44 ideas
| 7689 | The modal logic of C.I.Lewis was only interpreted by Kripke and Hintikka in the 1960s [Jacquette] |
| Full Idea: The modal syntax and axiom systems of C.I.Lewis (1918) were formally interpreted by Kripke and Hintikka (c.1965) who, using Z-F set theory, worked out model set-theoretical semantics for modal logics and quantified modal logics. | |
| From: Dale Jacquette (Ontology [2002], Ch. 2) | |
| A reaction: A historical note. The big question is always 'who cares?' - to which the answer seems to be 'lots of people', if they are interested in precision in discourse, in artificial intelligence, and maybe even in metaphysics. Possible worlds started here. |
| 17926 | Rejecting double negation elimination undermines reductio proofs [Colyvan] |
| Full Idea: The intuitionist rejection of double negation elimination undermines the important reductio ad absurdum proof in classical mathematics. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) |
| 17925 | Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan] |
| Full Idea: In intuitionist logic double negation elimination fails. After all, proving that there is no proof that there can't be a proof of S is not the same thing as having a proof of S. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
| A reaction: I do like people like Colyvan who explain things clearly. All of this difficult stuff is understandable, if only someone makes the effort to explain it properly. |
| 7681 | Logic describes inferences between sentences expressing possible properties of objects [Jacquette] |
| Full Idea: It is fundamental that logic depends on logical possibilities, in which logically possible properties are predicated of logically possible objects. Logic describes inferential structures among sentences expressing the predication of properties to objects. | |
| From: Dale Jacquette (Ontology [2002], Ch. 2) | |
| A reaction: If our imagination is the only tool we have for assessing possibilities, this leaves the domain of logic as being a bit subjective. There is an underlying Platonism to the idea, since inferences would exist even if nothing else did. |
| 7682 | Logic is not just about signs, because it relates to states of affairs, objects, properties and truth-values [Jacquette] |
| Full Idea: At one level logic can be regarded as a theory of signs and formal rules, but we cannot neglect the meaning of propositions as they relate to states of affairs, and hence to possible properties and objects... there must be the possibility of truth-values. | |
| From: Dale Jacquette (Ontology [2002], Ch. 2) | |
| A reaction: Thus if you define logical connectives by truth tables, you need the concept of T and F. You could, though, regard those too as purely formal (like 1 and 0 in electronics). But how do you decide which propositions are 1, and which are 0? |
| 17924 | Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan] |
| Full Idea: The law of excluded middle (for every proposition P, either P or not-P) must be carefully distinguished from its semantic counterpart bivalence, that every proposition is either true or false. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
| A reaction: So excluded middle makes no reference to the actual truth or falsity of P. It merely says P excludes not-P, and vice versa. |
| 7697 | On Russell's analysis, the sentence "The winged horse has wings" comes out as false [Jacquette] |
| Full Idea: It is infamous that on Russell's analysis the sentences "The winged horse has wings" and "The winged horse is a horse" are false, because in the extant domain of actual existent entities there contingently exist no winged horses | |
| From: Dale Jacquette (Ontology [2002], Ch. 6) | |
| A reaction: This is the best objection I have heard to Russell's account of definite descriptions. The connected question is whether 'quantifies over' is really a commitment to existence. See Idea 6067. |
| 17929 | Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan] |
| Full Idea: Löwenheim proved that if a first-order sentence has a model at all, it has a countable model. ...Skolem generalised this result to systems of first-order sentences. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2) |
| 17930 | Axioms are 'categorical' if all of their models are isomorphic [Colyvan] |
| Full Idea: A set of axioms is said to be 'categorical' if all models of the axioms in question are isomorphic. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2) | |
| A reaction: The best example is the Peano Axioms, which are 'true up to isomorphism'. Set theory axioms are only 'quasi-isomorphic'. |
| 7701 | Can a Barber shave all and only those persons who do not shave themselves? [Jacquette] |
| Full Idea: The Barber Paradox refers to the non-existent property of being a barber who shaves all and only those persons who do not shave themselves. | |
| From: Dale Jacquette (Ontology [2002], Ch. 9) | |
| A reaction: [Russell spotted this paradox, and it led to his Theory of Types]. This paradox may throw light on the logic of indexicals. What does "you" mean when I say to myself "you idiot!"? If I can behave as two persons, so can the barber. |
| 17928 | Ordinal numbers represent order relations [Colyvan] |
| Full Idea: Ordinal numbers represent order relations. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.2.3 n17) |
