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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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22808
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Liberalism is minimal government, or individual rights, or equality [Avineri/De-Shalit]
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Full Idea:
Liberalism has been defended as a theory of minimal government, or as a theory of basic individual rights, or as an egalitarian philosophy.
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From:
Avineri,S/De-Shalit,A (Intro to 'Communitarianism and Individualism' [1992], §5)
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A reaction:
Minimal government tends towards anarchist liberalism, but then what grounds the right to be free of government? Presumably any sensible theory of rights has to be egalitarian. What could ground unequal rights?
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22806
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Communitarians avoid oppression for the common good, by means of small mediating communities [Avineri/De-Shalit]
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Full Idea:
Because of the mediating structures of small communities, communitarians are less fearful [than liberals] of the emergence of an oppressive government as a result of the politics of the common good.
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From:
Avineri,S/De-Shalit,A (Intro to 'Communitarianism and Individualism' [1992], §5)
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A reaction:
A politics of the common good has an obvious implicit conservatism because the central consensus is always likely to disapprove of errant individuals, of all sorts. Only individual rights can block an oppressive government.
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22807
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If our values are given to us by society then we have no grounds to criticise them [Avineri/De-Shalit]
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Full Idea:
If communitarians are right that we are not free to choose, but rather that our values are determined by our community, the individualists say, then there is no reason to criticise the values of one's society.
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From:
Avineri,S/De-Shalit,A (Intro to 'Communitarianism and Individualism' [1992], §5)
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A reaction:
This is an obvious challenge, but if one's concept of community is a forum for free debate then it can be overcome. There is no avoiding the fact, though, that a good community always needs a high degree of consensus.
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