126 ideas
| 11283 | There is pure deductive reasoning, and explanatory demonstration reasoning [Aristotle, by Politis] |
| Full Idea: Aristotle distinguishes between deductive reasoning (sullogismos) and demonstration (apodeixis). All demonstration is deductive reasoning, but not all deductive reasoning is demonstration. | |
| From: report of Aristotle (Posterior Analytics [c.327 BCE], Bk I.2) by Vassilis Politis - Aristotle and the Metaphysics 5.3 | |
| A reaction: This sounds not far off the distinction between single-turnstile (formal proof) and double-turnstile (semantic consequence). Politis says, though, that the key point is the demonstration is explanatory. |
| 1672 | Maybe everything could be demonstrated, if demonstration can be reciprocal or circular [Aristotle] |
| Full Idea: Some optimists think understanding arises only through demonstration, but say there could be demonstration of everything, for it is possible to demonstrate in a circle or reciprocally. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 72b16) | |
| A reaction: I'm an optimist in this sense, though what is being described would probably best be called 'large-scale coherence'. Two reciprocal arguments look bad, but a hundred look good. |
| 1684 | Two falsehoods can be contrary to one another [Aristotle] |
| Full Idea: There are falsehoods which are contrary to one another and cannot be the case together e.g. that a man is a horse or a cow. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 88a29) |
| 12145 | Definitions are of what something is, and that is universal [Aristotle] |
| Full Idea: Definitions are thought to be of what something is, and what something is is in every case universal and positive. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 90b05) | |
| A reaction: This is exhibit A for those who think that Aristotelian essences concern the genus, rather than the particular. I suspect that this idea is best expressed as 'all we can say by way of definition of a particular thing involves the use of universals'. |
| 12075 | An Aristotelian definition is causal [Aristotle, by Witt] |
| Full Idea: An Aristotelian definition is causal. | |
| From: report of Aristotle (Posterior Analytics [c.327 BCE], Bk II.2) by Charlotte Witt - Substance and Essence in Aristotle 1.5 | |
| A reaction: [She refers us to Posterior Analytics II.2] This is important if we are tempted to follow a modern line of saying that we want Aristotelian essences, and that these are definitions. We ain't thinking of dictionaries. |
| 12384 | Definition by division needs predicates, which are well ordered and thorough [Aristotle] |
| Full Idea: To establish a definition through division, you must aim for three things: you must take what is predicated in what the thing is; you must order these items as first or second; and you must ensure that these are all there are. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 97a23) | |
| A reaction: This gives an indication of the thoroughness that Aristotle expects from a definition. They aren't like dictionary definitions of words. He expects definitions to often be very lengthy (see Idea 12292). |
| 9066 | You can define objects by progressively identifying what is the same and what is different [Aristotle] |
| Full Idea: Find what is in common among items similar and undifferentiated, then do the same for items of the same kind as the first group but a different form, and so on, till you come to a single account: this will be the definition of the object. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 97b07-14) | |
| A reaction: [His example is distinguishing 'magnanimity' from 'indifference to fortune' among people] Presumably this process works for the formation of new concepts (e.g. in biology), as well as for the definition of familiars in terms of other familiars. |
| 12382 | What it is and why it is are the same; screening defines and explains an eclipse [Aristotle] |
| Full Idea: What it is and why it is are the same. What is an eclipse? Privation of light from the moon by screening of the earth. Why is there an eclipse? ...What is a harmony? A numerical ratio between high and low. Why do the high and low harmonize? The ratio. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 90a15) | |
| A reaction: This is right at the heart of Aristotelian essentialism, and (I take it) modern scientific essentialism. If you fully know what cigarette tars are, and what human cell structure is, you understand immediately why cigarettes cause cancer. |
| 13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
| Full Idea: In a sense, satisfaction is the notion of 'truth in a model', and (as Hodes 1984 elegantly puts it) 'truth in a model' is a model of 'truth'. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1) | |
| A reaction: So we can say that Tarski doesn't offer a definition of truth itself, but replaces it with a 'model' of truth. |
| 13643 | Aristotelian logic is complete [Shapiro] |
| Full Idea: Aristotelian logic is complete. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.5) | |
| A reaction: [He cites Corcoran 1972] |
| 1668 | An axiom is a principle which must be understood if one is to learn anything [Aristotle] |
| Full Idea: An axiom is a principle which must be grasped if anyone is going to learn anything whatever. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 72a17) |
| 13651 | A set is 'transitive' if contains every member of each of its members [Shapiro] |
| Full Idea: If, for every b∈d, a∈b entails that a∈d, the d is said to be 'transitive'. In other words, d is transitive if it contains every member of each of its members. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.2) | |
| A reaction: The alternative would be that the members of the set are subsets, but the members of those subsets are not themselves members of the higher-level set. |
| 13647 | Choice is essential for proving downward Löwenheim-Skolem [Shapiro] |
| Full Idea: The axiom of choice is essential for proving the downward Löwenheim-Skolem Theorem. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1) |
| 13631 | Are sets part of logic, or part of mathematics? [Shapiro] |
| Full Idea: Is there a notion of set in the jurisdiction of logic, or does it belong to mathematics proper? | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: It immediately strikes me that they might be neither. I don't see that relations between well-defined groups of things must involve number, and I don't see that mapping the relations must intrinsically involve logical consequence or inference. |
| 13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro] |
| Full Idea: In set theory it is central to the iterative conception that the membership relation is well-founded, ...which means there are no infinite descending chains from any relation. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.4) |
| 13640 | Russell's paradox shows that there are classes which are not iterative sets [Shapiro] |
| Full Idea: The argument behind Russell's paradox shows that in set theory there are logical sets (i.e. classes) that are not iterative sets. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.3) | |
