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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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7519
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Many mental phenomena are totally unexplained by folk psychology [Churchland,PM]
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Full Idea:
Folk psychology fails utterly to explain a considerable variety of central psychological phenomena: mental illness, sleep, creativity, memory, intelligence differences, and many forms of learning, to cite just a few.
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From:
Paul M. Churchland (Folk Psychology [1996], III)
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A reaction:
If folk psychology is a theory, it will have been developed to predict behaviour, rather than as a full-blown psychological map. The odd thing is that some people seem to be very bad at folk psychology.
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7520
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Folk psychology never makes any progress, and is marginalised by modern science [Churchland,PM]
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Full Idea:
Folk psychology has not progressed significantly in the last 2500 years; if anything, it has been steadily in retreat during this period; it does not integrate with modern science, and its emerging wallflower status bodes ill for its future.
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From:
Paul M. Churchland (Folk Psychology [1996], III)
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A reaction:
[compressed] However, while shares in alchemy and astrology have totally collapsed, folk psychology shows not the slightest sign of going away, and it is unclear how it ever could. See Idea 3177.
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