29 ideas
| 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. |
| 15201 | That Queen Anne is dead is a 'general fact', not a fact about Queen Anne [Prior,AN] |
| Full Idea: The fact that Queen Anne has been dead for some years is not, in the strict sense of 'about', a fact about Queen Anne; it is not a fact about anyone or anything - it is a general fact. | |
| From: Arthur N. Prior (Changes in Events and Changes in Things [1968], p.13), quoted by Robin Le Poidevin - Past, Present and Future of Debate about Tense 1 b | |
| A reaction: He distinguishes 'general facts' (states of affairs, I think) from 'individual facts', involving some specific object. General facts seem to be what are expressed by negative existential truths, such as 'there is no Loch Ness Monster'. Useful. |
| 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) |
| 7590 | Consequentialism emphasises value rather than obligation in morality [Scruton] |
| Full Idea: According to consequentialism, the fundamental concept of morality is not obligation (deontological ethics) but value (axiological ethics). | |
| From: Roger Scruton (A Dictionary of Political Thought [1982], 'consequentialism') | |
| A reaction: These two views could come dramatically apart, in wartime, or in big ecological crises, or in a family breakup, or in religious disputes. Having identified the pair so clearly, why can we not aim for a civilised (virtuous) balance between the two? |
| 7589 | Altruism is either emotional (where your interests are mine) or moral (where they are reasons for me) [Scruton] |
| Full Idea: Two distinct motives go by the name of altruism: the emotions of liking, love and friendship, making another's interest automatically mine; and the moral motive of respect or considerateness, where another's interests become reasons for me, but not mine. | |
| From: Roger Scruton (A Dictionary of Political Thought [1982], 'altruism') | |
| A reaction: The second one has a strongly Kantian flavour, with its notion of impersonal duty. Virtue theorists will aspire to achieve the first state rather than the second, because good actions are then actively desired, and give pleasure to the doer. |
| 7595 | The idea of a right seems fairly basic; justice may be the disposition to accord rights to people [Scruton] |
| Full Idea: The idea of a right seems to be as basic as any other; we might even define justice in terms of it, as the disposition to accord to every person his rights. | |
| From: Roger Scruton (A Dictionary of Political Thought [1982], 'rights') | |
| A reaction: I am inclined to think that a set of fairly pure values (such as equality, kindness, sympathy, respect) must be in place before the idea of a right would occur to anyone. Aristotle has a powerful moral sense, but rights for slaves don't cross his mind. |
| 7588 | Allegiance is fundamental to the conservative view of society [Scruton] |
| Full Idea: Conservatives have made the concept of allegiance, conceived as a power, fundamental to their description of the experience of society | |
| From: Roger Scruton (A Dictionary of Political Thought [1982], 'allegiance') | |
| A reaction: This provokes the famous slogan of "My country - right or wrong!" However, the issue here is not going to be decided by a consequentialist analysis, but by a view a of human nature. I think I would want to carefully prise allegiance apart from loyalty. |
| 7594 | Democrats are committed to a belief and to its opposite, if the majority prefer the latter [Scruton] |
| Full Idea: The paradox of democracy (emphasised by Rousseau) is that I am compelled by my belief in democracy to embrace conflicting - perhaps even contradictory - opinions. If I believe A, and the majority vote for B, I am committed to enacting them both. | |
| From: Roger Scruton (A Dictionary of Political Thought [1982], 'paradox of democracy') | |
| A reaction: The paradox would have to be resolved by qualifying what exactly one is committed to by being a democrat. I would say I am committed to the right of my opponents to enact a policy with which I disagree. |
| 7593 | Liberals focus on universal human freedom, natural rights, and tolerance [Scruton, by PG] |
| Full Idea: Liberalism believes (roughly) in the supremacy of the individual, who has freedom and natural rights; it focuses on human, not divine affairs; it claims rights and duties are universal; and it advocates tolerance in religion and morality. | |
| From: report of Roger Scruton (A Dictionary of Political Thought [1982], 'liberalism') by PG - Db (ideas) | |
| A reaction: I find it hard to disagree with these principles, but the upshot in practice is often an excessive commitment to freedom and tolerance, because people fail to realise the subtle long-term erosions of society that can result. |
| 7592 | For positivists law is a matter of form, for naturalists it is a matter of content [Scruton] |
| Full Idea: For the positivist, law is law by virtue of its form; for the naturalist, by virtue of its content. | |
| From: Roger Scruton (A Dictionary of Political Thought [1982], 'law') | |
| A reaction: Clearly a perverse and 'unnatural' social rule (backed by government and implied force) is a 'law' in some sense of the word. It is hard to see how you could gain social consensus for a law if it didn't appear in some way to be 'natural justice'. |
| 7587 | The issue of abortion seems insoluble, because there is nothing with which to compare it [Scruton] |
| Full Idea: The issue of abortion is intractable, partly because of the absence of any other case to which it can be assimilated. | |
| From: Roger Scruton (A Dictionary of Political Thought [1982], 'abortion') | |
| A reaction: This is the legalistic approach to the problem, which always looks for precedents and comparisons. All problems must hav solutions, though (mustn't they?). The problem, though, is not the value of the foetus, but the unique form of 'ownership'. |
| 22899 | 'Thank goodness that's over' is not like 'thank goodness that happened on Friday' [Prior,AN] |
| Full Idea: One says 'thank goodness that is over', ..and it says something which it is impossible which any use of any tenseless copula with a date should convey. It certainly doesn't mean the same as 'thank goodness that occured on Friday June 15th 1954'. | |
| From: Arthur N. Prior (Changes in Events and Changes in Things [1968]), quoted by Adrian Bardon - Brief History of the Philosophy of Time 4 'Pervasive' | |
| A reaction: [Ref uncertain] This seems to be appealing to ordinary usage, in which tenses have huge significance. If we take time (with its past, present and future) as primitive, then tenses can have full weight. Did tenses mean anything at all to Einstein? |