| A reaction: In his preface, Shapiro expresses doubts about the idea of a 'logical set'. Hence the theorists like the iterative hierarchy because it is well-founded and under control, not because it is comprehensive in scope. See all of pp.19-20. |
| 13666 | Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro] |
| Full Idea: Iterative sets do not exhibit a Boolean structure, because the complement of an iterative set is not itself an iterative set. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1) |
| 13653 | 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro] |
| Full Idea: A 'well-ordering' of a set X is an irreflexive, transitive, and binary relation on X in which every non-empty subset of X has a least element. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.3) | |
| A reaction: So there is a beginning, an ongoing sequence, and no retracing of steps. |
| 13627 | There is no 'correct' logic for natural languages [Shapiro] |
| Full Idea: There is no question of finding the 'correct' or 'true' logic underlying a part of natural language. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: One needs the context of Shapiro's defence of second-order logic to see his reasons for this. Call me romantic, but I retain faith that there is one true logic. The Kennedy Assassination problem - can't see the truth because drowning in evidence. |
| 13642 | Logic is the ideal for learning new propositions on the basis of others [Shapiro] |
| Full Idea: A logic can be seen as the ideal of what may be called 'relative justification', the process of coming to know some propositions on the basis of others. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.3.1) | |
| A reaction: This seems to be the modern idea of logic, as opposed to identification of a set of 'logical truths' from which eternal necessities (such as mathematics) can be derived. 'Know' implies that they are true - which conclusions may not be. |
| 13668 | Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro] |
| Full Idea: Bernays (1918) formulated and proved the completeness of propositional logic, the first precise solution as part of the Hilbert programme. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2.1) |
| 13669 | Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro] |
| Full Idea: In 1910 Weyl observed that set theory seemed to presuppose natural numbers, and he regarded numbers as more fundamental than sets, as did Fraenkel. Dedekind had developed set theory independently, and used it to formulate numbers. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2.2) |
| 13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro] |
| Full Idea: Skolem and Gödel were the main proponents of first-order languages. The higher-order language 'opposition' was championed by Zermelo, Hilbert, and Bernays. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2) |
| 13662 | First-order logic was an afterthought in the development of modern logic [Shapiro] |
| Full Idea: Almost all the systems developed in the first part of the twentieth century are higher-order; first-order logic was an afterthought. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1) |
| 13624 | The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro] |
| Full Idea: The 'triumph' of first-order logic may be related to the remnants of failed foundationalist programmes early this century - logicism and the Hilbert programme. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: Being complete must also be one of its attractions, and Quine seems to like it because of its minimal ontological commitment. |
| 13660 | Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro] |
| Full Idea: Tharp (1975) suggested that compactness, semantic effectiveness, and the Löwenheim-Skolem properties are consequences of features one would want a logic to have. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5) | |
| A reaction: I like this proposal, though Shapiro is strongly against. We keep extending our logic so that we can prove new things, but why should we assume that we can prove everything? That's just what Gödel suggests that we should give up on. |
| 13673 | The notion of finitude is actually built into first-order languages [Shapiro] |
| Full Idea: The notion of finitude is explicitly 'built in' to the systems of first-order languages in one way or another. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1) | |
| A reaction: Personally I am inclined to think that they are none the worse for that. No one had even thought of all these lovely infinities before 1870, and now we are supposed to change our logic (our actual logic!) to accommodate them. Cf quantum logic. |
| 12376 | Demonstrations by reductio assume excluded middle [Aristotle] |
| Full Idea: Demonstrations by reduction to the impossible assume that everything is asserted or denied. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 77a23) | |
| A reaction: This sounds like the lynchpin of classical logic. |
| 15944 | Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine] |
| Full Idea: Shapiro preferred second-order logic to set theory because second-order logic refers only to the relations and operations in a domain, and not to the other things that set-theory brings with it - other domains, higher-order relations, and so forth. | |
| From: report of Stewart Shapiro (Foundations without Foundationalism [1991]) by Shaughan Lavine - Understanding the Infinite VII.4 |
| 13629 | Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro] |
| Full Idea: Three systems of semantics for second-order languages: 'standard semantics' (variables cover all relations and functions), 'Henkin semantics' (relations and functions are a subclass) and 'first-order semantics' (many-sorted domains for variable-types). | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: [my summary] |
| 13650 | Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro] |
|
Full Idea:
In 'Henkin' semantics, in a given model |
|
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 3.3) |
| 13645 | In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro] |
| Full Idea: In the standard semantics of second-order logic, by fixing a domain one thereby fixes the range of both the first-order variables and the second-order variables. There is no further 'interpreting' to be done. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 3.3) | |
| A reaction: This contrasts with 'Henkin' semantics (Idea 13650), or first-order semantics, which involve more than one domain of quantification. |
| 13649 | Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro] |
| Full Idea: The counterparts of Completeness, Compactness and the Löwenheim-Skolem theorems all fail for second-order languages with standard semantics, but hold for Henkin or first-order semantics. Hence such logics are much like first-order logic. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1) | |
| A reaction: Shapiro votes for the standard semantics, because he wants the greater expressive power, especially for the characterization of infinite structures. |
| 12373 | Something holds universally when it is proved of an arbitrary and primitive case [Aristotle] |
| Full Idea: Something holds universally when it is proved of an arbitrary and primitive case. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 73b33) | |
