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All the ideas for 'The Boundary Stones of Thought', 'A Free Will' and 'Philosophy of Arithmetic'

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51 ideas

1. Philosophy / C. History of Philosophy / 2. Ancient Philosophy / e. Late classical philosophy
In the third century Stoicism died out, replaced by Platonism, with Aristotelian ethics [Frede,M]
     Full Idea: By the second century CE Aristotelianism and Platonism had begun to eclipse Stoicism, and by the end of the third century Stoicism had no followers. All philosophers now opted for some form of Platonism, but including Aristotle's ethical principles.
     From: Michael Frede (A Free Will [1997], 04)
     A reaction: The idea that Aristotelian ethics dominated that period is new to me. Stoic influence remained strong in Augustine, and hence in Christianity.
In late antiquity nearly all philosophers were monotheists [Frede,M]
     Full Idea: I am inclined to think that almost all philosophers in late antiquity were monotheists.
     From: Michael Frede (A Free Will [1997], 08)
     A reaction: I'm not sure when late antiquity begins, in this remark. Maybe as early as 100 CE. Epictetus talks of Zeus as if he is supreme.
1. Philosophy / E. Nature of Metaphysics / 6. Metaphysics as Conceptual
Logic doesn't have a metaphysical basis, but nor can logic give rise to the metaphysics [Rumfitt]
     Full Idea: There is surely no metaphysical basis for logic, but equally there is no logical basis for metaphysics, if that implies that we can settle the choice of logic in advance of settling any seriously contested metaphysical-cum-semantic issues.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.5)
     A reaction: Is this aimed at Tim Williamson's book on treating modal logic as metaphysics? I agree with the general idea that logic won't deliver a metaphysics. I might want to defend a good metaphysics giving rise to a good logic.
3. Truth / A. Truth Problems / 1. Truth
The idea that there are unrecognised truths is basic to our concept of truth [Rumfitt]
     Full Idea: The realist principle that a statement may be true even though no one is able to recognise its truth is so deeply embedded in our ordinary conception of truth that any account that flouts it is liable to engender confusion.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 5.1)
3. Truth / B. Truthmakers / 7. Making Modal Truths
'True at a possibility' means necessarily true if what is said had obtained [Rumfitt]
     Full Idea: A statement is 'true at a possibility' if, necessarily, things would have been as the statement (actually) says they are, had the possibility obtained.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 6.6)
     A reaction: This is deliberately vague about what a 'possibility' is, but it is intended to be more than a property instantiation, and less than a possible world.
4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
Semantics for propositions: 1) validity preserves truth 2) non-contradition 3) bivalence 4) truth tables [Rumfitt]
     Full Idea: The classical semantics of natural language propositions says 1) valid arguments preserve truth, 2) no statement is both true and false, 3) each statement is either true or false, 4) the familiar truth tables.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1)
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
'Absolute necessity' would have to rest on S5 [Rumfitt]
     Full Idea: If there is such a notion as 'absolute necessity', its logic is surely S5.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 3.3)
     A reaction: There are plenty of people (mainly in the strict empiricist tradition) who don't believe in 'absolute' necessity.
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
It is the second-order part of intuitionistic logic which actually negates some classical theorems [Rumfitt]
     Full Idea: Although intuitionistic propositional and first-order logics are sub-systems of the corresponding classical systems, intuitionistic second-order logic affirms the negations of some classical theorems.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1)
Intuitionists can accept Double Negation Elimination for decidable propositions [Rumfitt]
     Full Idea: Double Negation Elimination is a rule of inference which the classicist accepts without restriction, but which the intuitionist accepts only for decidable propositions.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1)
     A reaction: This cures me of my simplistic understanding that intuitionists just reject the rules about double negation.
