18909
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Aristotelian sentences are made up by one of four 'formative' connectors [Aristotle, by Engelbretsen]
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Full Idea:
For Aristotle there are four formatives for sentences: 'belongs to some', 'belongs to every', 'belongs to no', and 'does not belong to every'. These are 'copulae'. Aristotle would have written 'wise belongs to some man'.
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From:
report of Aristotle (Prior Analytics [c.328 BCE]) by George Engelbretsen - Trees, Terms and Truth 3
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A reaction:
A rather set-theoretic reading. This invites a Quinean scepticism about whether wisdom is some entity which can 'belong' to a person. It makes trope theory sound attractive, offering a unique wisdom that is integrated into that particular person.
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8080
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Aristotelian identified 256 possible syllogisms, saying that 19 are valid [Aristotle, by Devlin]
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Full Idea:
Aristotle identified four 'figures' of argument, based on combinations of Subject (S) and Predicate (P) and Middle term (M). The addition of 'all' and 'some', and 'has' and 'has not' got the property, resulted in 256 possible syllogisms, 19 of them valid.
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From:
report of Aristotle (Prior Analytics [c.328 BCE]) by Keith Devlin - Goodbye Descartes Ch.2
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A reaction:
[Compressed version of Devlin] What Aristotle did was astonishing, and must be one of the key ideas of western civilization, even though a lot of his assumptions have been revised or rejected.
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13912
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Aristotle replaced Plato's noun-verb form with unions of pairs of terms by one of four 'copulae' [Aristotle, by Engelbretsen/Sayward]
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Full Idea:
Aristotle replaced the Platonic noun-verb account of logical syntax with a 'copular' account. A sentence is a pair of terms bound together logically (not necessarily grammatically) by one of four 'logical copulae' (every, none, some, not some).
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From:
report of Aristotle (Prior Analytics [c.328 BCE]) by Engelbretsen,G/Sayward,C - Philosophical Logic: Intro to Advanced Topics 8
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A reaction:
So the four copulas are are-all, are-never, are-sometimes, and are-sometime-not. Consider 'men' and 'mortal'. Alternatively, Idea 18909.
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9403
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There are three different deductions for actual terms, necessary terms and possible terms [Aristotle]
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Full Idea:
Since to belong, to belong of necessity, and to be possible to belong are different, ..there will be different deductions for each; one deduction will be from necessary terms, one from terms which belong, and one from possible terms.
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From:
Aristotle (Prior Analytics [c.328 BCE], 29b29-35)
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A reaction:
Fitting and Mendelsohn cite this as the earliest thoughts on modal logic. but Kneale and Kneale say that Aristotle got into a muddle, and so was unable to create a workable system.
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11148
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Deduction is when we suppose one thing, and another necessarily follows [Aristotle]
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Full Idea:
A deduction is a discourse in which, certain things having been supposed, something different from the things supposed results of necessity because these things are so.
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From:
Aristotle (Prior Analytics [c.328 BCE], 24b18)
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A reaction:
Notice that it is modal ('suppose', rather than 'know'), that necessity is involved, which is presumably metaphysical necessity, and that there are assumptions about what would be true, and not just what follows from what.
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8079
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Aristotelian logic has two quantifiers of the subject ('all' and 'some') [Aristotle, by Devlin]
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Full Idea:
Aristotelian logic has two quantifiers of the subject ('all' and 'some'), and two ways to combine the subject with the predicate ('have', and 'have not'), giving four propositions: all-s-have-p, all-s-have-not-p, some-s-have-p, and some-s-have-not-p.
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From:
report of Aristotle (Prior Analytics [c.328 BCE]) by Keith Devlin - Goodbye Descartes Ch.2
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A reaction:
Frege seems to have switched from 'some' to 'at-least-one'. Since then other quantifiers have been proposed. See, for example, Ideas 7806 and 6068.
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14234
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If you only refer to objects one at a time, you need sets in order to refer to a plurality [Oliver/Smiley]
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Full Idea:
A 'singularist', who refers to objects one at a time, must resort to the language of sets in order to replace plural reference to members ('Henry VIII's wives') by singular reference to a set ('the set of Henry VIII's wives').
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From:
Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro)
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A reaction:
A simple and illuminating point about the motivation for plural reference. Null sets and singletons give me the creeps, so I would personally prefer to avoid set theory when dealing with ontology.
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14237
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We can use plural language to refer to the set theory domain, to avoid calling it a 'set' [Oliver/Smiley]
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Full Idea:
Plurals earn their keep in set theory, to answer Skolem's remark that 'in order to treat of 'sets', we must begin with 'domains' that are constituted in a certain way'. We can speak in the plural of 'the objects', not a 'domain' of objects.
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From:
Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro)
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A reaction:
[Skolem 1922:291 in van Heijenoort] Zermelo has said that the domain cannot be a set, because every set belongs to it.
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14246
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If mathematics purely concerned mathematical objects, there would be no applied mathematics [Oliver/Smiley]
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Full Idea:
If mathematics was purely concerned with mathematical objects, there would be no room for applied mathematics.
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From:
Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1)
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A reaction:
Love it! Of course, they are using 'objects' in the rather Fregean sense of genuine abstract entities. I don't see why fictionalism shouldn't allow maths to be wholly 'pure', although we have invented fictions which actually have application.
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14247
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Sets might either represent the numbers, or be the numbers, or replace the numbers [Oliver/Smiley]
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Full Idea:
Identifying numbers with sets may mean one of three quite different things: 1) the sets represent the numbers, or ii) they are the numbers, or iii) they replace the numbers.
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From:
Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.2)
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A reaction:
Option one sounds the most plausible to me. I will take numbers to be patterns embedded in nature, and sets are one way of presenting them in shorthand form, in order to bring out what is repeated.
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18911
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Linguistic terms form a hierarchy, with higher terms predicable of increasing numbers of things [Aristotle, by Engelbretsen]
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Full Idea:
According to Aristotle, the terms of a language form a finite hierarchy, where the higher terms are predicable of more things than are lower terms.
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From:
report of Aristotle (Prior Analytics [c.328 BCE]) by George Engelbretsen - Trees, Terms and Truth 3
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A reaction:
I would be a bit cautious about placing something precisely in a hierarchy according to how many things it can be predicated of. It is a start, though, in trying to give a decent account of generality, which is a major concept in philosophy.
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7903
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The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna]
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Full Idea:
The six perfections are of giving, morality, patience, vigour, meditation, and wisdom.
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From:
Nagarjuna (Mahaprajnaparamitashastra [c.120], 88)
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A reaction:
What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate').
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