32 ideas
| 6222 | If a decision is in accord with right reason, everyone can agree with it [Cumberland] |
| Full Idea: No decision can be in accord with right reason unless all can agree on it. | |
| From: Richard Cumberland (De Legibus Naturae [1672], Ch.V.XLVI) | |
| A reaction: Personally I think anyone who disagrees with this should get out of philosophy (and into sociology, fantasy fiction, ironic game-playing, crime…). Of course 'can' agree is not the same as 'will' agree. You must have faith that good reasons are persuasive. |
| 3035 | Dialectic involves conversations with short questions and brief answers [Diog. Laertius] |
| Full Idea: Dialectic is when men converse by putting short questions and giving brief answers to those who question them. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 3.1.52) |
| 17749 | Post proved the consistency of propositional logic in 1921 [Walicki] |
| Full Idea: A proof of the consistency of propositional logic was given by Emil Post in 1921. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2.1) |
| 17765 | Propositional language can only relate statements as the same or as different [Walicki] |
| Full Idea: Propositional language is very rudimentary and has limited powers of expression. The only relation between various statements it can handle is that of identity and difference. As are all the same, but Bs can be different from As. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 7 Intro) | |
| A reaction: [second sentence a paraphrase] In predicate logic you could represent two statements as being the same except for one element (an object or predicate or relation or quantifier). |
| 17764 | Boolean connectives are interpreted as functions on the set {1,0} [Walicki] |
| Full Idea: Boolean connectives are interpreted as functions on the set {1,0}. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 5.1) | |
| A reaction: 1 and 0 are normally taken to be true (T) and false (F). Thus the functions output various combinations of true and false, which are truth tables. |
| 17752 | The empty set is useful for defining sets by properties, when the members are not yet known [Walicki] |
| Full Idea: The empty set is mainly a mathematical convenience - defining a set by describing the properties of its members in an involved way, we may not know from the very beginning what its members are. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1) |
| 17753 | The empty set avoids having to take special precautions in case members vanish [Walicki] |
| Full Idea: Without the assumption of the empty set, one would often have to take special precautions for the case where a set happened to contain no elements. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1) | |
| A reaction: Compare the introduction of the concept 'zero', where special precautions are therefore required. ...But other special precautions are needed without zero. Either he pays us, or we pay him, or ...er. Intersecting sets need the empty set. |
| 17759 | Ordinals play the central role in set theory, providing the model of well-ordering [Walicki] |
| Full Idea: Ordinals play the central role in set theory, providing the paradigmatic well-orderings. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
| A reaction: When you draw the big V of the iterative hierarchy of sets (built from successive power sets), the ordinals are marked as a single line up the middle, one ordinal for each level. |
| 17741 | To determine the patterns in logic, one must identify its 'building blocks' [Walicki] |
| Full Idea: In order to construct precise and valid patterns of arguments one has to determine their 'building blocks'. One has to identify the basic terms, their kinds and means of combination. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History Intro) | |
| A reaction: A deceptively simple and important idea. All explanation requires patterns and levels, and it is the idea of building blocks which makes such things possible. It is right at the centre of our grasp of everything. |
| 17747 | A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki] |
| Full Idea: A specification of a domain of objects, and of the rules for interpreting the symbols of a logical language in this domain such that all the theorems of the logical theory are true is said to be a 'model' of the theory. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.1.3) | |
| A reaction: The basic ideas of this emerged 1915-30, but it needed Tarski's account of truth to really get it going. |
| 17748 | The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki] |
| Full Idea: The L-S Theorem is ...a shocking result, since it implies that any consistent formal theory of everything - even about biology, physics, sets or the real numbers - can just as well be understood as being about natural numbers. It says nothing more. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2) | |
| A reaction: Illuminating. Particularly the point that no theory about the real numbers can say anything more than a theory about the natural numbers. So the natural numbers contain all the truths we can ever express? Eh????? |
| 17761 | A compact axiomatisation makes it possible to understand a field as a whole [Walicki] |
| Full Idea: Having such a compact [axiomatic] presentation of a complicated field [such as Euclid's], makes it possible to relate not only to particular theorems but also to the whole field as such. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
| 17763 | Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki] |
| Full Idea: Axiomatic systems, their primitive terms and proofs, are purely syntactic, that is, do not presuppose any interpretation. ...[142] They never address the world directly, but address a possible semantic model which formally represents the world. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
| 17758 | Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki] |
| Full Idea: An ordinal can be defined as a transitive set of transitive sets, or else, as a transitive set totally ordered by set inclusion. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
| 17755 | Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki] |
| Full Idea: The collection of ordinals is defined inductively: Basis: the empty set is an ordinal; Ind: for an ordinal x, the union with its singleton is also an ordinal; and any arbitrary (possibly infinite) union of ordinals is an ordinal. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
| A reaction: [symbolism translated into English] Walicki says they are called 'ordinal numbers', but are in fact a set. |
| 17756 | The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki] |
| Full Idea: We can form infinite ordinals by taking unions of ordinals. We can thus form 'limit ordinals', which have no immediate predecessor. ω is the first (the union of all finite ordinals), ω + ω = sω is second, 3ω the third.... | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
| 17760 | Two infinite ordinals can represent a single infinite cardinal [Walicki] |
