58 ideas
| 3240 | There is more insight in fundamental perplexity about problems than in their supposed solutions [Nagel] |
| Full Idea: Certain forms of perplexity (say about freedom, knowledge and the meaning of life) seem to me to embody more insight than any of the supposed solutions to those problems. | |
| From: Thomas Nagel (The View from Nowhere [1986], Intro) | |
| A reaction: Obviously false solutions won't embody much insight. This sounds good, but I suspect that the insight is in the recognition of the facts which give rise to the perplexity. I can't think of anything in favour of perplexity for its own sake. |
| 3242 | Philosophy is the childhood of the intellect, and a culture can't skip it [Nagel] |
| Full Idea: Philosophy is the childhood of the intellect, and a culture that tries to skip it will never grow up. | |
| From: Thomas Nagel (The View from Nowhere [1986], Intro) | |
| A reaction: Can he really mean that a mature culture doesn't need philosophy? |
| 3241 | It seems mad, but the aim of philosophy is to climb outside of our own minds [Nagel] |
| Full Idea: We are trying to climb outside of our own minds, an effort that some would regard as insane and that I regard as philosophically fundamental. | |
| From: Thomas Nagel (The View from Nowhere [1986], Intro) | |
| A reaction: It is not only philosophers who do this. It is an essential feature of the mind, and is inherent in the concept of truth. |
| 3248 | Realism invites scepticism because it claims to be objective [Nagel] |
| Full Idea: The search for objective knowledge, because of its commitment to realism, cannot refute scepticism and must proceed under its shadow, and scepticism is only a problem because of the realist claims of objectivity. | |
| From: Thomas Nagel (The View from Nowhere [1986], V.1) |
| 20989 | Views are objective if they don't rely on a person's character, social position or species [Nagel] |
| Full Idea: A view or form of thought is more objective than another if it relies less on the specifics of the individual's makeup and position in the world, or on the character of the particular type of creature he is. | |
| From: Thomas Nagel (The View from Nowhere [1986], Intro) | |
| A reaction: Notice that this defines comparative objectivity, rather than an absolute. I take it that something must be entirely objective to qualify as a 'fact', and so anything about which there is a consensus that it is a fact can be taken as wholly objective. |
| 22354 | Things cause perceptions, properties have other effects, hence we reach a 'view from nowhere' [Nagel, by Reiss/Sprenger] |
| Full Idea: First we realise that perceptions are caused by things, second we realise that properties have other effects (as well as causing perceptions), and third we conceive of a thing's true nature without perspectives. That is the 'view from nowhere'. | |
| From: report of Thomas Nagel (The View from Nowhere [1986], p.14) by Reiss,J/Spreger,J - Scientific Objectivity 2.1 | |
| A reaction: [My summary of their summary] This is obviously an optimistic view. I''m not sure how he can justify three precise stages, given than animals probably jump straight to the third stage, and engage with the nature's of things. |
| 9724 | Until the 1960s the only semantics was truth-tables [Enderton] |
| Full Idea: Until the 1960s standard truth-table semantics were the only ones that there were. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.10.1) | |
| A reaction: The 1960s presumably marked the advent of possible worlds. |
| 9705 | 'fld R' indicates the 'field' of all objects in the relation [Enderton] |
| Full Idea: 'fld R' indicates the 'field' of a relation, that is, the set of all objects that are members of ordered pairs on either side of the relation. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9704 | 'ran R' indicates the 'range' of objects being related to [Enderton] |
| Full Idea: 'ran R' indicates the 'range' of a relation, that is, the set of all objects that are members of ordered pairs and that are related to by the first objects. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9703 | 'dom R' indicates the 'domain' of objects having a relation [Enderton] |
| Full Idea: 'dom R' indicates the 'domain' of a relation, that is, the set of all objects that are members of ordered pairs and that have that relation. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9710 | We write F:A→B to indicate that A maps into B (the output of F on A is in B) [Enderton] |