| 17923 | Intuitionists only accept a few safe infinities [Colyvan] |
| Full Idea: For intuitionists, all but the smallest, most well-behaved infinities are rejected. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
| A reaction: The intuitionist idea is to only accept what can be clearly constructed or proved. |
| 17941 | Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan] |
| Full Idea: The problem with infinitesimals is that in some places they behaved like real numbers close to zero but in other places they behaved like zero. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 7.1.2) | |
| A reaction: Colyvan gives an example, of differentiating a polynomial. |
| 17922 | Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan] |
| Full Idea: Given Dedekind's reduction of real numbers to sequences of rational numbers, and other known reductions in mathematics, it was tempting to see basic arithmetic as the foundation of mathematics. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.1) | |
| A reaction: The reduction is the famous Dedekind 'cut'. Nowadays theorists seem to be more abstract (Category Theory, for example) instead of reductionist. |
| 17936 | Transfinite induction moves from all cases, up to the limit ordinal [Colyvan] |
| Full Idea: Transfinite inductions are inductive proofs that include an extra step to show that if the statement holds for all cases less than some limit ordinal, the statement also holds for the limit ordinal. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1 n11) |
| 17940 | Most mathematical proofs are using set theory, but without saying so [Colyvan] |
| Full Idea: Most mathematical proofs, outside of set theory, do not explicitly state the set theory being employed. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 7.1.1) |
| 17931 | Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan] |
| Full Idea: Structuralism is able to explain why mathematicians are typically only interested in describing the objects they study up to isomorphism - for that is all there is to describe. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 3.1.2) |
| 17932 | If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan] |
| Full Idea: In re structuralism does not posit anything other than the kinds of structures that are in fact found in the world. ...The problem is that the world may not provide rich enough structures for the mathematics. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 3.1.2) | |
| A reaction: You can perceive a repeating pattern in the world, without any interest in how far the repetitions extend. |
| 7707 | To grasp being, we must say why something exists, and why there is one world [Jacquette] |
| Full Idea: We grasp the concept of being only when we have satisfactorily answered the question why there is something rather than nothing and why there is only one logically contingent actual world. | |
| From: Dale Jacquette (Ontology [2002], Conclusion) | |
| A reaction: See Ideas 7688 and 7692 for a glimpse of Jacquette's answer. Personally I don't yet have a full grasp of the concept of being, but I'm sure I'll get there if I only work a bit harder. |
| 7692 | Being is maximal consistency [Jacquette] |
| Full Idea: Being is maximal consistency. | |
| From: Dale Jacquette (Ontology [2002], Ch. 2) | |
| A reaction: You'll have to read Ch.2 of Jacquette to see what this is all about, but as it stands it is a lovely slogan, and a wonderful googly/curve ball to propel at Parmenides or Heidegger. |
| 7687 | Existence is completeness and consistency [Jacquette] |
| Full Idea: A combinatorial ontology holds that existence is nothing more or less than completeness and consistency, or what is also called 'maximal consistency'. | |
| From: Dale Jacquette (Ontology [2002], Ch. 2) | |
| A reaction: You'll have to read Jacquette to understand this one! The claim is that existence is to be defined in terms of logic (and whatever is required for logic). I take this to be a bit Platonist (rather than conventionalist) about logic. |
| 7679 | Ontology is the same as the conceptual foundations of logic [Jacquette] |
| Full Idea: The principles of pure philosophical ontology are indistinguishable ... from the conceptual foundations of logic. | |
| From: Dale Jacquette (Ontology [2002], Pref) | |
| A reaction: I would take Russell to be an originator of this view. If the young Wittgenstein showed that the foundations of logic are simply conventional (truth tables), this seems to make ontology conventional too, which sounds very odd indeed (to me). |
| 7678 | Ontology must include the minimum requirements for our semantics [Jacquette] |
| Full Idea: The entities included in a theoretical ontology are those minimally required for an adequate philosophical semantics. ...These are the objects that we say exist, to which we are ontologically committed. | |
| From: Dale Jacquette (Ontology [2002], Pref) | |
| A reaction: Worded with exquisite care! He does not say that ontology is reducible to semantics (which is a silly idea). We could still be committed, as in a ghost story, to existence of some 'nameless thing'. Things utterly beyond our ken might exist. |
| 7683 | Logic is based either on separate objects and properties, or objects as combinations of properties [Jacquette] |
