25 ideas
| 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) |
| 4568 | If 'Queen of England' does not refer if there is no queen, its meaning can't refer if there is one [Cooper,DE] |
| Full Idea: If 'the Queen of England' is not a referring expression when there is no queen, nor can it be one when there is a queen - since the meaning of the expression is the same in either case. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §4.1) | |
| A reaction: I'm not convinced. Does this mean that since I can point with my finger at nothing, I therefore do not indicate anything when there is an object at which I am pointing. Sounds silly to me. |
| 4574 | If some peoples do not have categories like time or cause, they can't be essential features of rationality [Cooper,DE] |
| Full Idea: If our most basic concepts, like time, space, substance or causality, are not shared by some peoples, it puts paid to the cherished ideal of philosophers to discover a set of concepts or categories which any rational human must employ in his thinking. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §5.2) | |
| A reaction: This seems to be a place where a priori philosophy (Aristotle,Kant,Hegel) meets empirical research (Whorf). However, interpreting the research is so fraught with problems it drives you back to the a priori… |
| 4573 | If it is claimed that language correlates with culture, we must be able to identify the two independently [Cooper,DE] |
| Full Idea: If it is claimed that linguistic differences significantly correlate with cultural differences, it must therefore be possible to identify the linguistic differences independently from the cultural ones. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §5.1) | |
| A reaction: This is a basic objection to any extreme relativist version of the S-P hypothesis. They are part of the conspiracy to overemphasise language in philosophy, and they are wrong. |
| 4575 | A person's language doesn't prove their concepts, but how are concepts deduced apart from language? [Cooper,DE] |
| Full Idea: It would be absurd to say the Hopi lack the concept of time because they lack tensed verbs, ..but how do we find out what a man's concepts are except in terms of his language? | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §5.2) | |
| A reaction: Presumably we should look at animals, where concepts must be inferred in order to explain behaviour. I don't see why introspection (scientifically wicked) should not also be employed to detect our own non-verbal concepts. How are new words invented? |
| 4561 | Many sentences set up dispositions which are irrelevant to the meanings of the sentences [Cooper,DE] |
| Full Idea: Many sentences set up dispositions which are irrelevant to the meanings of the sentences. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §2.3) | |
| A reaction: Yet another telling objection to behaviourism. When I look at broccoli I may have a disposition to be sick, but that isn't part of the concept of broccoli. |
| 19087 | The meaning or purport of a symbol is all the rational conduct it would lead to [Peirce] |
| Full Idea: The entire intellectual purport of any symbol consists in the total of all modes of rational conduct which, conditionally upon all the possible different circumstances and desires, would ensue upon the acceptance of the symbol. | |
| From: Charles Sanders Peirce (Issues of Pragmaticism [1905], EP ii.246), quoted by Danielle Macbeth - Pragmatism and Objective Truth p.169 n1 | |
| A reaction: Macbeth says pragmatism is founded on this theory of meaning, rather than on a theory of truth. I don't see why the causes of a symbol shouldn't be as much a part of its meaning as the consequences are. |
| 4564 | I can meaningfully speculate that humans may have experiences currently impossible for us [Cooper,DE] |
| Full Idea: It is not meaningless for me to postulate the potential for humans to sense in a manner which is at present unimaginable and indescribable. There is no reason to believe me, but I might be right. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §3.1) | |
| A reaction: The key counterexample to verificationist theories of meaning is wild speculations, which are clearly meaningful, though frequently far beyond any likely human experience. Logical positivists are allergic to imagination. |
| 4565 | The verification principle itself seems neither analytic nor verifiable [Cooper,DE] |
| Full Idea: It seems that the positivists must admit that there is at least one statement which is meaningful, but which is neither verifiable nor analytic - namely, the statement of the principle of verification itself. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §3.1) | |
| A reaction: Some people think this objection is decisive, but I think any theory must be permitted a few metatheoretic assertions or axioms which are beyond discussion. Ayer thought the VP might be treated as analytic. Everyone has to start somewhere. |
| 4562 | Most people know how to use the word "Amen", but they do not know what it means [Cooper,DE] |
| Full Idea: Most people know how to use the word "Amen", but they do not know what it means. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §2.4) | |
| A reaction: Personally I find examples like this decisive against the 'use' theory of meaning. Maybe the defence is that the theory works for sentences, and individual words (like passwords) are peripheral. |
| 4563 | 'How now brown cow?' is used for elocution, but this says nothing about its meaning [Cooper,DE] |
| Full Idea: The sentence 'How now brown cow?' has its use in elocutions classes, yet this aspect of its use tells us nothing about its meaning. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §2.4) | |
| A reaction: Indeed, and also there are weird sentence of which we can assemble a meaning, but cannot think of any conceivable use ('rats swim in purple marmalade'). |
| 4571 | Reference need not be a hit-or-miss affair [Cooper,DE] |
| Full Idea: Reference need not be a hit-or-miss affair. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §4.2) | |
| A reaction: Sounds right. If the basic scenario is picking someone out in a crowd, your listener may think they know which person you are talking about, with a high degree of probability. |
| 4566 | Any thesis about reference is also a thesis about what exists to be referred to [Cooper,DE] |
| Full Idea: Any thesis about reference is also going to be a thesis about what there is in existence to refer to. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §4) | |
| A reaction: I see the point, but we must not put the cart before the horse. I may have an intuition that something exists, but not know how to refer to it (because of my small vocabulary). |
| 4572 | If predicates name things, that reduces every sentence to a mere list of names [Cooper,DE] |
| Full Idea: If predicates are names of entities, then subject/predicate sentences are pairs of names, since subjects are names (or referring expressions). But a pair of names is not a sentence at all, it is a mere list. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §4.4) | |
| A reaction: If that is meant to demolish universals it is too quick. Concatenating names is not the same as listing them. A relationship is asserted. There is a (mysterious) Platonic 'partaking' between form and particular. Perhaps. |
| 4576 | An analytic truth is one which becomes a logical truth when some synonyms have been replaced [Cooper,DE] |
| Full Idea: The definition of analytic truth which has, I believe, the most chance of success is one in terms of synonymy; ..an analytic truth is one which can be transformed into a logical truth once synonyms are replaced by synonyms. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §7.1) | |
| A reaction: Sounds promising, though there is Quine's notorious problem of circularity in all these concepts. If synonymy is conventional, then so is analyticity. I personally feel that the circle can be broken. |