9193
|
ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality [Dummett]
|
|
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.
|
|
From:
Michael Dummett (The Philosophy of Mathematics [1998], 7)
|
|
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.
|
9195
|
Intuitionists reject excluded middle, not for a third value, but for possibility of proof [Dummett]
|
|
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.
|
|
From:
Michael Dummett (The Philosophy of Mathematics [1998], 8.1)
|
|
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.
|
9186
|
First-order logic concerns objects; second-order adds properties, kinds, relations and functions [Dummett]
|
|
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.
|
|
From:
Michael Dummett (The Philosophy of Mathematics [1998], 3.1)
|
|
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.
|
9187
|
Logical truths and inference are characterized either syntactically or semantically [Dummett]
|
|
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.
|
|
From:
Michael Dummett (The Philosophy of Mathematics [1998], 3.1)
|
|
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?
|
16975
|
Essences are used to explain natural kinds, modality, and causal powers [Tahko]
|
|
Full Idea:
Essences are supposed to do a lot of explanatory work: natural kinds can be identified in terms of their essences, metaphysical modality can be reduced to essence, the causal power of objects can be explained with the help of essence.
|
|
From:
Tuomas E. Tahko (The Epistemology of Essence (draft) [2013], 1)
|
|
A reaction:
Natural kinds and modality are OK with me, but I'm dubious about the third one. If an essence explains something's causal powers, I have no idea what an essence might be. Essence are largely characterised in terms of causal powers.
|
7903
|
The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna]
|
|
Full Idea:
The six perfections are of giving, morality, patience, vigour, meditation, and wisdom.
|
|
From:
Nagarjuna (Mahaprajnaparamitashastra [c.120], 88)
|
|
A reaction:
What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate').
|
16977
|
If essence is modal and laws are necessary, essentialist knowledge is found by scientists [Tahko]
|
|
Full Idea:
If essence is conceived in terms of modality and the laws of nature are metaphysically necessary, it seems that the laws of nature constitute essentialist knowledge, so the discovery of essences is mostly due to scientists.
|
|
From:
Tuomas E. Tahko (The Epistemology of Essence (draft) [2013], 2.1)
|
|
A reaction:
This seems muddled to me. The idea that the laws themselves are essences is way off target. No one thinks all knowledge of necessities is essentialist. Mumford, for example, doesn't even believe in laws.
|