| Full Idea: Logic involves the possibilities of predicating properties of objects in a conceptual scheme wherein either objects and properties are included in altogether separate categories, or objects are reducible to combinations of properties. | |
| From: Dale Jacquette (Ontology [2002], Ch. 2) | |
| A reaction: In the first view, he says that objects are just 'logical pegs' for properties. Objects can't be individuated without properties. But combinations of properties would seem to need essences, or else they are too unstable to count as objects. |
| 7684 | Reduce states-of-affairs to object-property combinations, and possible worlds to states-of-affairs [Jacquette] |
| Full Idea: We can reduce references to states-of-affairs to object-property combinations, and we can reduce logically possible worlds to logically possible states-of-affairs combinations. | |
| From: Dale Jacquette (Ontology [2002], Ch. 2) | |
| A reaction: If we further reduce object-property combinations to mere combinations of properties (Idea 7683), then we have reduced our ontology to nothing but properties. Wow. We had better be very clear, then, about what a property is. I'm not. |
| 7703 | If classes can't be eliminated, and they are property combinations, then properties (universals) can't be either [Jacquette] |
| Full Idea: If classes alone cannot be eliminated from ontology on Quine's terms, and if classes are defined as property combinations, then neither are all properties, universals in the tradition sense, entirely eliminable. | |
| From: Dale Jacquette (Ontology [2002], Ch. 9) | |
| A reaction: If classes were totally conventional (and there was no such things as a 'natural' class) then you might admit something to a class without knowing its properties (as 'the thing in the box'). |
| 7685 | An object is a predication subject, distinguished by a distinctive combination of properties [Jacquette] |
| Full Idea: To be an object is to be a predication subject, and to be this as opposed to that particular object, whether existent or not, is to have a distinctive combination of properties. | |
| From: Dale Jacquette (Ontology [2002], Ch. 2) | |
| A reaction: The last part depends on Leibniz's Law. The difficulty is that two objects may only be distinguishable by being in different places, and location doesn't look like a property. Cf. Idea 5055. |
| 7699 | Numbers, sets and propositions are abstract particulars; properties, qualities and relations are universals [Jacquette] |
| Full Idea: Roughly, numbers, sets and propositions are assumed to be abstract particulars, while properties, including qualities and relations, are usually thought to be universals. | |
| From: Dale Jacquette (Ontology [2002], Ch. 9) | |
| A reaction: There is an interesting nominalist project of reducing all of these to particulars. Numbers to patterns, sets to their members, propositions to sentences, properties to causal powers, relations to, er, something else. |
| 7691 | The actual world is a consistent combination of states, made of consistent property combinations [Jacquette] |
| Full Idea: The actual world is a maximally consistent state-of-affairs combination involving all and only the existent objects, which in turn exist because they are maximally consistent property combinations. | |
| From: Dale Jacquette (Ontology [2002], Ch. 2) | |
| A reaction: [This extends Idea 7688]. This seems to invite the standard objections to the coherence theory of truth, such as Ideas 5422 and 4745. Is 'maximal consistency' merely a test for actuality, rather than an account of what actuality is? |
| 7688 | The actual world is a maximally consistent combination of actual states of affairs [Jacquette] |
| Full Idea: The actual world can be defined as a maximally consistent combination of actual states of affairs, or maximally consistent states-of-affairs combination. | |
| From: Dale Jacquette (Ontology [2002], Ch. 2) | |
| A reaction: A key part of Jacquette's program of deriving ontological results from the foundations of logic. Is the counterfactual situation of my pen being three centimetres to the left of its current position a "less consistent" situation than the actual one? |
| 7695 | Do proposition-structures not associated with the actual world deserve to be called worlds? [Jacquette] |
| Full Idea: Many modal logicians in their philosophical moments have raised doubts about whether structures of propositions not associated with the actual world deserved to be called worlds at all. | |
| From: Dale Jacquette (Ontology [2002], Ch. 2) | |
| A reaction: A good question. Consistency is obviously required, but we also need a lot of propositions before we would consider it a 'world'. Very remote but consistent worlds quickly become unimaginable. Does that matter? |
| 7694 | We must experience the 'actual' world, which is defined by maximally consistent propositions [Jacquette] |
| Full Idea: Conventional modal semantics, in which all logically possible worlds are defined in terms of maximally consistent proposition sets, has no choice except to allow that the actual world is the world we experience in sensation, or that we inhabit. | |
| From: Dale Jacquette (Ontology [2002], Ch. 9) | |
| A reaction: Jacquette dislikes this because he is seeking an account of ontology that doesn't, as so often, merely reduce to epistemology (e.g. Berkeley). See Idea 7691 for his preferred account. |
