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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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7720
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Two things can only resemble one another in some respect, and that may reintroduce a universal [Lowe]
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Full Idea:
A problem for resemblance nominalism is that in saying that two particulars 'resemble' one another, it is necessary to specify in what respect they do so (e.g. colour, shape, size), and this threatens to reintroduce what appears to be talk of universals.
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From:
E.J. Lowe (Locke on Human Understanding [1995], Ch.7)
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A reaction:
We see resemblance between faces instantly, long before we can specify the 'respects' of the resemblance. This supports the Humean hard-wired view of resemblance, rather than some appeal to Platonic universals.
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7714
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Personal identity is a problem across time (diachronic) and at an instant (synchronic) [Lowe]
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Full Idea:
There is the question of the identity of a person over or across time ('diachronic' personal identity), and there is also the question of what makes for personal identity at a time ('synchronic' personal identity).
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From:
E.J. Lowe (Locke on Human Understanding [1995], Ch.5)
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A reaction:
This seems to me to be the first and most important distinction in the philosophy of personal identity, and they regularly get run together. Locke, for example, has an account of synchronic identity, which is often ignored. It applies to objects too.
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7715
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Mentalese isn't a language, because it isn't conventional, or a means of public communication [Lowe]
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Full Idea:
'Mentalese' would be neither conventional nor a means of public communication so that even to call it a language is seriously misleading.
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From:
E.J. Lowe (Locke on Human Understanding [1995], Ch.7)
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A reaction:
It is, however, supposed to contain symbolic representations which are then used as tokens for computation, so it seems close to a language, if (for example) symbolic logic or mathematics were accepted as languages. But who understands it?
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21119
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Power is only legitimate if it is reasonable for free equal citizens to endorse the constitution [Rawls]
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Full Idea:
Exercise of political power is fully proper only when it is exercised in accordance with a constitution the essentials of which all citizens as free and equal may reasonably be expected to endorse in light of principles and ideals acceptable to reason.
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From:
John Rawls (Political Liberalism [1993], p.217), quoted by Andrew Shorten - Contemporary Political Theory 02
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A reaction:
This is not the actual endorsement of Rousseau, or the tacit endorsement of Locke (by living there), but adds a Kantian appeal to a rational consensus, on which rational people should converge. Very Enlightenment. 'Hypothetical consent'.
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