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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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5953
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For the Cyrenaics experience was not enough to give certainty about reality [Aristippus young, by Plutarch]
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Full Idea:
The Cyrenaics, placing all experience within themselves, thought such evidence was insufficient warrant for certainty about reality, and withdrew as in a siege from the world, admitting that objects 'appear', but refusing to pronounce the word 'are'.
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From:
report of Aristippus the younger (fragments/reports [c.335 BCE]) by Plutarch - 74: Reply to Colotes §1120
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A reaction:
This seems to be the most extreme position found in ancient thought. It accompanies their extreme hedonism, based on the reality of experience and lack of interest in anything external. A bit daft, really.
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3026
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Actions are influenced by circumstances, so Cyrenaics say felons should be reformed, not hated [Aristippus young, by Diog. Laertius]
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Full Idea:
Cyrenaics say errors should be pardoned, because men do not err intentionally but are influenced by circumstances; one should not hate a person, but only teach him better.
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From:
report of Aristippus the younger (fragments/reports [c.335 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.7.9
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A reaction:
A very appealing suggestion, and rather wonderful for its time. There is still implied agreement about what is 'error', and what counts as 'better'.
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3024
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Cyrenaics teach that honour, justice and shame are all based on custom and fashion [Aristippus young, by Diog. Laertius]
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Full Idea:
The Cyrenaics taught that there was nothing naturally and intrinsically just, or honourable, or disgraceful; but that things were considered so because of law and fashion.
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From:
report of Aristippus the younger (fragments/reports [c.335 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.7.8
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A reaction:
As we would say now, values and virtues are 'cultural constructs'. This obviously contains a lot of truth, but I don't think our opposition of genocide is just 'fashion'.
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