18 ideas
| 19053 | Logic would be more natural if negation only referred to predicates [Dummett] |
| Full Idea: A better proposal for a formal logic closer to natural language would be one that had a negation-operator only for (simple) predicates. | |
| From: Michael Dummett (Presupposition [1960], p.27) | |
| A reaction: Dummett observes that classical formal logic was never intended to be close to natural language. Term logic does have that aim, but the meta-question is whether that end is desirable, and why. |
| 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) |
| 19052 | Natural language 'not' doesn't apply to sentences [Dummett] |
| Full Idea: Natural language does not possess a sentential negation-operator. | |
| From: Michael Dummett (Presupposition [1960], p.27) | |
| A reaction: This is a criticism of Strawson, who criticises logic for not following natural language, but does it himself with negation. In the question of how language and logic connect, this idea seems important. Term Logic aims to get closer to natural language. |
| 8430 | Causal statements are used to explain, to predict, to control, to attribute responsibility, and in theories [Kim] |
| Full Idea: The function of causal statements is 1) to explain events, 2) for predictive usefulness, 3) to help control events, 4) with agents, to attribute moral responsibility, 5) in physical theory. We should judge causal theories by how they account for these. | |
| From: Jaegwon Kim (Causes and Counterfactuals [1973], p.207) | |
| A reaction: He suggests that Lewis's counterfactual theory won't do well on this test. I think the first one is what matters. Philosophy aims to understand, and that is achieved through explanation. Regularity and counterfactual theories explain very little. |
| 8396 | Many counterfactuals have nothing to do with causation [Kim, by Tooley] |
| Full Idea: Kim has pointed out that there are a number of counterfactuals that have nothing to do with causation. If John marries Mary, then if John had not existed he would not have married Mary, but that is not the cause of their union. | |
| From: report of Jaegwon Kim (Causes and Counterfactuals [1973], 5.2) by Michael Tooley - Causation and Supervenience | |
| A reaction: One might not think that this mattered, but it leaves the problem of distinguishing between the causal counterfactuals and the rest (and you mustn't mention causation when you are doing it!). |
| 8429 | Counterfactuals can express four other relations between events, apart from causation [Kim] |
| Full Idea: Counterfactuals can express 'analytical' dependency, or the fact that one event is part of another, or an action done by doing another, or (most interestingly) an event can determine another without causally determining it. | |
| From: Jaegwon Kim (Causes and Counterfactuals [1973], p.205) | |
| A reaction: [Kim gives example of each case] Counterfactuals can even express a relation that involves no dependency. Or they might just involve redescription, as in 'If Scott were still alive, then the author of "Waverley" would be too'. |
| 8428 | Causation is not the only dependency relation expressed by counterfactuals [Kim] |
| Full Idea: The sort of dependency expressed by counterfactual relations is considerably broader than strictly causal dependency, and causal dependency is only one among the heterogeneous group of dependency relationships counterfactuals can express. | |
| From: Jaegwon Kim (Causes and Counterfactuals [1973], p.205) | |
| A reaction: In 'If pigs could fly, one and one still wouldn't make three' there isn't even a dependency. Kim has opened up lines of criticism which make the counterfactual analysis of causation look very implausible to me. |
| 4781 | Many counterfactual truths do not imply causation ('if yesterday wasn't Monday, it isn't Tuesday') [Kim, by Psillos] |
| Full Idea: Kim gives a range of examples of counterfactual dependence without causation, as: 'if yesterday wasn't Monday, today wouldn't be Tuesday', and 'if my sister had not given birth, I would not be an uncle'. | |
| From: report of Jaegwon Kim (Causes and Counterfactuals [1973]) by Stathis Psillos - Causation and Explanation §3.3 | |
| A reaction: This is aimed at David Lewis. The objection seems like commonsense. "If you blink, the cat gets it". Causal claims involve counterfactuals, but they are not definitive of what causation is. |