| Full Idea: There may be several ordinals for the same cardinality. ...Two ordinals can represent different ways of well-ordering the same number (aleph-0) of elements. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
| A reaction: This only applies to infinite ordinals and cardinals. For the finite, the two coincide. In infinite arithmetic the rules are different. |
| 17757 | Members of ordinals are ordinals, and also subsets of ordinals [Walicki] |
| Full Idea: Every member of an ordinal is itself an ordinal, and every ordinal is a transitive set (its members are also its subsets; a member of a member of an ordinal is also a member of the ordinal). | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
| 17762 | In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki] |
| Full Idea: Since non-Euclidean geometry preserves all Euclid's postulates except the fifth one, all the theorems derived without the use of the fifth postulate remain valid. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
| 17754 | Inductive proof depends on the choice of the ordering [Walicki] |
| Full Idea: Inductive proof is not guaranteed to work in all cases and, particularly, it depends heavily on the choice of the ordering. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.1.1) | |
| A reaction: There has to be an well-founded ordering for inductive proofs to be possible. |
| 17742 | Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki] |
| Full Idea: The link between time and modality was severed by Duns Scotus, who proposed a notion of possibility based purely on the notion of semantic consistency. 'Possible' means for him logically possible, that is, not involving contradiction. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History B.4) |
| 1816 | Sceptics say demonstration depends on self-demonstrating things, or indemonstrable things [Diog. Laertius] |
| Full Idea: Sceptics say that every demonstration depends on things which demonstrates themselves, or on things which can't be demonstrated. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 9.Py.11) | |
| A reaction: This refers to two parts of Agrippa's Trilemma (the third being that demonstration could go on forever). He makes the first option sound very rationalist, rather than experiential. |
| 1819 | Scepticism has two dogmas: that nothing is definable, and every argument has an opposite argument [Diog. Laertius] |
| Full Idea: Sceptics actually assert two dogmas: that nothing should be defined, and that every argument has an opposite argument. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 9.Py.11) |
| 3064 | When sceptics say that nothing is definable, or all arguments have an opposite, they are being dogmatic [Diog. Laertius] |
| Full Idea: When sceptics say that they define nothing, and that every argument has an opposite argument, they here give a positive definition, and assert a positive dogma. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 9.11.11) |
| 3033 | Induction moves from some truths to similar ones, by contraries or consequents [Diog. Laertius] |
| Full Idea: Induction is an argument which by means of some admitted truths establishes naturally other truths which resemble them; there are two kinds, one proceeding from contraries, the other from consequents. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 3.1.23) |
| 6217 | Natural law is supplied to the human mind by reality and human nature [Cumberland] |
| Full Idea: Some truths of natural law, concerning guides to moral good and evil, and duties not laid down by civil law and government, are necessarily supplied ot the human mind by the nature of things and of men. | |
| From: Richard Cumberland (De Legibus Naturae [1672], Ch.I.I) | |
| A reaction: I agree that some moral truths have the power of self-evidence. If you say they are built into the mind, we now ask what did the building, and evolution is the only answer, and hence we distance ourselves from the truths, seeing them as strategies. |
| 6221 | If there are different ultimate goods, there will be conflicting good actions, which is impossible [Cumberland] |
| Full Idea: If there be posited different ultimate ends, whose causes are opposed to each other, then there will be truly good actions likewise opposed to each other, which is impossible. | |
| From: Richard Cumberland (De Legibus Naturae [1672], Ch.V.XVI) | |
| A reaction: A very interesting argument for there being one good rather than many, and an argument which I don't recall in any surviving Greek text. A response might be to distinguish between what is 'right' and what is 'good'. See David Ross. |
| 1838 | Cyrenaic pleasure is a motion, but Epicurean pleasure is a condition [Diog. Laertius] |
| Full Idea: Cyrenaics place pleasure wholly in motion, whereas Epicurus admits it as a condition. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 10.28) | |
| A reaction: Not a distinction we meet in modern discussions. Do events within the mind count as 'motion'? If so, these two agree. If not, I'd vote for Epicurus. |
| 1769 | Cynics believe that when a man wishes for nothing he is like the gods [Diog. Laertius] |
| Full Idea: Cynics believe that when a man wishes for nothing he is like the gods. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 6.Men.3) |
| 6218 | The happiness of individuals is linked to the happiness of everyone (which is individuals taken together) [Cumberland] |
| Full Idea: The happiness of each person cannot be separated from the happiness of all, because the whole is no different from the parts taken together. | |
| From: Richard Cumberland (De Legibus Naturae [1672], Ch.I.VI) | |
| A reaction: Sounds suspiciously like the fallacy of composition (Idea 6219). An objection to utilitarianism is its assumption that a group of people have a 'total happiness' that is different from their individual states. Still, Cumberland is on to utilitarianism. |
| 6220 | The happiness of all contains the happiness of each, and promotes it [Cumberland] |
| Full Idea: The common happiness of all contains the greatest happiness for each, and most effectively promotes it. …There is no path leading anyone to his own happiness, other than the path which leads all to the common happiness. | |
| From: Richard Cumberland (De Legibus Naturae [1672], Ch.I.VI) | |
| A reaction: I take this as a revolutionary idea, which leads to utilitarianism. It is doing what seemed to the Greeks unthinkable, which is combining hedonism with altruism. There is no proof for it, but it is a wonderful clarion call for building a civil society. |
| 6216 | Natural law is immutable truth giving moral truths and duties independent of society [Cumberland] |
| Full Idea: Natural law is certain propositions of immutable truth, which guide voluntary actions about the choice of good and avoidance of evil, and which impose an obligation to act, even without regard to civil laws, and ignoring compacts of governments. | |
| From: Richard Cumberland (De Legibus Naturae [1672], Ch.I.I) | |
| A reaction: Not a popular view, but I am sympathetic. If you are in a foreign country and find a person lying in pain, there is a terrible moral deficiency in anyone who just ignores such a thing. No legislation can take away a person's right of self-defence. |