4. Formal Logic / F. Set Theory ST / 1. Set Theory
Most set theorists doubt bivalence for the Continuum Hypothesis, but still use classical logic [Rumfitt]
     Full Idea: Many set theorists doubt if the Generalised Continuum Hypothesis must be either true or false; certainly, its bivalence is far from obvious. All the same, almost all set theorists use classical logic in their proofs.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.2)
     A reaction: His point is that classical logic is usually taken to rest on bivalence. He offers the set theorists a helping hand, by defending classical logic without resorting to bivalence.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
The iterated conception of set requires continual increase in axiom strength [Rumfitt]
     Full Idea: We are doomed to postulate an infinite sequence of successively stronger axiom systems as we try to spell out what is involved in iterating the power set operation 'as far as possible'.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.3)
     A reaction: [W.W. Tait is behind this idea] The problem with set theory, then, especially as a foundation of mathematics, is that it doesn't just expand, but has to keep reinventing itself. The 'large cardinal axioms' are what is referred to.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
A set may well not consist of its members; the empty set, for example, is a problem [Rumfitt]
     Full Idea: There seem strong grounds for rejecting the thesis that a set consists of its members. For one thing, the empty set is a perpetual embarrassment for the thesis.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.4)
     A reaction: Rumfitt also says that if 'red' has an extension, then membership of that set must be vague. Extensional sets are precise because their objects are decided in advance, but intensional (or logical) sets, decided by a predicate, can be vague.
A set can be determinate, because of its concept, and still have vague membership [Rumfitt]
     Full Idea: Vagueness in respect of membership is consistent with determinacy of the set's identity, so long as a set's identity is taken to consist, not in its having such-and-such members, but in its being the extension of the concept A.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.4)
     A reaction: To be determinate, it must be presumed that there is some test which will decide what falls under the concept. The rule can say 'if it is vague, reject it' or 'if it is vague, accept it'. Without one of those, how could the set have a clear identity?
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
If the totality of sets is not well-defined, there must be doubt about the Power Set Axiom [Rumfitt]
     Full Idea: Someone who is sympathetic to the thesis that the totality of sets is not well-defined ought to concede that we have no reason to think that the Power Set Axiom is true.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.6)
     A reaction: The point is that it is only this Axiom which generates the vast and expanding totality. In principle it is hard, though, to see what is intrinsically wrong with the operation of taking the power set of a set. Hence 'limitation of size'?
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
Logic is higher-order laws which can expand the range of any sort of deduction [Rumfitt]
     Full Idea: On the conception of logic recommended here, logical laws are higher-order laws that can be applied to expand the range of any deductive principles.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 3.3)
     A reaction: You need the concept of a 'deductive principle' to get this going, but I take it that might be directly known, rather than derived from a law.
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
The case for classical logic rests on its rules, much more than on the Principle of Bivalence [Rumfitt]
     Full Idea: I think it is a strategic mistake to rest the case for classical logic on the Principle of Bivalence: the soundness of the classical logic rules is far more compelling than the truth of Bivalence.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1)
     A reaction: The 'rules' to which he is referring are those of 'natural deduction', which make very few assumptions, and are intended to be intuitively appealing.
Classical logic rules cannot be proved, but various lines of attack can be repelled [Rumfitt]
     Full Idea: There is not the slightest prospect of proving that the rules of classical logic are sound. ….All that the defender of classical logic can do is scrutinize particular attacks and try to repel them.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1)
     A reaction: This is the agenda for Rumfitt's book.
If truth-tables specify the connectives, classical logic must rely on Bivalence [Rumfitt]
     Full Idea: If we specify the senses of the connectives by way of the standard truth-tables, then we must justify classical logic only by appeal to the Principle of Bivalence.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7)
     A reaction: Rumfitt proposes to avoid the truth-tables, and hence not to rely on Bivalence for his support of classical logic. He accepts that Bivalence is doubtful, citing the undecidability of the Continuum Hypothesis as a problem instance.
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
Logical consequence is a relation that can extended into further statements [Rumfitt]
     Full Idea: Logical consequence, I argue, is distinguished from other implication relations by the fact that logical laws may be applied in extending any implication relation so that it applies among some complex statements involving logical connectives.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 3.3)
     A reaction: He offers implication in electronics as an example of a non-logical implication relation. This seems to indicate that logic must be monotonic, that consequence is transitive, and that the Cut Law always applies.
