28 ideas
| 6161 | Structuralism is neo-Kantian idealism, with language playing the role of categories of understanding [Rowlands] |
| Full Idea: Structuralism is a form of neo-Kantian idealism, in which the job of creating Kant's phenomenal world has been taken over by language instead of forms of sensibility and categories of the understanding. | |
| From: Mark Rowlands (Externalism [2003], Ch.3) | |
| A reaction: A helpful connection, which explains my aversion to any attempt at understanding the world simply by analysing language, either in its ordinary usage, or in its underlying logical form. |
| 13030 | Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y) [Kunen] |
| Full Idea: Axiom of Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y). That is, a set is determined by its members. If every z in one set is also in the other set, then the two sets are the same. | |
| From: Kenneth Kunen (Set Theory [1980], §1.5) |
| 13032 | Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z) [Kunen] |
| Full Idea: Axiom of Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z). Any pair of entities must form a set. | |
| From: Kenneth Kunen (Set Theory [1980], §1.6) | |
| A reaction: Repeated applications of this can build the hierarchy of sets. |
| 13033 | Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A) [Kunen] |
| Full Idea: Axiom of Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A). That is, the union of a set (all the members of the members of the set) must also be a set. | |
| From: Kenneth Kunen (Set Theory [1980], §1.6) |
| 13037 | Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x) [Kunen] |
| Full Idea: Axiom of Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x). That is, there is a set which contains zero and all of its successors, hence all the natural numbers. The principal of induction rests on this axiom. | |
| From: Kenneth Kunen (Set Theory [1980], §1.7) |
| 13038 | Power Set: ∀x ∃y ∀z(z ⊂ x → z ∈ y) [Kunen] |
| Full Idea: Power Set Axiom: ∀x ∃y ∀z(z ⊂ x → z ∈ y). That is, there is a set y which contains all of the subsets of a given set. Hence we define P(x) = {z : z ⊂ x}. | |
| From: Kenneth Kunen (Set Theory [1980], §1.10) |
| 13034 | Replacement: ∀x∈A ∃!y φ(x,y) → ∃Y ∀X∈A ∃y∈Y φ(x,y) [Kunen] |
| Full Idea: Axiom of Replacement Scheme: ∀x ∈ A ∃!y φ(x,y) → ∃Y ∀X ∈ A ∃y ∈ Y φ(x,y). That is, any function from a set A will produce another set Y. | |
| From: Kenneth Kunen (Set Theory [1980], §1.6) |
| 13039 | Foundation:∀x(∃y(y∈x) → ∃y(y∈x ∧ ¬∃z(z∈x ∧ z∈y))) [Kunen] |
| Full Idea: Axiom of Foundation: ∀x (∃y(y ∈ x) → ∃y(y ∈ x ∧ ¬∃z(z ∈ x ∧ z ∈ y))). Aka the 'Axiom of Regularity'. Combined with Choice, it means there are no downward infinite chains. | |
| From: Kenneth Kunen (Set Theory [1980], §3.4) |
| 13036 | Choice: ∀A ∃R (R well-orders A) [Kunen] |
| Full Idea: Axiom of Choice: ∀A ∃R (R well-orders A). That is, for every set, there must exist another set which imposes a well-ordering on it. There are many equivalent versions. It is not needed in elementary parts of set theory. | |
| From: Kenneth Kunen (Set Theory [1980], §1.6) |
| 13029 | Set Existence: ∃x (x = x) [Kunen] |
| Full Idea: Axiom of Set Existence: ∃x (x = x). This says our universe is non-void. Under most developments of formal logic, this is derivable from the logical axioms and thus redundant, but we do so for emphasis. | |
| From: Kenneth Kunen (Set Theory [1980], §1.5) |
| 13031 | Comprehension: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ) [Kunen] |
| Full Idea: Comprehension Scheme: for each formula φ without y free, the universal closure of this is an axiom: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ). That is, there must be a set y if it can be defined by the formula φ. | |
| From: Kenneth Kunen (Set Theory [1980], §1.5) | |
| A reaction: Unrestricted comprehension leads to Russell's paradox, so restricting it in some way (e.g. by the Axiom of Specification) is essential. |
| 13040 | Constructibility: V = L (all sets are constructible) [Kunen] |
| Full Idea: Axiom of Constructability: this is the statement V = L (i.e. ∀x ∃α(x ∈ L(α)). That is, the universe of well-founded von Neumann sets is the same as the universe of sets which are actually constructible. A possible axiom. | |
| From: Kenneth Kunen (Set Theory [1980], §6.3) |
| 9463 | Classical logic is bivalent, has excluded middle, and only quantifies over existent objects [Jacquette] |