| Full Idea: We write F : A → B to indicate that A maps into B, that is, the domain of relating things is set A, and the things related to are all in B. If we add that F = B, then A maps 'onto' B. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9707 | 'F(x)' is the unique value which F assumes for a value of x [Enderton] |
|
Full Idea:
F(x) is a 'function', which indicates the unique value which y takes in |
|
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9700 | Two sets are 'disjoint' iff their intersection is empty [Enderton] |
| Full Idea: Two sets are 'disjoint' iff their intersection is empty (i.e. they have no members in common). | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9699 | The 'powerset' of a set is all the subsets of a given set [Enderton] |
| Full Idea: The 'powerset' of a set is all the subsets of a given set. Thus: PA = {x : x ⊆ A}. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9701 | A 'relation' is a set of ordered pairs [Enderton] |
| Full Idea: A 'relation' is a set of ordered pairs. The ordering relation on the numbers 0-3 is captured by - in fact it is - the set of ordered pairs {<0,1>,<0,2>,<0,3>,<1,2>,<1,3>,<2,3>}. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) | |
| A reaction: This can't quite be a definition of order among numbers, since it relies on the notion of a 'ordered' pair. |
| 9702 | A 'domain' of a relation is the set of members of ordered pairs in the relation [Enderton] |
| Full Idea: The 'domain' of a relation is the set of all objects that are members of ordered pairs that are members of the relation. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9708 | A function 'maps A into B' if the relating things are set A, and the things related to are all in B [Enderton] |
| Full Idea: A function 'maps A into B' if the domain of relating things is set A, and the things related to are all in B. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9709 | A function 'maps A onto B' if the relating things are set A, and the things related to are set B [Enderton] |
| Full Idea: A function 'maps A onto B' if the domain of relating things is set A, and the things related to are set B. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9711 | A relation is 'reflexive' on a set if every member bears the relation to itself [Enderton] |
| Full Idea: A relation is 'reflexive' on a set if every member of the set bears the relation to itself. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9712 | A relation is 'symmetric' on a set if every ordered pair has the relation in both directions [Enderton] |
| Full Idea: A relation is 'symmetric' on a set if every ordered pair in the set has the relation in both directions. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9713 | A relation is 'transitive' if it can be carried over from two ordered pairs to a third [Enderton] |
| Full Idea: A relation is 'transitive' on a set if the relation can be carried over from two ordered pairs to a third. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9706 | A 'function' is a relation in which each object is related to just one other object [Enderton] |
| Full Idea: A 'function' is a relation which is single-valued. That is, for each object, there is only one object in the function set to which that object is related. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9714 | A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects [Enderton] |
| Full Idea: A relation satisfies 'trichotomy' on a set if every ordered pair is related (in either direction), or the objects are identical. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9717 | A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second [Enderton] |
| Full Idea: A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9716 | We 'partition' a set into distinct subsets, according to each relation on its objects [Enderton] |
| Full Idea: Equivalence classes will 'partition' a set. That is, it will divide it into distinct subsets, according to each relation on the set. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9715 | An 'equivalence relation' is a reflexive, symmetric and transitive binary relation [Enderton] |
| Full Idea: An 'equivalence relation' is a binary relation which is reflexive, and symmetric, and transitive. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9722 | Inference not from content, but from the fact that it was said, is 'conversational implicature' [Enderton] |
| Full Idea: The process is dubbed 'conversational implicature' when the inference is not from the content of what has been said, but from the fact that it has been said. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.7.3) |
| 9718 | Validity is either semantic (what preserves truth), or proof-theoretic (following procedures) [Enderton] |
