42 ideas
| 6575 | Philosophy may never find foundations, and may undermine our lives in the process [Fogelin] |
| Full Idea: Not only is traditional philosophy incapable of discovering the foundations it seeks, but the philosophical enterprise may itself dislodge the contingent, de facto supports that our daily life depends upon. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.2) | |
| A reaction: In the end Fogelin is not so pessimistic, but he is worried by the concern of philosophers with paradox and contradiction. I don't remotely consider this a reason to reject philosophy, but it might be a reason to keep it sealed off from daily life. |
| 6585 | Rationality is threatened by fear of inconsistency, illusions of absolutes or relativism, and doubt [Fogelin] |
| Full Idea: The three main threats to our rational lives are fear of inconsistency, illusions (of absolutism and relativism) and doubt. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.4) | |
| A reaction: This is a very nice analysis of the forces that can destroy the philosopher's aspiration to the rational life. Personally I still suffer from a few illusions about the possibility of absolutes, but I may grow out of it. The other three don't bother me. |
| 6557 | Humans may never be able to attain a world view which is both rich and consistent [Fogelin] |
| Full Idea: It might be wholly unreasonable to suppose that human beings will ever be able to attain a view of the world that is both suitably rich and completely consistent. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Intro) | |
| A reaction: Fogelin's lectures develop this view very persuasively. I think all philosophers must believe that the gods could attain a 'rich and consistent' view. Our problem is that we are a badly organised team, whose members keep dying. |
| 6568 | A game can be played, despite having inconsistent rules [Fogelin] |
| Full Idea: The presence of an inconsistency in the rules that govern a game need not destroy the game. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.2) | |
| A reaction: He only defends this thesis if the inconsistency is away from the main centre of the action. You can't have an inconsistent definition of scoring a goal or a touchdown. |
| 6560 | The law of noncontradiction is traditionally the most basic principle of rationality [Fogelin] |
| Full Idea: Traditionally many philosophers (Aristotle among them) have considered the law of noncontradiction to be the deepest, most fundamental principle of rationality. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.1) | |
| A reaction: For Aristotle, see Idea 1601 (and 'Metaphysics' 1005b28). The only denier of the basic character of the law that I know of is Nietzsche (Idea 4531). Fogelin, despite many qualifications, endorses the law, and so do I. |
| 6565 | The law of noncontradiction makes the distinction between asserting something and denying it [Fogelin] |
| Full Idea: People who reject the law of noncontradiction obliterate any significant difference between asserting something and denying it; …this will not move anyone who genuinely opts either for silence or for madness. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.1) | |
| A reaction: This seems a sufficiently firm and clear assertion of the basic nature of this law. The only rival view seems to be that of Nietzsche (Idea 4531), but then you wonder how Nietzsche is in a position to assert the relativity of the law. |
| 6574 | Legal reasoning is analogical, not deductive [Fogelin] |
| Full Idea: There is almost universal agreement that legal reasoning is fundamentally analogical, not deductive, in character. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.2) | |
| A reaction: This raises the question of whether analogy can be considered as 'reasoning' in itself. How do you compare the examples? Could you compare two examples if you lacked language, or rules, or a scale of values? |
| 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. |
| 21598 | Austin revealed many meanings for 'vague': rough, ambiguous, general, incomplete... [Austin,JL, by Williamson] |
| Full Idea: Austin's account brought out the variety of features covered by 'vague' in different contexts: roughness, ambiguity, imprecision, lack of detail, generality, inaccuracy, incompleteness. Even 'vague' is vague. | |
| From: report of J.L. Austin (Sense and Sensibilia [1962], p.125-8) by Timothy Williamson - Vagueness 3.1 | |
| A reaction: Some of these sound the same. Maybe Austin distinguishes them. |
| 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) |
| 6582 | Conventions can only work if they are based on something non-conventional [Fogelin] |
| Full Idea: Convention, to exist at all, must have a basis in something that is not conventional; conventions, to work, need something nonconventional to build upon and shape. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.3) | |
| A reaction: Fogelin attributes his point to Hume. I agree entirely. No convention could ever possibly catch on in a society unless there were some point to it. If you can't see a point to a convention (like wearing ties) then start looking, because it's there. |
| 6576 | My view is 'circumspect rationalism' - that only our intellect can comprehend the world [Fogelin] |
| Full Idea: My own view might be called 'circumspect rationalism' - the view that our intellectual faculties provide our only means for comprehending the world in which we find oruselves. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.3) | |
| A reaction: He needs to say more than that to offer a theory, but I like the label, and it fits the modern revival of rationalism, with which I sympathise, and which rests, I think, on Russell's point that self-evidence comes in degrees, not as all-or-nothing truth. |
| 6589 | Knowledge is legitimate only if all relevant defeaters have been eliminated [Fogelin] |
| Full Idea: In general a knowledge claim is legitimate only if all relevant defeaters have been eliminated. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.4) | |
| A reaction: The problem here is what is 'relevant'. Fogelin's example is 'Are you sure the suspect doesn't have a twin brother?' If virtual reality is relevant, most knowledge is defeated. Certainly, imaginative people feel that they know less than others. |
| 6596 | For coherentists, circularity is acceptable if the circle is large, rich and coherent [Fogelin] |
| Full Idea: Coherentists argue that if the circle of justifications is big enough, rich enough, coherent enough, and so on, then there is nothing wrong circularity. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.4) | |
