
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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.
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10756
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A model is a domain, and an interpretation assigning objects, predicates, relations etc. [Rossberg]
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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.
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From:
Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3)
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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.
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10758
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If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model [Rossberg]
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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.
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From:
Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3)
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A reaction:
So the term 'categorical' is gradually replacing the much-used phrase 'up to isomorphism'.
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6007
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If you know your father, but don't recognise your father veiled, you know and don't know the same person [Eubulides, by Dancy,R]
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Full Idea:
The 'undetected' or 'veiled' paradox of Eubulides says: if you know your father, and don't know the veiled person before you, but that person is your father, you both know and don't know the same person.
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From:
report of Eubulides (fragments/reports [c.390 BCE]) by R.M. Dancy - Megarian School
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A reaction:
Essentially an uninteresting equivocation on two senses of "know", but this paradox comes into its own when we try to give an account of how linguistic reference works. Frege's distinction of sense and reference tried to sort it out (Idea 4976).
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6008
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Removing one grain doesn't destroy a heap, so a heap can't be destroyed [Eubulides, by Dancy,R]
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Full Idea:
The 'sorites' paradox of Eubulides says: if you take one grain of sand from a heap (soros), what is left is still a heap; so no matter how many grains of sand you take one by one, the result is always a heap.
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From:
report of Eubulides (fragments/reports [c.390 BCE]) by R.M. Dancy - Megarian School
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A reaction:
(also Cic. Acad. 2.49) This is a very nice paradox, which goes to the heart of our bewilderment when we try to fully understand reality. It homes in on problems of identity, as best exemplified in the Ship of Theseus (Ideas 1212 + 1213).
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