18 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. |
| 17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
| Full Idea: One feature of the Axiom of Choice that troubled many mathematicians was the so-called Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres the same size as the original. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) | |
| A reaction: (The key is that the parts are non-measurable). To an outsider it is puzzling that the Axiom has been universally accepted, even though it produces such a result. Someone can explain that, I'm sure. |
| 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. |
| 17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
| Full Idea: If-thenism denies that mathematics is in the business of discovering truths about abstracta. ...[their opponents] obviously don't regard any starting point, even a consistent one, as equally worthy of investigation. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.3) | |
| A reaction: I have some sympathy with if-thenism, in that you can obviously study the implications of any 'if' you like, but deep down I agree with the critics. |
| 17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
| Full Idea: At the end of the nineteenth century there was a renewed emphasis on rigor, the central tool of which was axiomatization, along the lines of Hilbert's axioms for geometry and Dedekind's axioms for real numbers. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) |
| 17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
| Full Idea: The fact that two apparently fruitful mathematical themes turn out to coincide makes it all the more likely that they're tracking a genuine strain of mathematical depth. | |
| From: Penelope Maddy (Defending the Axioms [2011], 5.3ii) |
| 17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
| Full Idea: One form of the Continuum Hypothesis is the claim that every infinite set of reals is either countable or of the same size as the full set of reals. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.4 n40) |
| 17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
| Full Idea: Our set-theoretic methods track the underlying contours of mathematical depth. ...What sets are, most fundamentally, is markers for these contours ...they are maximally effective trackers of certain trains of mathematical fruitfulness. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.4) | |
| A reaction: This seems to make it more like a map of mathematics than the actual essence of mathematics. |
| 17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
| Full Idea: Ordinary perceptual cognition is most likely involved in our grasp of elementary arithmetic, but ...this connection to the physical world has long since been idealized away in the infinitary structures of contemporary pure mathematics. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.3) | |
| A reaction: Despite this, Maddy's quest is for a 'naturalistic' account of mathematics. She ends up defending 'objectivity' (and invoking Tyler Burge), rather than even modest realism. You can't 'idealise away' the counting of objects. I blame Cantor. |
| 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. |
| 16661 | There are two sorts of category - referring to things, and to circumstances of things [Boethius] |
| Full Idea: Is it not now clear what the difference is between items in the categories? Some serve to refer to a thing, whereas others serve to refer to the circumstances of a thing. | |
| From: Boethius (Concerning the Trinity [c.518], Ch. 4), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 12.5 |
| 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. |