| A reaction: There must always be something wrong with circularity, and no god would put up with it, but we might have to. Of course, two pieces of evidence might be unconnected, such as an equation and an observation. |
| 6597 | A rule of justification might be: don't raise the level of scrutiny without a good reason [Fogelin] |
| Full Idea: One rule for the justification of knowledge might be: Do not raise the level of scrutiny in the absence of a particular reason that triggers it. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.4) | |
| A reaction: That won't decide the appropriate level of scrutiny from which to start. One of my maxims is 'don't set the bar too high', but it seems tough that one should have to justify moving it. The early scientists tried raising it, and were amazed by the results. |
| 6588 | Scepticism is cartesian (sceptical scenarios), or Humean (future), or Pyrrhonian (suspend belief) [Fogelin] |
| Full Idea: The three forms of scepticism are cartesian, Humean and Pyrrhonian. The first challenges belief by inventing sceptical scenarios; the second doubts the future; the third aims to suspend belief. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.4) | |
| A reaction: A standard distinction is made between methodological and global scepticism. The former seems to be Cartesian, and the latter Pyrrhonian. The interest here is see Hume placed in a distinctive category, because of his views on induction. |
| 6590 | Scepticism deals in remote possibilities that are ineliminable and set the standard very high [Fogelin] |
| Full Idea: Sceptical scenarios deal in wildly remote defeating possibilities, so that the level of scrutiny becomes unrestrictedly high, and they also usually deal with defeators that are in principle ineliminable. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.4) | |
| A reaction: The question of how high we 'set the bar' seems to me central to epistemology. There is clearly an element of social negotiation involved, centring on what is appropriate. If, though, scepticism is 'ineliminable', we must face up to that. |
| 6583 | Radical perspectivism replaces Kant's necessary scheme with many different schemes [Fogelin] |
| Full Idea: We reach radical perspectivism by replacing Kant's single, necessary categorial scheme with a plurality of competing categorial schemes. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.3) | |
| A reaction: It certainly looks as if Kant sent us down a slippery slope into the dafter aspects of twentieth century relativism. The best antidote I know of is Davidson's (e.g. Idea 6398). But then it seems unimaginative to say that only one scheme is possible. |
| 6555 | We are also irrational, with a unique ability to believe in bizarre self-created fictions [Fogelin] |
| Full Idea: We as human beings are also irrational animals, unique among animals in our capacity to place faith in bizarre fictions of our own construction. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Intro) | |
| A reaction: This is glaringly true, and a very nice corrective to the talk of Greeks and others about man as the 'rational animal'. From a distance we might be described by Martians as the 'mad animal'. Is the irrational current too strong to swim against? |
| 6605 | Critics must be causally entangled with their subject matter [Fogelin] |
| Full Idea: Critics must become causally entangled with their subject matter. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.6) | |
| A reaction: This remark is built on Hume's views. You may have a strong view about a singer, but it may be hard to maintain when someone plays you six rival versions of the same piece. I agree entirely with the remark. It means there are aesthetic experts. |
| 6607 | The word 'beautiful', when deprived of context, is nearly contentless [Fogelin] |
| Full Idea: Like the word 'good', the word 'beautiful', when deprived of contextual support, is nearly contentless. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.6) | |
| A reaction: If I say with, for example, Oscar Wilde that beauty is the highest ideal in life, this doesn't strike me as contentless, but I still sympathise with Fogelin's notion that beauty is rooted in particulars. |
| 6604 | Saying 'It's all a matter to taste' ignores the properties of the object discussed [Fogelin] |
| Full Idea: "It is all a matter of taste" may be an all-purpose stopper of discussions of aesthetic values, but it also completely severs the connection with the actual properties of the object under consideration. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.6) | |
| A reaction: This remark grows out of his discussion of Hume. I like this remark, which ties in with Particularism in morality, and with the central role of experiments in science. The world forces beliefs on us. |
| 6586 | Cynics are committed to morality, but disappointed or disgusted by human failings [Fogelin] |
| Full Idea: Cynics are usually unswerving in their commitment to a moral ideal, but disappointed or disgusted by humanity's failure to meet it. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.3) | |
| A reaction: I felt quite suicidal the other day when I saw someone park diagonally across two parking spaces. They can't seem to grasp the elementary Kantian slogan 'What if everybody did that?' It's all hopeless. I wonder if I am becoming a bit of a Cynic? |
| 6572 | Deterrence, prevention, rehabilitation and retribution can come into conflict in punishments [Fogelin] |
| Full Idea: The purposes of punishment include deterrence, prevention, rehabilitation, and retribution, but they don't always sit well together. Deterrence is best served by making prisons miserable places, but this may run counter to rehabilitation. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.2) | |
| A reaction: It seems to most educated people that retribution should be pushed far down the list if we are to be civilised (see Idea 1659), and yet personal revenge for a small act of aggression seems basic, normal and acceptable. We dream of rehabilitation. |
| 6573 | Retributivists say a crime can be 'paid for'; deterrentists still worry about potential victims [Fogelin] |
| Full Idea: A strict retributivist is likely to say that once a crime is paid for, that's that; a deterrence theorist is likely to say that the protection of potential victims overrides the released convict's right to a free and fresh start. | |
| From: Robert Fogelin (Walking the Tightrope of Reason [2003], Ch.2) | |
| A reaction: Interesting since the retributivist here has the more liberal attitude. Reformists will also have a dilemma when years in prison have failed to reform the convict. Virtue theorists like balance, and sensitively consider our relations with the criminals. |