| Full Idea: Classical logic (of Whitehead, Russell, Gödel, Church) is a two-valued system of propositional and predicate logic, in which all propositions are exclusively true or false, and quantification and predication are over existent objects only. | |
| From: Dale Jacquette (Intro to I: Classical Logic [2002], p.9) | |
| A reaction: All of these get challenged at some point, though the existence requirement is the one I find dubious. |
| 6163 | If bivalence is rejected, then excluded middle must also be rejected [Rowlands] |
| Full Idea: If you reject the principle of bivalence (that a proposition is either determinately true or false), then statements are also not subject to the Law of Excluded Middle (P or not-P). | |
| From: Mark Rowlands (Externalism [2003], Ch.3) | |
| A reaction: I think Rowlands is wrong about this. Excluded Middle could be purely syntacti, or its semantics could be 'True or Not-True'. Only bivalent excluded middle introduces 'True or False'. Compare Idea 4752. |
| 6155 | Supervenience is a one-way relation of dependence or determination between properties [Rowlands] |
| Full Idea: Supervenience is essentially a one-way relation of dependence or determination, …which holds, in the first instance, between properties. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: This definition immediately shows why supervenient properties are in danger of being epiphenomenal (i.e. causally irrelevant). Carefully thought about the notion of a 'one-way' relation will, I think, make it more obscure rather than clearer. |
| 6154 | It is argued that wholes possess modal and counterfactual properties that parts lack [Rowlands] |
| Full Idea: Some have argued that a mereological whole should not be identified with the sum of its parts on the grounds that the former possess certain properties - specifically modal and (perhaps) counterfactual properties - that the latter lacks. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: I am not convinced that modal and counterfactual claims should count as properties. If my pen is heated it melts (a property), but if my pen were intelligent it could do philosophy. Intelligence is a property, but the situation isn't. |
| 6157 | Tokens are dated, concrete particulars; types are their general properties or kinds [Rowlands] |
| Full Idea: Tokens are dated, concrete, particular occurrences or instances; types are the general properties that these occurrences exemplify or the kinds to which they belong. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: It might be said that types are sets, of which tokens are the members. The question of 'general properties' raises the question of whether universals must exist to make kinds possible. |
| 6159 | Strong idealism is the sort of mess produced by a Cartesian separation of mind and world [Rowlands] |
| Full Idea: Neo-Kantian idealism, and the excesses of recent versions of it, are precisely the sort of mess one can get oneself into through an uncritical acceptance of the dichotomizing of mind and world along Cartesian internalist lines. | |
| From: Mark Rowlands (Externalism [2003], Ch.3) | |
| A reaction: I am unconvinced that internalism about the mind (that its contents can be defined without reference to anything external) leads to this disastrous split. We don't have to abandon the links between an internal mind and the world. |
| 6152 | Minds are rational, conscious, subjective, self-knowing, free, meaningful and self-aware [Rowlands] |
| Full Idea: The apparent features of mind which are not obviously physical include: rationality, thought, consciousness, subjectivity, infallible first-person knowledge, freedom, meaning and self-awareness. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: A helpful list, some of which can be challenged. Ryle challenges first-person infallibility. Hume challenges self-awareness. Quine challenges meaning. Lots of people (e.g. Spinoza) challenge freedom. The Churchlands seem to challenge consciousness. |
| 6173 | Content externalism implies that we do not have privileged access to our own minds [Rowlands] |
| Full Idea: Content externalism threatens the idea of first-person authority in all its forms, and does so because it calls into question the idea that the access we have to our own mental states is privileged in the way required for such authority. | |
| From: Mark Rowlands (Externalism [2003], Ch.7) | |
| A reaction: I am inclined to respond by saying that since we clearly have privileged access to our own minds, that means there must be something wrong with content externalism. |