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
Normal deduction presupposes the Cut Law [Rumfitt]
     Full Idea: Our deductive practices seem to presuppose the Cut Law.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 2.3)
     A reaction: That is, if you don't believe that deductions can be transitive (and thus form a successful chain of implications), then you don't really believe in deduction. It remains a well known fact that you can live without the Cut Law.
5. Theory of Logic / D. Assumptions for Logic / 1. Bivalence
When faced with vague statements, Bivalence is not a compelling principle [Rumfitt]
     Full Idea: I do not regard Bivalence, when applied to vague statements, as an intuitively compelling principle which we ought to try to preserve.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.7)
     A reaction: The point of Rumfitt's book is to defend classical logic despite failures of bivalence. He also cites undecidable concepts such as the Continuum Hypothesis.
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
In specifying a logical constant, use of that constant is quite unavoidable [Rumfitt]
     Full Idea: There is no prospect whatever of giving the sense of a logical constant without using that very constant, and much else besides, in the metalinguistic principle that specifies that sense.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1)
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
Introduction rules give deduction conditions, and Elimination says what can be deduced [Rumfitt]
     Full Idea: 'Introduction rules' state the conditions under which one may deduce a conclusion whose dominant logical operator is the connective. 'Elimination rules' state what may be deduced from some premises, where the major premise is dominated by the connective.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1)
     A reaction: So Introduction gives conditions for deduction, and Elimination says what can actually be deduced. If my magic wand can turn you into a frog (introduction), and so I turn you into a frog, how does that 'eliminate' the wand?
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
Logical truths are just the assumption-free by-products of logical rules [Rumfitt]
     Full Idea: Gentzen's way of formalising logic has accustomed people to the idea that logical truths are simply the by-products of logical rules, that arise when all the assumptions on which a conclusion rests have been discharged.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 2.5)
     A reaction: This is the key belief of those who favour the natural deduction account of logic. If you really believe in separate logic truths, then you can use them as axioms.
5. Theory of Logic / K. Features of Logics / 10. Monotonicity
Monotonicity means there is a guarantee, rather than mere inductive support [Rumfitt]
     Full Idea: Monotonicity seems to mark the difference between cases in which a guarantee obtains and those where the premises merely provide inductive support for a conclusion.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 2.3)
     A reaction: Hence it is plausible to claim that 'non-monotonic logic' is a contradiction in terms.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
Maybe an ordinal is a property of isomorphic well-ordered sets, and not itself a set [Rumfitt]
     Full Idea: Menzel proposes that an ordinal is something isomorphic well-ordered sets have in common, so while an ordinal can be represented as a set, it is not itself a set, but a 'property' of well-ordered sets.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.2)
     A reaction: [C.Menzel 1986] This is one of many manoeuvres available if you want to distance mathematics from set theory.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero
0 is not a number, as it answers 'how many?' negatively [Husserl, by Dummett]
     Full Idea: Husserl contends that 0 is not a number, on the grounds that 'nought' is a negative answer to the question 'how many?'.
     From: report of Edmund Husserl (Philosophy of Arithmetic [1894], p.144) by Michael Dummett - Frege philosophy of mathematics Ch.8
     A reaction: I seem to be in a tiny minority in thinking that Husserl may have a good point. One apple is different from one orange, but no apples are the same as no oranges. That makes 0 a very peculiar number. See Idea 9838.
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
Multiplicity in general is just one and one and one, etc. [Husserl]
     Full Idea: Multiplicity in general is no more than something and something and something, etc.; ..or more briefly, one and one and one, etc.
     From: Edmund Husserl (Philosophy of Arithmetic [1894], p.85), quoted by Gottlob Frege - Review of Husserl's 'Phil of Arithmetic'
     A reaction: Frege goes on to attack this idea fairly convincingly. It seems obvious that it is hard to say that you have seventeen items, if the only numberical concept in your possession is 'one'. How would you distinguish 17 from 16? What makes the ones 'multiple'?