| Full Idea: The point of logic is to give an account of the notion of validity,..in two standard ways: the semantic way says that a valid inference preserves truth (symbol |=), and the proof-theoretic way is defined in terms of purely formal procedures (symbol |-). | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.3..) | |
| A reaction: This division can be mirrored in mathematics, where it is either to do with counting or theorising about things in the physical world, or following sets of rules from axioms. Language can discuss reality, or play word-games. |
| 9721 | A logical truth or tautology is a logical consequence of the empty set [Enderton] |
| Full Idea: A is a logical truth (tautology) (|= A) iff it is a semantic consequence of the empty set of premises (φ |= A), that is, every interpretation makes A true. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.3.4) | |
| A reaction: So the final column of every line of the truth table will be T. |
| 9994 | A truth assignment to the components of a wff 'satisfy' it if the wff is then True [Enderton] |
| Full Idea: A truth assignment 'satisfies' a formula, or set of formulae, if it evaluates as True when all of its components have been assigned truth values. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.2) | |
| A reaction: [very roughly what Enderton says!] The concept becomes most significant when a large set of wff's is pronounced 'satisfied' after a truth assignment leads to them all being true. |
| 9719 | A proof theory is 'sound' if its valid inferences entail semantic validity [Enderton] |
| Full Idea: If every proof-theoretically valid inference is semantically valid (so that |- entails |=), the proof theory is said to be 'sound'. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.7) |
| 9720 | A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity [Enderton] |
| Full Idea: If every semantically valid inference is proof-theoretically valid (so that |= entails |-), the proof-theory is said to be 'complete'. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.7) |
| 9995 | Proof in finite subsets is sufficient for proof in an infinite set [Enderton] |
| Full Idea: If a wff is tautologically implied by a set of wff's, it is implied by a finite subset of them; and if every finite subset is satisfiable, then so is the whole set of wff's. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5) | |
| A reaction: [Enderton's account is more symbolic] He adds that this also applies to models. It is a 'theorem' because it can be proved. It is a major theorem in logic, because it brings the infinite under control, and who doesn't want that? |
| 9996 | Expressions are 'decidable' if inclusion in them (or not) can be proved [Enderton] |
| Full Idea: A set of expressions is 'decidable' iff there exists an effective procedure (qv) that, given some expression, will decide whether or not the expression is included in the set (i.e. doesn't contradict it). | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.7) | |
| A reaction: This is obviously a highly desirable feature for a really reliable system of expressions to possess. All finite sets are decidable, but some infinite sets are not. |
| 9997 | For a reasonable language, the set of valid wff's can always be enumerated [Enderton] |
| Full Idea: The Enumerability Theorem says that for a reasonable language, the set of valid wff's can be effectively enumerated. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5) | |
| A reaction: There are criteria for what makes a 'reasonable' language (probably specified to ensure enumerability!). Predicates and functions must be decidable, and the language must be finite. |
| 13768 | Validity can preserve certainty in mathematics, but conditionals about contingents are another matter [Edgington] |
| Full Idea: If your interest in logic is confined to applications to mathematics or other a priori matters, it is fine for validity to preserve certainty, ..but if you use conditionals when arguing about contingent matters, then great caution will be required. | |
| From: Dorothy Edgington (Conditionals [2001], 17.2.1) |
| 13770 | There are many different conditional mental states, and different conditional speech acts [Edgington] |
| Full Idea: As well as conditional beliefs, there are conditional desires, hopes, fears etc. As well as conditional statements, there are conditional commands, questions, offers, promises, bets etc. | |
| From: Dorothy Edgington (Conditionals [2001], 17.3.4) |
| 13764 | Are conditionals truth-functional - do the truth values of A and B determine the truth value of 'If A, B'? [Edgington] |
| Full Idea: Are conditionals truth-functional - do the truth values of A and B determine the truth value of 'If A, B'? Are they non-truth-functional, like 'because' or 'before'? Do the values of A and B, in some cases, leave open the value of 'If A,B'? | |
| From: Dorothy Edgington (Conditionals [2001], 17.1) | |
| A reaction: I would say they are not truth-functional, because the 'if' asserts some further dependency relation that goes beyond the truth or falsity of A and B. Logical ifs, causal ifs, psychological ifs... The material conditional ⊃ is truth-functional. |
| 13765 | 'If A,B' must entail ¬(A & ¬B); otherwise we could have A true, B false, and If A,B true, invalidating modus ponens [Edgington] |
| Full Idea: If it were possible to have A true, B false, and If A,B true, it would be unsafe to infer B from A and If A,B: modus ponens would thus be invalid. Hence 'If A,B' must entail ¬(A & ¬B). | |
| From: Dorothy Edgington (Conditionals [2001], 17.1) | |
| A reaction: This is a firm defence of part of the truth-functional view of conditionals, and seems unassailable. The other parts of the truth table are open to question, though, if A is false, or they are both true. |
| 9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton] |
| Full Idea: Not all sentences using 'if' are conditionals. Consider 'if you want a banana, there is one in the kitchen'. The rough test is that a conditional can be rewritten as 'that A implies that B'. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.6.4) |
| 22429 | We achieve objectivity by dropping secondary qualities, to focus on structural primary qualities [Nagel] |
| Full Idea: At the end [of the three stages of objectivity] the secondary qualities drop out of our picture of the external world, and the underlyiing primary qualities such as shape, size, weight, and motion are thought of structurally. | |
| From: Thomas Nagel (The View from Nowhere [1986], II) | |
| A reaction: This is the orthodox view for realists about the external world, and I largely agree. The only problem I see is that secondary qualities contain information, such as the colour of rotting fruit - but then colour is not an essential feature of rot. |
| 3249 | Modern science depends on the distinction between primary and secondary qualities [Nagel] |
| Full Idea: The distinction between primary and secondary qualities is the precondition for the development of modern physics and chemistry. | |
| From: Thomas Nagel (The View from Nowhere [1986], V.3) |
| 3247 | Epistemology is centrally about what we should believe, not the definition of knowledge [Nagel] |
| Full Idea: The central problem of epistemology is what to believe and how to justify one's beliefs, not the impersonal problem of whether my beliefs can be said to be knowledge. | |
| From: Thomas Nagel (The View from Nowhere [1986], V.1) | |
| A reaction: Wrong. The question of whether what one has is 'knowledge' is not impersonal at all - it is having the social status of a knower or expert. |
| 3252 | Scepticism is based on ideas which scepticism makes impossible [Nagel] |
| Full Idea: The sceptic reaches scepticism through thoughts that scepticism makes unthinkable. | |
| From: Thomas Nagel (The View from Nowhere [1986], V.6) |
| 3251 | Observed regularities are only predictable if we assume hidden necessity [Nagel] |
| Full Idea: Observed regularities provide reason to believe that they will be repeated only to the extent that they provide evidence of hidden necessary connections, which hold timelessly. | |
| From: Thomas Nagel (The View from Nowhere [1986], V.5) |
| 3244 | Personal identity cannot be fully known a priori [Nagel] |
| Full Idea: The full conditions of personal identity cannot be extracted from the concept of a person at all: they cannot be arrived at a priori. | |
| From: Thomas Nagel (The View from Nowhere [1986], III.2) | |
| A reaction: However, if you turn to experience to get the hang of what a person is, it is virtually impossible to disentangle the essentials from the accidental features of being a person. How essential are memories or reasoning or hopes or understandings or plans? |
| 3245 | The question of whether a future experience will be mine presupposes personal identity [Nagel] |
| Full Idea: The identity of the self must have some sort of objectivity, otherwise the subjective question whether a future experience will be mine or not will be contentless. | |
| From: Thomas Nagel (The View from Nowhere [1986], III.3) | |
