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9193
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ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality [Dummett]
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Full Idea:
ZF set theory is a first-order axiomatization. Variables range over sets, there are no second-order variables, and primitive predicates are just 'equals' and 'member of'. The axiom of extensionality says sets with the same members are identical.
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From:
Michael Dummett (The Philosophy of Mathematics [1998], 7)
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A reaction:
If the eleven members of the cricket team are the same as the eleven members of the hockey team, is the cricket team the same as the hockey team? Our cricket team is better than our hockey team, so different predicates apply to them.
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9195
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Intuitionists reject excluded middle, not for a third value, but for possibility of proof [Dummett]
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Full Idea:
It must not be concluded from the rejection of excluded middle that intuitionistic logic operates with three values: true, false, and neither true nor false. It does not make use of true and false, but only with a construction being a proof.
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From:
Michael Dummett (The Philosophy of Mathematics [1998], 8.1)
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A reaction:
This just sounds like verificationism to me, with all its problems. It seems to make speculative statements meaningless, which can't be right. Realism has lots of propositions which are assumed to be true or false, but also unknowable.
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9186
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First-order logic concerns objects; second-order adds properties, kinds, relations and functions [Dummett]
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Full Idea:
First-order logic is distinguished by generalizations (quantification) only over objects: second-order logic admits generalizations or quantification over properties or kinds of objects, and over relations between them, and functions defined over them.
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From:
Michael Dummett (The Philosophy of Mathematics [1998], 3.1)
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A reaction:
Second-order logic was introduced by Frege, but is (interestingly) rejected by Quine, because of the ontological commitments involved. I remain unconvinced that quantification entails ontological commitment, so I'm happy.
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9187
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Logical truths and inference are characterized either syntactically or semantically [Dummett]
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Full Idea:
There are two ways of characterizing logical truths and correct inference. Proof-theoretic or syntactic characterizations, if the formalization admits of proof or derivation; and model-theoretic or semantic versions, being true in all interpretations.
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From:
Michael Dummett (The Philosophy of Mathematics [1998], 3.1)
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A reaction:
Dummett calls this distinction 'fundamental'. The second one involves truth, and hence meaning, where the first one just responds to rules. ..But how can you have a notion of correctly following a rule, without a notion of truth?
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16369
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There is a single file per object, memorised, reactivated, consolidated and expanded [Papineau, by Recanati]
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Full Idea:
For Papineau there is just one file, which is initialised on the first encounter with the object, stored in memory, reactivated on further encounters, and consolidated with familiarity. Accumulation of information shows it is the same file.
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From:
report of David Papineau (Phenomenal and Perceptual Concepts [2006]) by François Recanati - Mental Files 7.2
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A reaction:
Recanati attempts to refute this view, defending a more complex taxonomy of files. I'm sympathetic to Papineau, as distinct shift in file type doesn't sound very plausible. Simplicity suggests Papineau as a better starting-point.
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6030
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Each part of the soul has its virtue - pleasure for appetite, success for competition, and rectitude for reason [Galen]
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Full Idea:
We have by nature these three appropriate relationships, corresponding to each form of the soul's parts - to pleasure because of the appetitive part, to success because of the competitive part, and to rectitude because of the rational part.
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From:
Galen (On Hippocrates and Plato [c.170], 5.5.8)
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A reaction:
This is a nice combination of Plato's tripartite theory of soul (in 'Republic') and Aristotle's derivation of virtues from functions. Presumably, though, reason should master the other two, and there is nothing in Galen's idea to explain this.
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