19 ideas
| 8952 | We reach 'reflective equilibrium' when intuitions and theory completely align [Fisher] |
| Full Idea: A state of 'reflective equilibrium' is when our theory and our intuitions become completely aligned | |
| From: Jennifer Fisher (On the Philosophy of Logic [2008], 12.IV) | |
| A reaction: [Rawls made this concept famous] This is a helpful concept in trying to spell out the ideal which is the dream of believers in 'pure reason' - that there is a goal in which everything comes right. The problem is when people have different intuitions! |
| 8943 | Three-valued logic says excluded middle and non-contradition are not tautologies [Fisher] |
| Full Idea: In three-valued logic (L3), neither the law of excluded middle (p or not-p), nor the law of non-contradiction (not(p and not-p)) will be tautologies. If p has the value 'indeterminate' then so will not-p. | |
| From: Jennifer Fisher (On the Philosophy of Logic [2008], 07.I) | |
| A reaction: I quite accept that the world is full of indeterminate propositions, and that excluded middle and non-contradiction can sometimes be uncertain, but I am reluctant to accept that what is being offered here should be called 'logic'. |
| 8945 | Fuzzy logic has many truth values, ranging in fractions from 0 to 1 [Fisher] |
| Full Idea: In fuzzy logic objects have properties to a greater or lesser degree, and truth values are given as fractions or decimals, ranging from 0 to 1. Not-p is defined as 1-p, and other formula are defined in terms of maxima and minima for sets. | |
| From: Jennifer Fisher (On the Philosophy of Logic [2008], 07.II) | |
| A reaction: The question seems to be whether this is actually logic, or a recasting of probability theory. Susan Haack attacks it. If logic is the study of how truth is preserved as we move between propositions, then 0 and 1 need a special status. |
| 17824 | The master science is physical objects divided into sets [Maddy] |
| Full Idea: The master science can be thought of as the theory of sets with the entire range of physical objects as ur-elements. | |
| From: Penelope Maddy (Sets and Numbers [1981], II) | |
| A reaction: This sounds like Quine's view, since we have to add sets to our naturalistic ontology of objects. It seems to involve unrestricted mereology to create normal objects. |
| 8951 | Classical logic is: excluded middle, non-contradiction, contradictions imply all, disjunctive syllogism [Fisher] |
| Full Idea: For simplicity, we can say that 'classical logic' amounts to the truth of four sentences: 1) either p or not-p; 2) it is not the case that both p and not-p; 3) from p and not-p, infer q; 4) from p or q and not-p, infer q. | |
| From: Jennifer Fisher (On the Philosophy of Logic [2008], 12.I) | |
| A reaction: [She says there are many ways of specifying classical logic] Intuition suggests that 2 and 4 are rather hard to dispute, while 1 is ignoring some grey areas, and 3 is totally ridiculous. There is, of course, plenty of support for 3! |
| 8950 | Logic formalizes how we should reason, but it shouldn't determine whether we are realists [Fisher] |
| Full Idea: Even if one is inclined to be a realist about everything, it is hard to see why our logic should be the determiner. Logic is supposed to formalize how we ought to reason, but whether or not we should be realists is a matter of philosophy, not logic. | |
| From: Jennifer Fisher (On the Philosophy of Logic [2008], 09.I) | |
| A reaction: Nice to hear a logician saying this. I do not see why talk in terms of an object is a commitment to its existence. We can discuss the philosopher's stone, or Arthur's sword, or the Loch Ness monster, or gravitinos, with degrees of commitment. |
| 17825 | Set theory (unlike the Peano postulates) can explain why multiplication is commutative [Maddy] |
| Full Idea: If you wonder why multiplication is commutative, you could prove it from the Peano postulates, but the proof offers little towards an answer. In set theory Cartesian products match 1-1, and n.m dots when turned on its side has m.n dots, which explains it. | |
| From: Penelope Maddy (Sets and Numbers [1981], II) | |
| A reaction: 'Turning on its side' sounds more fundamental than formal set theory. I'm a fan of explanation as taking you to the heart of the problem. I suspect the world, rather than set theory, explains the commutativity. |
| 17826 | Standardly, numbers are said to be sets, which is neat ontology and epistemology [Maddy] |
| Full Idea: The standard account of the relationship between numbers and sets is that numbers simply are certain sets. This has the advantage of ontological economy, and allows numbers to be brought within the epistemology of sets. | |
| From: Penelope Maddy (Sets and Numbers [1981], III) | |
| A reaction: Maddy votes for numbers being properties of sets, rather than the sets themselves. See Yourgrau's critique. |
| 17828 | Numbers are properties of sets, just as lengths are properties of physical objects [Maddy] |
| Full Idea: I propose that ...numbers are properties of sets, analogous, for example, to lengths, which are properties of physical objects. | |
| From: Penelope Maddy (Sets and Numbers [1981], III) | |
| A reaction: Are lengths properties of physical objects? A hole in the ground can have a length. A gap can have a length. Pure space seems to contain lengths. A set seems much more abstract than its members. |
| 17827 | Sets exist where their elements are, but numbers are more like universals [Maddy] |
