21 ideas
7396 | Hobbes created English-language philosophy [Hobbes, by Tuck] |
Full Idea: Hobbes created English-language philosophy. | |
From: report of Thomas Hobbes (The Elements of Law [1640]) by Richard Tuck - Hobbes Pref | |
A reaction: Tuck mentions Hooker as a predecessor in jurisprudence. Otherwise, an impressive label. |
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) |
16638 | The qualities of the world are mere appearances; reality is the motions which cause them [Hobbes] |
Full Idea: Whatsoever accidents or qualities our senses make us think there be in the world, they are not there, but are seemings and apparitions only. The things that really are in the world without us are those motions by which these seemings are caused. | |
From: Thomas Hobbes (The Elements of Law [1640], I.2.10), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 10.2 | |
A reaction: This seems to count as a sense-datum theory, rather than a representative theory of perception, since it makes no commitment to the qualities containing any accurate information at all. We just start from the qualities and try to work it out. |
16688 | Evidence is conception, which is imagination, which proceeds from the senses [Hobbes] |
Full Idea: All evidence is conception, as it is said, and all conception is imagination and proceeds from sense. And spirits we suppose to be those substances which work not upon the sense, and therefore not conceptible. | |
From: Thomas Hobbes (The Elements of Law [1640], I.11.5), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 16.2 | |
A reaction: This is exactly the same as Hume's claim that all ideas are the result of impressions, and is the very essence of empiricism. We see here that such an epistemology can have huge consequences. |
7405 | Experience can't prove universal truths [Hobbes] |
Full Idea: Experience concludeth nothing universally. | |
From: Thomas Hobbes (The Elements of Law [1640], I.4.10), quoted by Richard Tuck - Hobbes Ch.2 | |
A reaction: Empiricists seem proud to claim this limitation on human understanding, where rationalists like Leibniz use it as an argument against empiricism. Kripke says (e.g. Idea 4966) they are both wrong! I sympathise with Kripke. |
7408 | It is an error that reason should control the passions, which give right guidance on their own [Hobbes, by Tuck] |
Full Idea: Hobbes (and Descartes, and many contemporaries) argued that the traditional idea that reason should control the passions was an error, and that (properly understood) our emotions would guide us in the right direction. | |
From: report of Thomas Hobbes (The Elements of Law [1640]) by Richard Tuck - Hobbes Ch.2 | |
A reaction: I'm an intellectualist on this one. It strikes me as rather naïve and romantic to think that unthinking emotion could ever consistently approach what is right. A recipe for disaster. |
468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
7407 | Good and evil are what please us; goodness and badness the powers causing them [Hobbes] |
Full Idea: We call good and evil the things that please and displease us; and so we call goodness and badness, the qualities of powers whereby they do it. | |
From: Thomas Hobbes (The Elements of Law [1640], I.7.3), quoted by Richard Tuck - Hobbes Ch.2 | |
A reaction: It is pointed out by Tuck that this is just like his treatment of colour terms (values as secondary qualities). I would have thought it was obvious that I could say 'x pleases me, although I disapprove of it' (e.g. black humour). |
7410 | Self-preservation is basic, and people judge differently about that, implying ethical relativism [Hobbes, by Tuck] |
Full Idea: If men are their own judges of what conduces to their preservation, ..all men make different decisions about what counts as a danger, so (for Hobbes) the grimmest version of ethical relativism seems to be the only possible ethical vision. | |
From: report of Thomas Hobbes (The Elements of Law [1640]) by Richard Tuck - Hobbes Ch.2 | |
A reaction: This might depend on self-preservation being the only fundamental value. But if self-preservation is not a pressing issue, presumably other values might come into play, some of them less concerned with the individual's own interests. |
7409 | Hobbes shifted from talk of 'the good' to talk of 'rights' [Hobbes, by Tuck] |
Full Idea: Hobbes (like Grotius) shifted from talking about 'the good', which had been the traditional subject for both ancient and Renaissance moralists, to talking instead about 'rights'. | |
From: report of Thomas Hobbes (The Elements of Law [1640]) by Richard Tuck - Hobbes Ch.2 | |
A reaction: This is part of the crucial shift away from the Greek interest in excellence of character, towards the Enlightenment legalistic interest in right actions, as well as social rights. Bad move, well analysed by MacIntyre. |
7411 | The attributes of God just show our inability to conceive his nature [Hobbes] |
Full Idea: All the attributes of God signify our inability and defect of power to conceive any thing concerning his nature. | |
From: Thomas Hobbes (The Elements of Law [1640], I.10.2), quoted by Richard Tuck - Hobbes Ch.2 | |
A reaction: Presumably he means that 'omnipotence' should just be translated as 'mind-boggling power'. St Anselm's concept of God (Idea 1405) is helpful here, placing it at the upper limit of what can actually be conceived. |