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
Husserl said counting is more basic than Frege's one-one correspondence [Husserl, by Heck]
     Full Idea: Husserl famously argued that one should not explain number in terms of equinumerosity (or one-one correspondence), but should explain equinumerosity in terms of sameness of number, which should be characterised in terms of counting.
     From: report of Edmund Husserl (Philosophy of Arithmetic [1894]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 3
     A reaction: [Heck admits he hasn't read the Husserl] I'm very sympathetic to Husserl, though nearly all modern thinking favours Frege. Counting connects numbers to their roots in the world. Mathematicians seem oblivious of such things.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
Infinitesimals do not stand in a determinate order relation to zero [Rumfitt]
     Full Idea: Infinitesimals do not stand in a determinate order relation to zero: we cannot say an infinitesimal is either less than zero, identical to zero, or greater than zero. ….Infinitesimals are so close to zero as to be theoretically indiscriminable from it.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.4)
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry) [Rumfitt]
     Full Idea: One of the motivations behind Cantor's and Dedekind's pioneering explorations in the field was the ambition to give real analysis a new foundation in set theory - and hence a foundation independent of geometry.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.6)
     A reaction: Rumfitt is inclined to think that the project has failed, although a weaker set theory than ZF might do the job (within limits).
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
An object that is not clearly red or orange can still be red-or-orange, which sweeps up problem cases [Rumfitt]
     Full Idea: A borderline red-orange object satisfies the disjunctive predicate 'red or orange', even though it satisfies neither 'red' or 'orange'. When applied to adjacent bands of colour, the disjunction 'sweeps up' objects which are reddish-orange.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.5)
     A reaction: Rumfitt offers a formal principle in support of this. There may be a problem with 'adjacent'. Different colour systems will place different colours adjacent to red. In other examples the idea of 'adjacent' may make no sense. Rumfitt knows this!
The extension of a colour is decided by a concept's place in a network of contraries [Rumfitt]
     Full Idea: On Sainsbury's picture, a colour has an extension that it has by virtue of its place in a network of contrary colour classifications. Something is determined to be 'red' by being a colour incompatible with orange, yellow, green, blue, indigo and violet.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.5)
     A reaction: Along with Idea 18839, this gives quite a nice account of vagueness, by requiring a foil to the vague predicate, and using the disjunction of the predicate and its foil to handle anything caught in between them.
10. Modality / A. Necessity / 5. Metaphysical Necessity
Metaphysical modalities respect the actual identities of things [Rumfitt]
     Full Idea: The central characteristic mark of metaphysical necessity is that a metaphysical possibility respects the actual identities of things - in a capacious sense of 'thing'.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 3.4)
     A reaction: He contrast this with logical necessity, and concludes that some truths are metaphysically but not logically necessary, such as 'Hesperus is identical with Phosphorus'. Personally I like the idea of a 'necessity-maker', so that fits.
10. Modality / A. Necessity / 6. Logical Necessity
S5 is the logic of logical necessity [Rumfitt]
     Full Idea: I accept the widely held thesis that S5 is the logic of logical necessity.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 6.4 n16)
     A reaction: It seems plausible that S5 is also the logic of metaphysical necessity, but that does not make them the same thing. The two types of necessity have two different grounds.
10. Modality / B. Possibility / 1. Possibility
Since possibilities are properties of the world, calling 'red' the determination of a determinable seems right [Rumfitt]
     Full Idea: Some philosophers describe the colour scarlet as a determination of the determinable red; since the ways the world might be are naturally taken to be properties of the world, it helps to bear this analogy in mind.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 6.4)
     A reaction: This fits nicely with the disposition accounts of modality which I favour. Hence being 'coloured' is a real property of objects, even in the absence of the name of its specific colour.
If two possibilities can't share a determiner, they are incompatible [Rumfitt]
     Full Idea: Two possibilities are incompatible when no possibility determines both.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.1)
     A reaction: This strikes me as just the right sort of language for building up a decent metaphysical picture of the world, which needs to incorporate possibilities as well as actualities.
10. Modality / E. Possible worlds / 1. Possible Worlds / e. Against possible worlds
Possibilities are like possible worlds, but not fully determinate or complete [Rumfitt]
     Full Idea: Possibilities are things of the same general character as possible worlds, on one popular conception of the latter. They differ from worlds, though, in that they are not required to be fully determinate or complete.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 6)
     A reaction: A rather promising approach to such things, even though a possibility is fairly determinate at its core, but very vague at the edges. It is possible that the UK parliament might be located in Birmingham, for example. Is this world 'complete'?
11. Knowledge Aims / A. Knowledge / 2. Understanding
Medieval logicians said understanding A also involved understanding not-A [Rumfitt]
     Full Idea: Mediaeval logicians had a principle, 'Eadem est scientia oppositorum': in order to attain a clear conception of what it is for A to be the case, one needs to attain a conception of what it is for A not to be the case.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.2)
     A reaction: Presumably 'understanding' has to be a fairly comprehensive grasp of the matter, so understanding the negation sounds like a reasonable requirement for the real thing.
13. Knowledge Criteria / B. Internal Justification / 3. Evidentialism / a. Evidence
In English 'evidence' is a mass term, qualified by 'little' and 'more' [Rumfitt]
     Full Idea: In English, the word 'evidence' behaves as a mass term: we speak of someone's having little evidence for an assertion, and of one thinker's having more evidence than another for a claim. One the other hand, we also speak of 'pieces' of evidence.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 5.2)
     A reaction: And having 'more' evidence does not mean having a larger number of pieces of evidence, so it really is like an accumulated mass.
15. Nature of Minds / C. Capacities of Minds / 3. Abstraction by mind
Husserl identifies a positive mental act of unification, and a negative mental act for differences [Husserl, by Frege]
     Full Idea: Husserl identifies a 'unitary mental act' where several contents are connected or related to one another, and also a difference-relation where two contents are related to one another by a negative judgement.
     From: report of Edmund Husserl (Philosophy of Arithmetic [1894], p.73-74) by Gottlob Frege - Review of Husserl's 'Phil of Arithmetic' p.322
     A reaction: Frege is setting this up ready for a fairly vicious attack. Where Hume has a faculty for spotting resemblances, it is not implausible that we should also be hard-wired to spot differences. 'You look different; have you changed your hair style?'
16. Persons / F. Free Will / 2. Sources of Free Will
For Christians man has free will by creation in God's image (as in Genesis) [Frede,M]
     Full Idea: The Christian view, following Genesis, is that man is created in the image of God, and this is understood as crucially involving the idea that man has a free will in the image of God's will.
     From: Michael Frede (A Free Will [1997], 08)
     A reaction: The idea of free will evidently originated with Epictetus, but was taken up by Christians because it fitted doctrinal needs. Even Epictetus saw free will as originating in Zeus.
The idea of free will achieved universal acceptance because of Christianity [Frede,M]
     Full Idea: There is no doubt that the notion of a free will found almost universal acceptance owing to the influence of Christianity.
     From: Michael Frede (A Free Will [1997], 07)
     A reaction: This is presumably because a free will not only elevates us above the animals, qualifying us for immortality, but also gives us absolute and ultimate responsibility for our lives, which thus justifies either salvation or damnation.
The Stoics needed free will, to allow human choices in a divinely providential cosmos [Frede,M]
     Full Idea: The Stoics said that everything happens according to a divine providential plan, so they had to explain how this was compatible with human choices. They tried to do this with their doctrine of freedom and a free will.
     From: Michael Frede (A Free Will [1997], 10)
     A reaction: Epictetus made our ability to choose central to moral life, so he particularly needed (and thus created, it seems) this doctrine.