| Full Idea: A set of things is located where the aggregate of those things is located, ...but a number is simultaneously located at many different places (10 in my hand, and a baseball team) ...so numbers seem more like universals than particulars. | |
| From: Penelope Maddy (Sets and Numbers [1981], III) | |
| A reaction: My gut feeling is that Maddy's master idea (of naturalising sets by building them from ur-elements of natural objects) won't work. Sets can work fine in total abstraction from nature. |
| 17830 | Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy] |
| Full Idea: I am not suggesting a reduction of number theory to set theory ...There are only sets with number properties; number theory is part of the theory of finite sets. | |
| From: Penelope Maddy (Sets and Numbers [1981], V) |
| 17823 | If mathematical objects exist, how can we know them, and which objects are they? [Maddy] |
| Full Idea: The popular challenges to platonism in philosophy of mathematics are epistemological (how are we able to interact with these objects in appropriate ways) and ontological (if numbers are sets, which sets are they). | |
| From: Penelope Maddy (Sets and Numbers [1981], I) | |
| A reaction: These objections refer to Benacerraf's two famous papers - 1965 for the ontology, and 1973 for the epistemology. Though he relied too much on causal accounts of knowledge in 1973, I'm with him all the way. |
| 17829 | Number words are unusual as adjectives; we don't say 'is five', and numbers always come first [Maddy] |
| Full Idea: Number words are not like normal adjectives. For example, number words don't occur in 'is (are)...' contexts except artificially, and they must appear before all other adjectives, and so on. | |
| From: Penelope Maddy (Sets and Numbers [1981], IV) | |
| A reaction: [She is citing Benacerraf's arguments] |
| 8946 | We could make our intuitions about heaps precise with a million-valued logic [Fisher] |
| Full Idea: We could construct a 1,000,000-valued logic that would allow our intuitions concerning a heap to vary exactly with the amount of sand in the heap. | |
| From: Jennifer Fisher (On the Philosophy of Logic [2008]) | |
| A reaction: Presumably only an infinite number of grains of sand would then produce a true heap, and even one grain would count as a bit of a heap, which must both be wrong, so I can't see this helping much. |
| 8944 | Vagueness can involve components (like baldness), or not (like boredom) [Fisher] |
| Full Idea: Vague terms come in at least two different kinds: those whose constituent parts come in discrete packets (bald, rich, red) and those that don't (beauty, boredom, niceness). | |
| From: Jennifer Fisher (On the Philosophy of Logic [2008], 07.II) | |
| A reaction: The first group seem to be features of the external world, and the second all occur in the mind. Baldness may be vague, but presumably hairs are (on the whole) not. Nature doesn't care whether someone is actually 'bald' or not. |
| 8941 | We can't explain 'possibility' in terms of 'possible' worlds [Fisher] |
| Full Idea: Explaining 'it is possible that p' by saying p is true in at least one possible world doesn't get me very far. If I don't understand what possibility is, then appealing to possible worlds is not going to do me much good. | |
| From: Jennifer Fisher (On the Philosophy of Logic [2008], 06.III) | |
| A reaction: This seems so blatant that I assume friends of possible worlds will have addressed the problem. Note that you will also need to understand 'possible' to define necessity as 'true in all possible worlds'. Necessarily-p is not-possibly-not-p. |
| 8947 | If all truths are implied by a falsehood, then not-p might imply both q and not-q [Fisher] |
| Full Idea: If all truths are implied by a falsehood, then 'if there are no trees in the park then there is no shade' and 'if there are no trees in the park there is plenty of shade' both come out as true. Intuitively, though, the second one is false. | |
| From: Jennifer Fisher (On the Philosophy of Logic [2008], 08.I) | |
| A reaction: The rule that a falsehood implies all truths must be the weakest idea in classical logic, if it actually implies a contradiction. This means we must take an interest in relevance logics. |
| 8949 | In relevance logic, conditionals help information to flow from antecedent to consequent [Fisher] |
| Full Idea: A good account of relevance logic suggests that a conditional will be true when the flow of information is such that a conditional is the device that helps information to flow from the antecedent to the consequent. | |
| From: Jennifer Fisher (On the Philosophy of Logic [2008], 08.III) | |
| A reaction: Hm. 'If you are going out, you'll need an umbrella'. This passes on information about 'out', but also brings in new information. 'If you are going out, I'm leaving you'. What flows is an interpretation of the antecedent. Tricky. |
| 7902 | The Buddha made flowers float in the air, to impress people, and make them listen [Mahavastu] |
| Full Idea: When the young Brahmin threw her two lotuses, they stood suspended in the air. This was one of the miracles by which the Buddhas impress people, to make them listen to the truth. | |
| From: Mahavastu (The Great Event [c.200], I.231-9) | |
| A reaction: Presumably this is the reason that Jesus did miracles. It is hard to spot the truth among the myriad of lies, if there is no supporting miracle to give authority to the speaker. |