| A reaction: A key idea in mathematical logic, but it always puzzles me. If you snatch a random person in London, and they are extremely tall, does that prove that people of London are extremely tall? How do we know the arbitrary is representative? |
| 13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
| Full Idea: It follows from Gödel's incompleteness theorem that the semantic consequence relation of second-order logic is not effective. For example, the set of logical truths of any second-order logic is not recursively enumerable. It is not even arithmetic. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: I don't fully understand this, but it sounds rather major, and a good reason to avoid second-order logic (despite Shapiro's proselytising). See Peter Smith on 'effectively enumerable'. |
| 13637 | If a logic is incomplete, its semantic consequence relation is not effective [Shapiro] |
| Full Idea: Second-order logic is inherently incomplete, so its semantic consequence relation is not effective. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.2.1) |
| 12363 | Everything is either asserted or denied truly [Aristotle] |
| Full Idea: Of the fact that everything is either asserted or denied truly, we must believe that it is the case. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 71a14) | |
| A reaction: Presumably this means that every assertion which could possibly be asserted must come out as either true or false. This will have to include any assertions with vague objects or predicates, and any universal assertions, and negative assertions. |
| 13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro] |
| Full Idea: It is sometimes difficult to find a formula that is a suitable counterpart of a particular sentence of natural language, and there is no acclaimed criterion for what counts as a good, or even acceptable, 'translation'. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1) |
| 13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro] |
| Full Idea: The main role of substitutional semantics is to reduce ontology. As an alternative to model-theoretic semantics for formal languages, the idea is to replace the 'satisfaction' relation of formulas (by objects) with the 'truth' of sentences (using terms). | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4) | |
| A reaction: I find this very appealing, and Ruth Barcan Marcus is the person to look at. My intuition is that logic should have no ontology at all, as it is just about how inference works, not about how things are. Shapiro offers a compromise. |
| 13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro] |
| Full Idea: The 'satisfaction' relation may be thought of as a function from models, assignments, and formulas to the truth values {true,false}. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1) | |
| A reaction: This at least makes clear that satisfaction is not the same as truth. Now you have to understand how Tarski can define truth in terms of satisfaction. |
| 13644 | Semantics for models uses set-theory [Shapiro] |
| Full Idea: Typically, model-theoretic semantics is formulated in set theory. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.5.1) |
| 13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro] |
| Full Idea: An axiomatization is 'categorical' if all its models are isomorphic to one another; ..hence it has 'essentially only one' interpretation [Veblen 1904]. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.2.1) |
| 13670 | Categoricity can't be reached in a first-order language [Shapiro] |
| Full Idea: Categoricity cannot be attained in a first-order language. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.3) |
| 13658 | Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro] |
| Full Idea: A language has the Downward Löwenheim-Skolem property if each satisfiable countable set of sentences has a model whose domain is at most countable. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5) | |
| A reaction: This means you can't employ an infinite model to represent a fact about a countable set. |
| 13659 | Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro] |
| Full Idea: A language has the Upward Löwenheim-Skolem property if for each set of sentences whose model has an infinite domain, then it has a model at least as big as each infinite cardinal. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5) | |
| A reaction: This means you can't have a countable model to represent a fact about infinite sets. |
| 13648 | The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro] |
| Full Idea: The Löwenheim-Skolem theorems mean that no first-order theory with an infinite model is categorical. If Γ has an infinite model, then it has a model of every infinite cardinality. So first-order languages cannot characterize infinite structures. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1) | |
| A reaction: So much of the debate about different logics hinges on characterizing 'infinite structures' - whatever they are! Shapiro is a leading structuralist in mathematics, so he wants second-order logic to help with his project. |
| 13675 | Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro] |
| Full Idea: The Upward Löwenheim-Skolem theorem fails (trivially) with substitutional semantics. If there are only countably many terms of the language, then there are no uncountable substitution models. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4) | |
| A reaction: Better and better. See Idea 13674. Why postulate more objects than you can possibly name? I'm even suspicious of all real numbers, because you can't properly define them in finite terms. Shapiro objects that the uncountable can't be characterized. |
| 13004 | Aristotle's axioms (unlike Euclid's) are assumptions awaiting proof [Aristotle, by Leibniz] |
| Full Idea: Aristotle's way with axioms, rather than Euclid's, is as assumptions which we are willing to agree on while awaiting an opportunity to prove them | |
| From: report of Aristotle (Posterior Analytics [c.327 BCE], 76b23-) by Gottfried Leibniz - New Essays on Human Understanding 4.07 | |
| A reaction: Euclid's are understood as basic self-evident truths which will be accepted by everyone, though the famous parallel line postulate undermined that. The modern view of axioms is a set of minimum theorems that imply the others. I like Aristotle. |
| 13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro] |
| Full Idea: A logic is 'weakly sound' if every theorem is a logical truth, and 'strongly sound', or simply 'sound', if every deduction from Γ is a semantic consequence of Γ. Soundness indicates that the deductive system is faithful to the semantics. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1) | |
| A reaction: Similarly, 'weakly complete' is when every logical truth is a theorem. |
| 13628 | We can live well without completeness in logic [Shapiro] |
| Full Idea: We can live without completeness in logic, and live well. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: This is the kind of heady suggestion that American philosophers love to make. Sounds OK to me, though. Our ability to draw good inferences should be expected to outrun our ability to actually prove them. Completeness is for wimps. |
| 13630 | Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro] |
| Full Idea: It is sometimes said that non-compactness is a defect of second-order logic, but it is a consequence of a crucial strength - its ability to give categorical characterisations of infinite structures. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: The dispute between fans of first- and second-order may hinge on their attitude to the infinite. I note that Skolem, who was not keen on the infinite, stuck to first-order. Should we launch a new Skolemite Crusade? |
| 13646 | Compactness is derived from soundness and completeness [Shapiro] |
| Full Idea: Compactness is a corollary of soundness and completeness. If Γ is not satisfiable, then, by completeness, Γ is not consistent. But the deductions contain only finite premises. So a finite subset shows the inconsistency. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1) | |
| A reaction: [this is abbreviated, but a proof of compactness] Since all worthwhile logics are sound, this effectively means that completeness entails compactness. |
| 13661 | A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro] |
| Full Idea: A logical language is 'semantically effective' if the collection of logically true sentences is a recursively enumerable set of strings. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5) |
| 12377 | Mathematics is concerned with forms, not with superficial properties [Aristotle] |
| Full Idea: Mathematics is concerned with forms [eide]: its objects are not said of any underlying subject - for even if geometrical objects are said of some underlying subject, still it is not as being said of an underlying subject that they are studied. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 79a08) | |
| A reaction: Since forms turn out to be essences, in 'Metaphysics', this indicates an essentialist view of mathematics. |
| 12372 | The essence of a triangle comes from the line, mentioned in any account of triangles [Aristotle] |
| Full Idea: Something holds of an item in itself if it holds of it in what it is - e.g., line of triangles and point of lines (their essence comes from these items, which inhere in the account which says what they are). | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 73a35) | |
| A reaction: A helpful illustration of how a definition gives us the essence of something. You could not define triangles without mentioning straight lines. The lines are necessary features, but they are essential for any explanation, and for proper understanding. |
| 13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
| Full Idea: 'Definitions' of integers as pairs of naturals, rationals as pairs of integers, reals as Cauchy sequences of rationals, and complex numbers as pairs of reals are reductive foundations of various fields. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.1) | |
| A reaction: On p.30 (bottom) Shapiro objects that in the process of reduction the numbers acquire properties they didn't have before. |
| 13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro] |
| Full Idea: The main problem of characterizing the natural numbers is to state, somehow, that 0,1,2,.... are all the numbers that there are. We have seen that this can be accomplished with a higher-order language, but not in a first-order language. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4) |
| 13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro] |
| Full Idea: By convention, the natural numbers are the finite ordinals, the integers are certain equivalence classes of pairs of finite ordinals, etc. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.3) |
| 12369 | A unit is what is quantitatively indivisible [Aristotle] |
| Full Idea: Arithmeticians posit that a unit is what is quantitatively indivisible. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 72a22) | |
| A reaction: Presumably indeterminate stuff like water is non-quantitatively divisible (e.g. Moses divides the Red Sea), as are general abstracta (curved shapes from rectilinear ones). Does 'quantitative' presupposes units, making the idea circular? |
| 13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro] |
| Full Idea: The 'continuum' is the cardinality of the powerset of a denumerably infinite set. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.2) |
| 13657 | First-order arithmetic can't even represent basic number theory [Shapiro] |
| Full Idea: Few theorists consider first-order arithmetic to be an adequate representation of even basic number theory. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 5 n28) | |
| A reaction: This will be because of Idea 13656. Even 'basic' number theory will include all sorts of vast infinities, and that seems to be where the trouble is. |
| 13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro] |
| Full Idea: There are sets of natural numbers definable in set-theory but not in arithmetic. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.3.3) |
| 13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro] |
| Full Idea: It is claimed that aiming at a universal language for all contexts, and the thesis that logic does not involve a process of abstraction, separates the logicists from algebraists and mathematicians, and also from modern model theory. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1) | |
| A reaction: I am intuitively drawn to the idea that logic is essentially the result of a series of abstractions, so this gives me a further reason not to be a logicist. Shapiro cites Goldfarb 1979 and van Heijenoort 1967. Logicists reduce abstraction to logic. |
| 13625 | Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro] |
| Full Idea: I extend Quinean holism to logic itself; there is no sharp border between mathematics and logic, especially the logic of mathematics. One cannot expect to do logic without incorporating some mathematics and accepting at least some of its ontology. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: I have strong sales resistance to this proposal. Mathematics may have hijacked logic and warped it for its own evil purposes, but if logic is just the study of inferences then it must be more general than to apply specifically to mathematics. |
| 13663 | Some reject formal properties if they are not defined, or defined impredicatively [Shapiro] |
| Full Idea: Some authors (Poincaré and Russell, for example) were disposed to reject properties that are not definable, or are definable only impredicatively. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1) | |
| A reaction: I take Quine to be the culmination of this line of thought, with his general rejection of 'attributes' in logic and in metaphysics. |
| 18910 | To seek truth, study the real connections between subjects and attributes [Aristotle] |
| Full Idea: If, however, one is aiming at truth, one must be guided by the real connexions of subjects and attributes. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 81b22), quoted by George Engelbretsen - Trees, Terms and Truth 3 | |
| A reaction: I take this to be a warning that predicates that indicate mere 'Cambridge properties' (such as relations, locations, coincidences etc) have nothing to do with ontology. See Shoemaker on properties. |
| 13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
| Full Idea: Properties are often taken to be intensional; equiangular and equilateral are thought to be different properties of triangles, even though any triangle is equilateral if and only if it is equiangular. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.3) | |
| A reaction: Many logicians seem to want to treat properties as sets of objects (red being just the set of red things), but this looks like a desperate desire to say everything in first-order logic, where only objects are available to quantify over. |
| 1675 | Separate Forms aren't needed for logic, but universals (one holding of many) are essential [Aristotle] |
| Full Idea: There need be no forms (one item apart from the many) for demonstrations. But there must be universals, where one thing holds of the many. Without universals there are no middle terms, and so no demonstrations. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 77a05) |
| 1677 | We can forget the Forms, as they are irrelevant, and not needed in giving demonstrations [Aristotle] |
| Full Idea: We can say goodbye to the forms. They are nonny-noes; and if there are any they are irrelevant - for demonstrations are not concerned with them. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 83a34) |
| 1687 | Why are being terrestrial and a biped combined in the definition of man, but being literate and musical aren't? [Aristotle] |
| Full Idea: Why will a man be a two-footed terrestrial animal and not an animal and terrestrial? Assumptions do not make it necessary that what is predicated form a unity - rather, it is as if the same man were musical and literate. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 92a30) |
| 1681 | Units are positionless substances, and points are substances with position [Aristotle] |
| Full Idea: A unit is a positionless substance, and a point a substance having position. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 87a36) |
| 12146 | Definitions recognise essences, so are not themselves essences [Aristotle] |
| Full Idea: If a definition is the recognition of some essence, it is clear that such items are not essences. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 90b17) | |
| A reaction: So definitions are not themselves essences (as some modern thinkers claim). The idea seems obvious to me, but it is a warning against a simplistic view of Aristotelian essences, and a reminder that such things are real, not verbal. |
| 23647 | Objects have an essential constitution, producing its qualities, which we are too ignorant to define [Reid] |
| Full Idea: Individuals and objects have a real essence, or constitution of nature, from which all their qualities flow: but this essence our faculties do not comprehend. They are therefore incapable of definition. | |
| From: Thomas Reid (Essays on Intellectual Powers 4: Conception [1785], 1) | |
| A reaction: Aha - he's one of us! I prefer the phrase 'essential nature' of an object, which is understood, I think, by everyone. I especially like the last bit, directed at those who mistakenly think that Aristotle identified the essence with the definition. |
| 17039 | The predicates of a thing's nature are necessary to it [Aristotle] |
| Full Idea: Whatever is predicated in what something is is necessary. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 96b03) | |
| A reaction: This does NOT say that the essence is just the necessities. He goes on to say to say separately that certain properties of a triplet are part of the essence, as well as being necessary. This shows the nature of a thing is also necessary. |
| 11994 | Aristotelian essences are properties mentioned at the starting point of a science [Aristotle, by Kung] |
| Full Idea: As Aristotle uses the term 'essence', only those properties which are mentioned in or relatively close to the starting points of the science will be essential. | |
| From: report of Aristotle (Posterior Analytics [c.327 BCE]) by Joan Kung - Aristotle on Essence and Explanation II | |
| A reaction: I take this to be the correct way to understand Aristotelian essence - as something understood by its role in scientific explanations. We may, of course, work back to the starting point of a science, by disentangling the mess in the middle. |
| 12381 | What is necessary cannot be otherwise [Aristotle] |
| Full Idea: What is necessary cannot be otherwise. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 88b32) | |
| A reaction: If the next interesting question is the source of necessity, then the question seems to be 'what prevents it from being otherwise?'. |
| 1690 | A stone travels upwards by a forced necessity, and downwards by natural necessity [Aristotle] |
| Full Idea: There are two types of necessity, one according to nature and impulse, the other by force and contrary to impulse. A stone travels upwards and downwards from different necessities. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 94b38) |
| 11958 | Impossibilites are easily conceived in mathematics and geometry [Reid, by Molnar] |
| Full Idea: Reid pointed out how easily conceivable mathematical and geometric impossibilities are. | |
| From: report of Thomas Reid (Essays on Intellectual Powers 4: Conception [1785], IV.III) by George Molnar - Powers 11.3 | |
| A reaction: The defence would be that you have to really really conceive them, and the only way the impossible can be conceived is by blurring it at the crucial point, or by claiming to conceive more than you actually can |
| 12072 | For Aristotle knowledge is explanatory, involving understanding, and principles or causes [Aristotle, by Witt] |
| Full Idea: For Aristotle, knowledge is explanatory, for to know something is to understand it, and to understand something is to grasp its principles or causes. | |
| From: report of Aristotle (Posterior Analytics [c.327 BCE]) by Charlotte Witt - Substance and Essence in Aristotle 1.2 | |
| A reaction: Thus the kind of 'knowledge' displayed in quiz shows would not count as knowledge at all, if it was mere recall of facts. To know is to be able to explain, which is to be able to teach. See Idea 11241. |
| 12073 | 'Episteme' means grasping causes, universal judgments, explanation, and teaching [Aristotle, by Witt] |
| Full Idea: For Aristotle, a person who has 'episteme' grasps the cause of a given phenomenon, can make a universal judgment about it, can explain it, and can teach others about it. | |
| From: report of Aristotle (Posterior Analytics [c.327 BCE]) by Charlotte Witt - Substance and Essence in Aristotle 1.2 | |
| A reaction: This I take to be the context in which we should understand what Aristotle means by an 'essence' - it is the source of all of the above, so it both makes a thing what it is, and explains why it shares features with other such things. |
| 12378 | The reason why is the key to knowledge [Aristotle] |
| Full Idea: Study of the reason why has the most importance for knowledge. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 79a24) | |
| A reaction: I take the study of reasons for belief to be much more central to epistemology than finding ways to answer radical sceptics about the basic possibility of knowledge. |
| 12364 | We understand a thing when we know its explanation and its necessity [Aristotle] |
| Full Idea: We understand something simpliciter when we think we know of the explanation because of which the object holds that it is its explanation, and also that it is not possible for it to be otherwise. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 71b10) | |
| A reaction: The second half sounds odd, since we ought to understand that something could have been otherwise, and knowing whether or not it could have been otherwise is part of the understanding. It sounds like Spinozan determinism. |
| 12370 | Some understanding, of immediate items, is indemonstrable [Aristotle] |
| Full Idea: Not all understanding is demonstrative: rather, in the case of immediate items understanding is indemonstrable. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 72b19) | |
| A reaction: These are the foundations of Aristotle's epistemology, and I take it that they can be both empiricist and rationalist - sense experiences, and a priori intuitions. |
| 12366 | We only understand something when we know its explanation [Aristotle] |
| Full Idea: We only understand something when we know its explanation. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 71b30) | |
| A reaction: If we believe that the whole aim of philosophy is 'understanding' (Idea 543) - and if it isn't then I am not sure what the aim is, and alternative aims seem a lot less interesting - then we should care very much about explanations, as well as reasons. |
| 1685 | No one has mere belief about something if they think it HAS to be true [Aristotle] |
| Full Idea: No one holds something as an opinion when he thinks that it is impossible for it to be otherwise - for then he thinks he understands it. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 89a07) |
| 1673 | Knowledge proceeds from principles, so it is hard to know if we know [Aristotle] |
| Full Idea: It is difficult to know whether you know something or not. For it is difficult to know whether or not our knowledge of something proceeds from its principles - and this is what it is to know something. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 76a25) |
| 12379 | You cannot understand anything through perception [Aristotle] |
| Full Idea: You cannot understand anything through perception. Demonstrations are universal, and universals cannot be perceived. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 87b28) |
| 16725 | Some knowledge is lost if you lose a sense, and there is no way the knowledge can be replaced [Aristotle] |
| Full Idea: The loss of any one of the senses entails the loss of a corresponding portion of knowledge, and since we learn either by induction or by demonstration, this knowledge cannot be acquired. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 81a37) | |
| A reaction: This suggests Jackson's 'knowledge argument', that raw experience contains some genuine knowledge, for which there is no mechanistic substitute. Not that I accept…. |
| 23309 | Aristotle's concepts of understanding and explanation mean he is not a pure empiricist [Aristotle, by Frede,M] |
| Full Idea: It is a certain notion of understanding and, correspondingly, explanation which makes Aristotle think that knowledge, properly speaking, could not be a matter of mere experience. | |
| From: report of Aristotle (Posterior Analytics [c.327 BCE]) by Michael Frede - Aristotle's Rationalism p.160 | |
| A reaction: Frede says this means that Aristotle is a rationalist, though few empiricists think understanding is 'merely' a matter of experience. My own epistemology is Explanatory Empiricism, which I see as more empiricist than rationalist. |
| 1693 | Animals may have some knowledge if they retain perception, but understanding requires reasons to be given [Aristotle] |
| Full Idea: In some animals the perception is retained, and in some not. Without retention knowledge is impossible. Some animals go further and form an account based on the perception. This leads to memory and experience, and so to either skill or understanding. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 99b35-) |
| 9067 | Many memories of the same item form a single experience [Aristotle] |
| Full Idea: When it occurs often in connection with the same item, ..memories which are many in number form a single experience. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 100a05) | |
| A reaction: This is Aristotle at his most empirical. He is not describing an operation of the understanding, but a process of association. The process he alludes to is at the heart of the abstractionist view of concept-formation. |
| 1671 | Sceptics say justification is an infinite regress, or it stops at the unknowable [Aristotle] |
| Full Idea: Sceptics say that there is either an infinite regress of ideas based on one another, or things come to a stop at primitives which are unknowable (because they can't be demonstrated). | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 72b09) | |
| A reaction: This is one strand of what eventually becomes the classic Agrippa's Trilemma (Idea 8850). For Aristotle's view on this one, see Idea 562. |
| 1670 | When you understand basics, you can't be persuaded to change your mind [Aristotle] |
| Full Idea: Anyone who understands anything simpliciter (as basic) must be incapable of being persuaded to change his mind. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 72b04) | |
| A reaction: A typical Aristotle test which seems rather odd to us. Surely I can change my mind, and decide that something is not basic after all? But, says Aristotle, then you didn't really think it was basic. |
| 1691 | Aim to get definitions of the primitive components, thus establishing the kind, and work towards the attributes [Aristotle] |
| Full Idea: Divide a whole into its primitives, then try to get definitions of these. Thus you establish the kind, and then study the attributes through the primitive common items. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 96b16) |
| 12383 | There must be definitions before demonstration is possible [Aristotle] |
| Full Idea: There is no demonstration of anything of which there is no definition. Definitions are of what something is, i.e. of its essence, but all demonstrations clearly suppose and assume what a thing is. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 90b30) | |
| A reaction: Note that while essentialism rests on definitions, the job is not yet complete once the definitions are done. With good definitions, it should be easy to show how the pieces of the jigsaw fit together. |
| 1674 | All demonstration is concerned with existence, axioms and properties [Aristotle] |
| Full Idea: All demonstrative science [apodeiktike episteme] is concerned with three things: what it posits to exist (the kind), the axioms (primitives basic to demonstration), and the attributes. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 76b12) |
| 24068 | Demonstration is more than entailment, as the explanatory order must match the causal order [Aristotle, by Koslicki] |
| Full Idea: Aristotle's demonstration encompasses more than deductive entailment, in that the explanatory order of priority represented in a successful demonstration must mirror precisely the causal order of priority in the phenomena in question. | |
| From: report of Aristotle (Posterior Analytics [c.327 BCE]) by Kathrin Koslicki - Form, Matter and Substance 4.5 | |
| A reaction: Interesting. I presume this is correct, but is not an aspect I had registered. In Metaphysics his essentialist explanations are causal, so it all hangs together. |
| 17310 | Aristotle gets asymmetric consequence from demonstration, which reflects real causal priority [Aristotle, by Koslicki] |
| Full Idea: In Aristotle's system, the relevant notion of asymmetric consequence that is operative in his model of scientific explanation is that of demonstration. ...It is a theoretical/linguistic reflection of an asymmetric real-world relation of causal priority. | |
| From: report of Aristotle (Posterior Analytics [c.327 BCE]) by Kathrin Koslicki - Varieties of Ontological Dependence 7.3 n7 | |
| A reaction: The asymmetry is required for explanation, and for grounding. |
| 21359 | Aristotle doesn't actually apply his theory of demonstration to his practical science [Leroi on Aristotle] |
| Full Idea: There is a conflict between the syllogistic theory of demonstration of the Posterior Analytics, with its austere programme of certainties, and how Aristotle actually does science. | |
| From: comment on Aristotle (Posterior Analytics [c.327 BCE]) by Armand Marie LeRoi - The Lagoon: how Aristotle invented science 104 | |
| A reaction: Leroi observes that there are no demonstrations anywhere in the biological writings. Biology probably lends itself least to such an approach. |
| 1667 | Premises must be true, primitive and immediate, and prior to and explanatory of conclusions [Aristotle] |
| Full Idea: Demonstrative understanding must proceed from items which are true and primitive and immediate and more familiar and prior to and explanatory of the conclusions. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 71b22) |
| 12365 | We can know by demonstration, which is a scientific deduction leading to understanding [Aristotle] |
| Full Idea: We know things through demonstration, by which I mean a scientific deduction, and by 'scientific' I mean a deduction by possessing which we understand something. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 71b17) | |
| A reaction: This is a distinctively Aristotelian account of what science aims at, and which seems to have dropped out of modern accounts of science, which are still under the influence of logical positivism. Time to revive it. |
| 10918 | Demonstrative understanding rests on necessary features of the thing in itself [Aristotle] |
| Full Idea: If demonstrative understanding proceeds from necessary principles, and whatever holds of an object in itself is necessary, then it is clear that demonstrative deductions will proceed from certain items of this sort. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 74b05-) | |
| A reaction: This is the characterization of the essence of something in terms of what counts as a good explanation of that thing. Although explanation is a bit subjective, I like this approach, because you will dig down to the source of the powers of the thing. |
| 12374 | Demonstrations must be necessary, and that depends on the middle term [Aristotle] |
| Full Idea: If you understand something demonstratively, it must hold from necessity, so it is plain that your demonstration must proceed through a middle term which is necessary. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 75a13) | |
| A reaction: How can a middle 'term' be necessary, if it is not a proposition? Presumably Socrates is necessarily a man, and men are necessarily mortal, so it is the predication which is necessary. |
| 12148 | Demonstrations are syllogisms which give explanations [Aristotle] |
| Full Idea: Demonstrations are probative deductions [sullogismos] which give the explanation [aitias] and the reason why. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 85b24) | |
| A reaction: This notion seems to have slipped out of modern philosophy of science, because (while scientists have just pressed on) philosophers of science have raised so many sceptical questions that they have, I would say, lost the plot. |
| 1679 | Universal demonstrations are about thought; particular demonstrations lead to perceptions [Aristotle] |
| Full Idea: Universal demonstrations are objects of thought, particular demonstrations terminate in perception. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 86a30) |
| 1680 | Demonstration is better with fewer presuppositions, and it is quicker if these are familiar [Aristotle] |
| Full Idea: A demonstration is superior if it depends on fewer suppositions or propositions - for if these are familiar, knowledge will come more quickly, and this is preferable. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 86a35) |
| 12147 | The principles of demonstrations are definitions [Aristotle] |
| Full Idea: The principles of demonstrations are definitions. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 90b25) | |
| A reaction: This I take to be a key idea linking Aristotle's desire to understand the world, by using demonstrations to reach good explanations. Definitions turn out to rest on essences, so our understanding of the world rests on essences. |
| 12371 | A demonstration is a deduction which proceeds from necessities [Aristotle] |
| Full Idea: A demonstration is a deduction which proceeds from necessities. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 73a24) | |
| A reaction: Elsewhere he tells us that demonstration that brings understanding (Idea 12365), so this is an interesting gloss. He says that the middle term of the syllogism gives the understanding, but necessities reside in the whole propositions of the premisses. |
| 1683 | We learn universals from many particulars [Aristotle] |
| Full Idea: It is from many particulars that the universal becomes plain. Universals are valuable because they make the explanation plain. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 88a05) |
| 12367 | What is most universal is furthest away, and the particulars are nearest [Aristotle] |
| Full Idea: What is most universal is furthest away, and the particulars are nearest. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 72a05) | |
| A reaction: This is the puzzle that bother Aristotle about explanation, that we can only grasp the universals, when we want to explain the particulars. |
| 12385 | Are particulars explained more by universals, or by other particulars? [Aristotle] |
| Full Idea: Which of the middle terms is explanatory for the particulars - the one which is primitive in the direction of the universal, or the one which is primitive in the direction of the particular? | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 99b09) | |
| A reaction: I'm not clear about this, but it shows Aristotle wrestling with the issue of whether explanations are of particulars or universals, and whether they employ particulars as well as employing universals. The particular must be defined! |
| 12380 | Universals are valuable because they make the explanations plain [Aristotle] |
| Full Idea: Universals are valuable because they make the explanations plain. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 88a06) | |
| A reaction: Everything in Aristotle comes back to human capacity to understand. There seems to be an ideal explanation consisting entirely of particulars, but humans are not equipped to grasp it. We think in a broad brush way. |
| 1689 | Explanation is of the status of a thing, inferences to it, initiation of change, and purpose [Aristotle] |
| Full Idea: There are four sorts of explanation: what it is to be something, that if certain items hold it is necessary for this to hold, what initiated the change, and the purpose. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 94a21) | |
| A reaction: This might be summed up as: 'we want to know the essence, the necessary conditions, the cause, and the purpose'. Can anyone improve on that as the aims of explanation? The second explanation (necessary preconditions) isn't in 'Physics' - Idea 8332. |
| 1686 | What we seek and understand are facts, reasons, existence, and identity [Aristotle] |
| Full Idea: The things we seek are equal in number to those we understand: the fact, the reason why, if something is, and what something is. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 89b24) |
| 12357 | Explanation and generality are inseparable [Aristotle, by Wedin] |
| Full Idea: For Aristotle, explanation and generality are fellow-travellers. | |
| From: report of Aristotle (Posterior Analytics [c.327 BCE]) by Michael V. Wedin - Aristotle's Theory of Substance X.11 | |
| A reaction: This isn't 'lawlike' explanation, but it is interestingly close to it. It seems to be based on the fact that predicates are universals, so we can only state truths in general terms. |
| 1669 | The foundation or source is stronger than the thing it causes [Aristotle] |
| Full Idea: Something always holds better because of that because of which it holds - e.g. that because of which we love something is better loved. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 72a30) |
| 1678 | Universals give better explanations, because they are self-explanatory and primitive [Aristotle] |
| Full Idea: Universals are more explanatory (for something which holds in itself is itself explanatory of itself; and universals are primitive; hence universals are explanatory) - so universal demonstrations are better. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 85b25) |
| 9068 | Perception creates primitive immediate principles by building a series of firm concepts [Aristotle] |
| Full Idea: Primitive immediate principles ...come about from perception - as in a battle, when a rout has occurred, first one man makes a stand, then another, and then another, until a position of strength is reached. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 100a12) | |
| A reaction: Philosophers don't create imagery like that any more. This empiricist account of how concepts and universals are created is part of a campaign against Plato's theory of forms. [Idea 9069 continues his idea] |
| 9069 | A perception lodging in the soul creates a primitive universal, which becomes generalised [Aristotle] |
| Full Idea: When one undifferentiated item in perception makes a stand, there is a primitive universal in the soul; for although you perceive particulars, perception is of universals - e.g. of man, not of Callias the man. One animal makes a stand, until animal does. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 100a15-) | |
| A reaction: This is the quintessential account of abstractionism, with the claim that primitive universals arise directly in perception, but only in repeated perception. How the soul does it is a mystery to Aristotle, just as associations are a mystery to Hume. |
| 9070 | We learn primitives and universals by induction from perceptions [Aristotle] |
| Full Idea: We must get to know the primitives by induction; for this is the way in which perception instils universals. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 100b04) | |
| A reaction: This statement is so strongly empirical it could have come from John Stuart Mill. The modern post-Fregean view of universals is essentially platonist - that they have a life and logic of their own, and their method of acquisition is irrelevant. |
| 23646 | Reference is by name, or a term-plus-circumstance, or ostensively, or by description [Reid] |
| Full Idea: An individual is expressed by a proper name, or by a general word joined to distinguishing circumstances; if unknown, it may be pointed out to the senses; when beyond the reach of the senses it may be picked out by an imperfect but true description. | |
| From: Thomas Reid (Essays on Intellectual Powers 4: Conception [1785], 1) | |
| A reaction: [compressed] If Putnam, Kripke and Donnellan had read this paragraph they could have save themselves a lot of work! I take reference to be the activity of speakers and writers, and these are the main tools of the trade. |
| 23645 | A word's meaning is the thing conceived, as fixed by linguistic experts [Reid] |
| Full Idea: The meaning of a word (such as 'felony') is the thing conceived; and that meaning is the conception affixed to it by those who best understand the language. | |
| From: Thomas Reid (Essays on Intellectual Powers 4: Conception [1785], 1) | |
| A reaction: He means legal experts. This is precisely that same as Putnam's account of the meaning of 'elm tree'. His discussion here of reference is the earliest I have encountered, and it is good common sense (for which Reid is famous). |
| 12368 | Negation takes something away from something [Aristotle] |
| Full Idea: The part of a contradictory pair which says something of something is an affirmation; the part which takes something from something is a negation. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 72a14) | |
| A reaction: So affirmation is predication about an object ['Fa'], and negation is denial of predication. We have a scope problem: there is nothing which is F [¬∃x(Fx)], or there is a thing which is not-F [∃x(¬Fx)]. Aristotle seems to mean the latter. |
| 1692 | If you shouldn't argue in metaphors, then you shouldn't try to define them either [Aristotle] |
| Full Idea: If you should not argue in metaphors, it is plain too that you should neither define by metaphors nor define what is said in metaphors; for then you will necessarily argue in metaphors. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 97b37) | |
| A reaction: Impeccable logic, but seeing a similarity can be a wonderful shortcut to seeing a great truth. |
| 12375 | Whatever holds of a kind intrinsically holds of it necessarily [Aristotle] |
| Full Idea: In each kind, whatever holds of something in itself and as such holds of it from necessity. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 75a30) | |
| A reaction: This seems to confirm the view that essential properties are necessary, but it does not, of course, follow that all necessary properties are essential properties (e.g. trivial necessities are not essential). |
| 1688 | Properties must be proved, but not essence; but existents are not a kind, so existence isn't part of essence [Aristotle] |
| Full Idea: Everything which a thing is must be proved through a demonstration - except its essence. But existence is not the essence of anything; for the things that exist do not constitute a kind. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 92b14) |