18. Thought / D. Concepts / 4. Structure of Concepts / b. Analysis of concepts
We clarify concepts (e.g. numbers) by determining their psychological origin [Husserl, by Velarde-Mayol]
     Full Idea: Husserl said that the clarification of any concept is made by determining its psychological origin. He is concerned with the psychological origins of the operation of calculating cardinal numbers.
     From: report of Edmund Husserl (Philosophy of Arithmetic [1894]) by Victor Velarde-Mayol - On Husserl 2.2
     A reaction: This may not be the same as the 'psychologism' that Frege so despised, because Husserl is offering a clarification, rather than the intrinsic nature of number concepts. It is not a theory of the origin of numbers.
18. Thought / E. Abstraction / 8. Abstractionism Critique
Psychologism blunders in focusing on concept-formation instead of delineating the concepts [Dummett on Husserl]
     Full Idea: Husserl substitutes his account of the process of concept-formation for a delineation of the concept. It is above all in making this substitution that psychologism is objectionable (and Frege opposed it so vehemently).
     From: comment on Edmund Husserl (Philosophy of Arithmetic [1894]) by Michael Dummett - Frege philosophy of mathematics Ch.2
     A reaction: While this is a powerful point which is a modern orthodoxy, it hardly excludes a study of concept-formation from being of great interest for other reasons. It may not appeal to logicians, but it is crucial part of the metaphysics of nature.
Husserl wanted to keep a shadowy remnant of abstracted objects, to correlate them [Dummett on Husserl]
     Full Idea: Husserl saw that abstracted units, though featureless, must in some way retain their distinctness, some shadowy remnant of their objects. So he wanted to correlate like-numbered sets, not just register their identity, but then abstractionism fails.
     From: comment on Edmund Husserl (Philosophy of Arithmetic [1894]) by Michael Dummett - Frege philosophy of mathematics Ch.12
     A reaction: Abstractionism is held to be between the devil and the deep blue sea, of depending on units which are identifiable, when they are defined as devoid of all individuality. We seem forced to say that the only distinction between them is countability.
19. Language / A. Nature of Meaning / 4. Meaning as Truth-Conditions
We understand conditionals, but disagree over their truth-conditions [Rumfitt]
     Full Idea: It is striking that our understanding of conditionals is not greatly impeded by widespread disagreement about their truth-conditions.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 4.2)
     A reaction: Compare 'if you dig there you might find gold' with 'if you dig there you will definitely find gold'. The second but not the first invites 'how do you know that?', implying truth. Two different ifs.
19. Language / F. Communication / 3. Denial
The truth grounds for 'not A' are the possibilities incompatible with truth grounds for A [Rumfitt]
     Full Idea: The truth-grounds of '¬A' are precisely those possibilities that are incompatible with any truth-ground of A.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.1)
     A reaction: This is Rumfitt's proposal for the semantics of 'not', based on the central idea of a possibility, rather than a possible world. The incompatibility tracks back to an absence of shared grounding.
20. Action / B. Preliminaries of Action / 2. Willed Action / a. Will to Act
There is no will for Plato or Aristotle, because actions come directly from perception of what is good [Frede,M]
     Full Idea: Plato and Aristotle do not have a notion of a will, since for them a willing, a desire of reason, is a direct result of one's cognitive state: once one sees something to be good, one will it.
     From: Michael Frede (A Free Will [1997], 09)
     A reaction: The point is that their decisions are 'direct', whereas the will introduces the concept of a final arbiter which weighs up the desires, reasons and drives. The historical steps were first the meta-choosing, then the will as entity, then the will as free.
29. Religion / A. Polytheistic Religion / 4. Dualist Religion
The Gnostic demiurge (creator) is deluded, and doesn't care about us [Frede,M]
     Full Idea: The Gnostics thought the demiurge or creator pursues its own interests without regard for what this does to us, a being lacking in wisdom and goodness, as seen by its deluding itself into thinking that it is God, and demanding worship.
     From: Michael Frede (A Free Will [1997], Intro)
     A reaction: Frede mentions Irenaeus as a source of this view. The idea that the Great Being doesn't care about us seems a fairly accurate observation.