| 17943 | Probability supports Bayesianism better as degrees of belief than as ratios of frequencies [Colyvan] |
| Full Idea: Those who see probabilities as ratios of frequencies can't use Bayes's Theorem if there is no objective prior probability. Those who accept prior probabilities tend to opt for a subjectivist account, where probabilities are degrees of belief. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 9.1.8) | |
| A reaction: [compressed] |
| 17939 | Mathematics can reveal structural similarities in diverse systems [Colyvan] |
| Full Idea: Mathematics can demonstrate structural similarities between systems (e.g. missing population periods and the gaps in the rings of Saturn). | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 6.3.2) | |
| A reaction: [Colyvan expounds the details of his two examples] It is these sorts of results that get people enthusiastic about the mathematics embedded in nature. A misunderstanding, I think. |
| 17938 | Mathematics can show why some surprising events have to occur [Colyvan] |
| Full Idea: Mathematics can show that under a broad range of conditions, something initially surprising must occur (e.g. the hexagonal structure of honeycomb). | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 6.3.2) |
| 17934 | Proof by cases (by 'exhaustion') is said to be unexplanatory [Colyvan] |
| Full Idea: Another style of proof often cited as unexplanatory are brute-force methods such as proof by cases (or proof by exhaustion). | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) |
| 17933 | Reductio proofs do not seem to be very explanatory [Colyvan] |
| Full Idea: One kind of proof that is thought to be unexplanatory is the 'reductio' proof. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) | |
| A reaction: Presumably you generate a contradiction, but are given no indication of why the contradiction has arisen? Tracking back might reveal the source of the problem? Colyvan thinks reductio can be explanatory. |
| 17935 | If inductive proofs hold because of the structure of natural numbers, they may explain theorems [Colyvan] |
| Full Idea: It might be argued that any proof by induction is revealing the explanation of the theorem, namely, that it holds by virtue of the structure of the natural numbers. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) | |
| A reaction: This is because induction characterises the natural numbers, in the Peano Axioms. |
| 17942 | Can a proof that no one understands (of the four-colour theorem) really be a proof? [Colyvan] |
| Full Idea: The proof of the four-colour theorem raises questions about whether a 'proof' that no one understands is a proof. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 9.1.6) | |
| A reaction: The point is that the theorem (that you can colour countries on a map with just four colours) was proved with the help of a computer. |
| 7706 | If qualia supervene on intentional states, then intentional states are explanatorily fundamental [Jacquette] |
| Full Idea: If qualia supervene on intentional states, then intentionality is also more explanatorily fundamental than qualia. | |
| From: Dale Jacquette (Ontology [2002], Ch.10) | |
| A reaction: See Idea 7272 for opposite view. Maybe intentional states are large mental objects of which we are introspectively aware, but which are actually composed of innumerable fine-grained qualia. Intentional states would only explain qualia if they caused them. |
| 17937 | Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan] |
| Full Idea: One type of generalisation in mathematics extends a system to go beyond what is was originally set up for; another kind involves abstracting away from some details in order to capture similarities between different systems. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.2) |
| 7704 | Reduction of intentionality involving nonexistent objects is impossible, as reduction must be to what is actual [Jacquette] |
| Full Idea: If intentionality sometimes involves a relation to nonexistent objects, like my dreamed-of visit to a Greek island, then such thoughts cannot be explained physically or causally, because only actual physical entities and events can be mentioned. | |
| From: Dale Jacquette (Ontology [2002], Ch.10) | |
| A reaction: Unimpressive. Thoughts of a Greek island will obviously reduce to memories of islands and Greece and travel brochures. Memory clearly retains past events in the present, and hence past events can be part of the material used in reductive accounts. |
| 7702 | The extreme views on propositions are Frege's Platonism and Quine's extreme nominalism [Jacquette] |
| Full Idea: The extreme ontological alternatives with respect to the metaphysics of propositions are a Fregean Platonism (his "gedanken", 'thoughts'), and a radical nominalism or inscriptionalism, as in Quine, where they are just marks related to stimuli. | |
| From: Dale Jacquette (Ontology [2002], Ch. 9) | |
| A reaction: Personally I would want something between the two - that propositions are brain events of a highly abstract kind. I say that introspection reveals pre-linguistic thoughts, which are propositions. A proposition is an intentional state. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |