27 ideas
2676 | Didactic argument starts from the principles of the subject, not from the opinions of the learner [Aristotle] |
Full Idea: Didactic arguments are those which reason from the principles appropriate to each branch of learning and not from the opinions of the answerer (for he who is learning must take things on trust). | |
From: Aristotle (Sophistical Refutations [c.331 BCE], 165b01) |
2675 | Reasoning is a way of making statements which makes them lead on to other statements [Aristotle] |
Full Idea: Reasoning is based on certain statements made in such a way as necessarily to cause the assertion of things other than those statements and as a result of those statements. | |
From: Aristotle (Sophistical Refutations [c.331 BCE], 165a01) |
2677 | Dialectic aims to start from generally accepted opinions, and lead to a contradiction [Aristotle] |
Full Idea: Dialectical arguments are those which, starting from generally accepted opinions, reason to establish a contradiction. | |
From: Aristotle (Sophistical Refutations [c.331 BCE], 165b03) |
2674 | Competitive argument aims at refutation, fallacy, paradox, solecism or repetition [Aristotle] |
Full Idea: Those who compete and contend in argument aim at five objects: refutation, fallacy, paradox, solecism, and the reduction of one's opponent to a state of babbling, that is, making him say the same thing over and over again. | |
From: Aristotle (Sophistical Refutations [c.331 BCE], 165b15) |
10301 | The axiom of choice is controversial, but it could be replaced [Shapiro] |
Full Idea: The axiom of choice has a troubled history, but is now standard in mathematics. It could be replaced with a principle of comprehension for functions), or one could omit the variables ranging over functions. | |
From: Stewart Shapiro (Higher-Order Logic [2001], n 3) |
10588 | First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro] |
Full Idea: Early study of first-order logic revealed a number of important features. Gödel showed that there is a complete, sound and effective deductive system. It follows that it is Compact, and there are also the downward and upward Löwenheim-Skolem Theorems. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
10298 | Some say that second-order logic is mathematics, not logic [Shapiro] |
Full Idea: Some authors argue that second-order logic (with standard semantics) is not logic at all, but is a rather obscure form of mathematics. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) |
10299 | If the aim of logic is to codify inferences, second-order logic is useless [Shapiro] |
Full Idea: If the goal of logical study is to present a canon of inference, a calculus which codifies correct inference patterns, then second-order logic is a non-starter. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |
A reaction: This seems to be because it is not 'complete'. However, moves like plural quantification seem aimed at capturing ordinary language inferences, so the difficulty is only that there isn't a precise 'calculus'. |
10300 | Logical consequence can be defined in terms of the logical terminology [Shapiro] |
Full Idea: Informally, logical consequence is sometimes defined in terms of the meanings of a certain collection of terms, the so-called 'logical terminology'. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |
A reaction: This seems to be a compositional account, where we build a full account from an account of the atomic bits, perhaps presented as truth-tables. |
16967 | 'Are Coriscus and Callias at home?' sounds like a single question, but it isn't [Aristotle] |
Full Idea: If you ask 'Are Coriscus and Callias at home or not at home?', whether they are both at home or not there, the number of propositions is more than one. For if the answer is true, it does not follow that the question is a single one. | |
From: Aristotle (Sophistical Refutations [c.331 BCE], 176a08) | |
A reaction: [compressed] Aristotle is saying that some questions should not receive a 'yes' or 'no' answer, because they are equivocal. Arthur Prior cites this passage, on 'and'. Ordinary use of 'and' need not be the logical use of 'and'. |
10290 | Second-order variables also range over properties, sets, relations or functions [Shapiro] |
Full Idea: Second-order variables can range over properties, sets, or relations on the items in the domain-of-discourse, or over functions from the domain itself. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
10292 | Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro] |
Full Idea: Downward Löwenheim-Skolem: a finite or denumerable set of first-order formulas that is satisfied by a model whose domain is infinite is satisfied in a model whose domain is the natural numbers | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
10590 | Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro] |
Full Idea: Upward Löwenheim-Skolem: if a set of first-order formulas is satisfied by a domain of at least the natural numbers, then it is satisfied by a model of at least some infinite cardinal. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
10296 | The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro] |
Full Idea: Both of the Löwenheim-Skolem Theorems fail for second-order languages with a standard semantics | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.3.2) |
10297 | The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro] |
Full Idea: The Löwenheim-Skolem theorem is usually taken as a sort of defect (often thought to be inevitable) of the first-order logic. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |
A reaction: [He is quoting Wang 1974 p.154] |
10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro] |
Full Idea: Full second-order logic has all the expressive power needed to do mathematics, but has an unworkable model theory. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) | |
A reaction: [he credits Cowles for this remark] Having an unworkable model theory sounds pretty serious to me, as I'm not inclined to be interested in languages which don't produce models of some sort. Surely models are the whole point? |
10591 | Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro] |
Full Idea: In studying second-order logic one can think of relations and functions as extensional or intensional, or one can leave it open. Little turns on this here, and so words like 'property', 'class', and 'set' are used interchangeably. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.2.1) | |
A reaction: Important. Students of the metaphysics of properties, who arrive with limited experience of logic, are bewildered by this attitude. Note that the metaphysics is left wide open, so never let logicians hijack the metaphysical problem of properties. |
16149 | Generic terms like 'man' are not substances, but qualities, relations, modes or some such thing [Aristotle] |
Full Idea: 'Man', and every generic term, denotes not an individual substance but a quality or relation or mode or something of the kind. | |
From: Aristotle (Sophistical Refutations [c.331 BCE], 179a01) | |
A reaction: This is Aristotle's denial that species constitutes the essence of anything. I take 'man' to be a categorisation of individuals, and is ontologically nothing at all in its own right. |
11840 | Only if two things are identical do they have the same attributes [Aristotle] |
Full Idea: It is only to things which are indistinguishable and one in essence [ousia] that all the same attributes are generally held to belong. | |
From: Aristotle (Sophistical Refutations [c.331 BCE], 179a37) | |
A reaction: This simply IS Leibniz's Law (to which I shall from now on quietly refer to as 'Aristotle's Law'). It seems that it just as plausible to translate 'ousia' as 'being' rather than 'essence'. 'Indistinguishable' and 'one in ousia' are not the same. |
7357 | People who control others with fluent language often end up being hated [Kongzi (Confucius)] |
Full Idea: Of what use is eloquence? He who engages in fluency of words to control men often finds himself hated by them. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], V.5) | |
A reaction: I don't recall Socrates making this very good point to any of the sophists (such as Gorgias). The idea that if you battle or connive your way to dominance over others then you are successful is false. Life is a much longer game than that. |
7358 | All men prefer outward appearance to true excellence [Kongzi (Confucius)] |
Full Idea: I have yet to meet a man as fond of excellence as he is of outward appearances. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], IX.18) | |
A reaction: Interestingly, this cynical view of the love of virtue is put by Plato into the mouths of Glaucon and Adeimantus (in Bk II of 'Republic', e.g. Idea 12), and not into the mouth of Socrates, who goes on to defend the possibility of true virtue. |
7362 | Humans are similar, but social conventions drive us apart (sages and idiots being the exceptions) [Kongzi (Confucius)] |
Full Idea: In our natures we approximate one another; habits put us further and further apart. The only ones who do not change are sages and idiots. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XVII.2) | |
A reaction: I find most of Confucius rather uninteresting, but this is a splendid remark about the influence of social conventions on human nature. Sages can achieve universal morality if they rise above social convention, and seek the true virtues of human nature. |
7360 | Do not do to others what you would not desire yourself [Kongzi (Confucius)] |
Full Idea: Do not do to others what you would not desire yourself. Then you will have no enemies, either in the state or in your home. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XII.2) | |
A reaction: The Golden Rule, but note the second sentence. Logically, it leads to the absurdity of not giving someone an Elvis record for Christmas because you yourself don't like Elvis. Kant (Idea 3733) and Nietzsche (Idea 4560) offer good criticisms. |
7359 | Excess and deficiency are equally at fault [Kongzi (Confucius)] |
Full Idea: Excess and deficiency are equally at fault. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XI.16) | |
A reaction: This is the sort of wisdom we admire in Aristotle (and in any sensible person), but it may also be the deepest motto of conservatism, and it is a long way from romantic philosophy, and the clarion call of Nietzsche to greater excitement in life. |
7363 | The virtues of the best people are humility, maganimity, sincerity, diligence, and graciousness [Kongzi (Confucius)] |
Full Idea: He who in this world can practise five things may indeed be considered Man-at-his-best: humility, maganimity, sincerity, diligence, and graciousness. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XVII.5) | |
A reaction: A very nice list. Who could resist working with a colleague who had such virtues? Who could go wrong if they married a person who had them? I can't think of anything important that is missing. |
7361 | Men of the highest calibre avoid political life completely [Kongzi (Confucius)] |
Full Idea: Men of the highest calibre avoid political life completely. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XIV.37) | |
A reaction: Plato notes that such people tend to avoid political life (and a left sheltering, as if from a wild storm!), but he thinks they should be dragged into the political arena for the common good. Confucius seems to approve of the avoidance. Plato is right. |
23393 | Confucianism assumes that all good developments have happened, and there is only one Way [Norden on Kongzi (Confucius)] |
Full Idea: The two major limitations of Confucianism are that it assumes that all worthwhile cultural, social and ethical innovation has already occurred, and that it does not recognise the plurality of worthwhile ways of life. | |
From: comment on Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE]) by Bryan van Norden - Intro to Classical Chinese Philosophy 3.III | |
A reaction: In modern liberal terms that is about as conservative as it is possible to get. We think of it as the state of mind of an old person who can only long for the way things were when they were young. But 'hold fast to that which is good'! |