| 6174 | If someone is secretly transported to Twin Earth, others know their thoughts better than they do [Rowlands] |
| Full Idea: If someone knew that a thinker had, without realising it, been transported to Twin Earth, they would almost certainly be a higher authority on the content of the thinker's thoughts than would the thinker. | |
| From: Mark Rowlands (Externalism [2003], Ch.8) | |
| A reaction: They would certainly be a higher authority on the truth of the thinker's thoughts, but only in the way that you might think I hold a diamond when I know it is a club. If the thinker believes it is H2O, the fact that it isn't is irrelevant to content. |
| 6158 | Supervenience of mental and physical properties often comes with token-identity of mental and physical particulars [Rowlands] |
| Full Idea: One often finds a supervenience thesis concerning the relation between mental and physical properties combined with a token identity theory concerning the relation between mental and physical particulars. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: This brings out the important clarifying point that supervenience is said to be between properties, not substances. The point is that supervenience will always cry out for an explanation, preferably a sensible one. |
| 6168 | The content of a thought is just the meaning of a sentence [Rowlands] |
| Full Idea: The content of the thought that the sky is blue is simply the meaning of the sentence "The sky is blue". | |
| From: Mark Rowlands (Externalism [2003], Ch.5) | |
| A reaction: This seems to imply that it is logically impossible for a non-language-speaker, such as a chimpanzee, to think that the sky is the same colour as the water. If we allow propositions, we might be able to keep meanings without the sentences. |
| 6167 | Action is bodily movement caused by intentional states [Rowlands] |
| Full Idea: An action is a bodily movement that is caused by intentional states such as beliefs, desires and so on. | |
| From: Mark Rowlands (Externalism [2003], Ch.5) | |
| A reaction: A useful definition, and clearly one that has no truck with attempts at giving behaviourist definitions of action. The definition of a 'moral action' needs to be built on this one. Particular types of belief and desire, presumably. |
| 6177 | Moral intuition seems unevenly distributed between people [Rowlands] |
| Full Idea: The faculty of moral intuition seems to be unevenly distributed between people. | |
| From: Mark Rowlands (Externalism [2003], Ch.11) | |
| A reaction: This would be a good argument if it was thought that the source of moral intuitions was divine, but people vary enormously in their intuitions about maths, about character, about danger. If you believe in any intuition at all, you must accept its variety. |
| 6156 | The 17th century reintroduced atoms as mathematical modes of Euclidean space [Rowlands] |
| Full Idea: The seventeenth century revolution reintroduced the classical concept of the atom in somewhat new attire as an essentially mathematical entity whose primary qualities could be precisely quantified as modes or aspects of Euclidean space. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: Obviously this very abstract view of atoms didn't last, once they began to identify specific physical atoms, such as oxygen. This view fits in with Newton's use of pure (abstract) points such as the 'centre of gravity'. |
| 6170 | Natural kinds are defined by their real essence, as in gold having atomic number 79 [Rowlands] |
| Full Idea: Part of what it means to be a natural kind is that they are defined by a real essence, a constitution that marks them out as the substance they are (as water is essentially H2O, and gold essentially has atomic number 79). | |
| From: Mark Rowlands (Externalism [2003], Ch.6) | |
| A reaction: A 'real essence' would be the opposite of a 'conventional essence', which is just a human way of seeing things. |
| 6178 | It is common to see the value of nature in one feature, such as life, diversity, or integrity [Rowlands] |
| Full Idea: In recent environmental philosophy it is common to see the value of nature identified with one or another natural feature of the environment: life, diversity, ecosystemic integrity and so on. | |
| From: Mark Rowlands (Externalism [2003], Ch.11) | |
| A reaction: This thought seems to be asking for the Open Question argument. What is so good about life, or diversity? Our strongest intuition must be that the survival of the ecosystem, and whatever makes that possible, is the highest value. |