| A reaction: This sounds a bit circular and question-begging. If there is no objective self, then the question of whether a future experience will be mine would be a misconceived question. I sympathise with Nagel's attempt to show how personal identity is a priori. |
| 3246 | I can't even conceive of my brain being split in two [Nagel] |
| Full Idea: It is hard to think of myself as being identical with my brain. If my brain is to be split, with one half miserable and the other half euphoric, my expectations can take no form, as my idea of myself doesn't allow for divisibility. | |
| From: Thomas Nagel (The View from Nowhere [1986], III.4) | |
| A reaction: Nagel is trying to imply that there is some sort of conceptual impossibility here, but it may just be very difficult. I can think about my lovely lunch while doing my miserable job. Does Nagel want to hang on to a unified thing which doesn't exist? |
| 3257 | Total objectivity can't see value, but it sees many people with values [Nagel] |
| Full Idea: A purely objective view has no way of knowing whether anything has any value, but actually its data include the appearance of value to individuals with particular perspectives, including oneself. | |
| From: Thomas Nagel (The View from Nowhere [1986], VIII.2) | |
| A reaction: I would have thought that a very objective assessment of someone's health is an obvious revelation of value, irrespective of anyone's particular perspective. |
| 3265 | We don't worry about the time before we were born the way we worry about death [Nagel] |
| Full Idea: We do not regard the period before we were born in the same way that we regard the prospect of death. | |
| From: Thomas Nagel (The View from Nowhere [1986], XI.3) | |
| A reaction: This is a challenge to Epicurus, who said death is no worse than pre-birth. This idea may be true of the situation immediately post-death, but a thousand years from now it is hard to distinguish them. |
| 3263 | If our own life lacks meaning, devotion to others won't give it meaning [Nagel] |
| Full Idea: If no one's life has any meaning in itself, how can it acquire meaning through devotion to the meaningless lives of others? | |
| From: Thomas Nagel (The View from Nowhere [1986], XI.2) | |
| A reaction: This is one of the paradoxes of compassion. The other is that the virtue requires other people to be in need of help, which can't be a desirable situation. |
| 3256 | Pain doesn't have a further property of badness; it gives a reason for its avoidance [Nagel] |
| Full Idea: The objective badness of pain is not some mysterious further property that all pains have, but just the fact that there is reason for anyone capable of viewing the world objectively to want it to stop. | |
| From: Thomas Nagel (The View from Nowhere [1986], VIII.2) | |
| A reaction: Presumably all pains (e.g. of grief and of toothache) have something in common, to qualify as pains. It must be more than being disliked, because we can dislike a food. |
| 3261 | Something may be 'rational' either because it is required or because it is acceptable [Nagel] |
| Full Idea: "Rational" may mean rationally required or rationally acceptable | |
| From: Thomas Nagel (The View from Nowhere [1986], X.4) |
| 3258 | If cockroaches can't think about their actions, they have no duties [Nagel] |
| Full Idea: If cockroaches cannot think about what they should do, there is nothing they should do. | |
| From: Thomas Nagel (The View from Nowhere [1986], VIII.3) |
| 3254 | If we can decide how to live after stepping outside of ourselves, we have the basis of a moral theory [Nagel] |
| Full Idea: If we can make judgements about how we should live even after stepping outside of ourselves, they will provide the material for moral theory. | |
| From: Thomas Nagel (The View from Nowhere [1986], VIII.1) |
| 3264 | We should see others' viewpoints, but not lose touch with our own values [Nagel] |
| Full Idea: One should occupy a position far enough outside your own life to reduce the importance of the difference between yourself and other people, yet not so far outside that all human values vanish in a nihilistic blackout (i.e.aim for a form of humility). | |
| From: Thomas Nagel (The View from Nowhere [1986], XI.2) |
| 3255 | We find new motives by discovering reasons for action different from our preexisting motives [Nagel] |
| Full Idea: There are reasons for action, and we must discover them instead of deriving them from our preexisting motives - and in that way we can acquire new motives superior to the old. | |
| From: Thomas Nagel (The View from Nowhere [1986], VIII.1) |
| 3262 | Utilitarianism is too demanding [Nagel] |
| Full Idea: Utilitarianism is too demanding. | |
| From: Thomas Nagel (The View from Nowhere [1986], X.5) |