16 ideas
| 14352 | '¬', '&', and 'v' are truth functions: the truth of the compound is fixed by the truth of the components [Jackson] |
| Full Idea: It is widely agreed that '¬', '&', and 'v' are 'truth functions': the truth value of a compound sentence formed using them is fully determined by the truth value or values of the component sentences. | |
| From: Frank Jackson (Conditionals [2006], 'Equiv') | |
| A reaction: A candidate for not being a truth function might be a conditional →, where the arrow adds something over and above the propositions it connects. The relationship has an additional truth value? Does A depend on B? |
| 13152 | We can talk of 'innumerable number', about the infinite points on a line [Newton] |
| Full Idea: If any man shall take the words number and sum in a larger sense, to understand things which are numberless and sumless (such as the infinite points on a line), I could allow him the contradictious phrase 'innumerable number' without absurdity. | |
| From: Isaac Newton (Letters to Bentley [1692], 1693.02.25) | |
| A reaction: [compressed] I take the key point here to be the phrase of taking number 'in a larger sense'. Like the word 'atom' in physics, the word 'number' retains its traditional reference, but has considerably shifted its scope. Amateurs must live with this. |
| 13151 | Not all infinites are equal [Newton] |
| Full Idea: It is an error that all infinites are equal. | |
| From: Isaac Newton (Letters to Bentley [1692], 1693.01.17) | |
| A reaction: There follows a discussion of the mathematicians' view of infinity. Cantor was not the first to notice that there is more than one sort of of infinity. |
| 10855 | Actual infinities are not allowed in mathematics - only limits which may increase without bound [Gauss] |
| Full Idea: I protest against the use of an infinite quantity as an actual entity; this is never allowed in mathematics. The infinite is only a manner of speaking, in which one properly speaks of limits ...which are permitted to increase without bound. | |
| From: Carl Friedrich Gauss (Letter to Shumacher [1831]), quoted by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.7 |
| 14360 | Possible worlds for subjunctives (and dispositions), and no-truth for indicatives? [Jackson] |
| Full Idea: Subjunctive conditionals are intimately connected with dispositional properties and causation. ...Consequently, a position some find attractive is that possible worlds theory applies to subjunctives, while the no-truth theory applies to indicatives. | |
| From: Frank Jackson (Conditionals [2006], 'Indicative') | |
| A reaction: My intuitions are to reject this and favour a unified account, where both sorts of conditionals are mappings of the relationships among the facts of actuality. Nice slogan! |
| 14353 | Modus ponens requires that A→B is F when A is T and B is F [Jackson] |
| Full Idea: Modus ponens is intuitively valid, but in A,A→B|B if A is true and B is false that must be because A→B is false. So A→B is false when A is true and B is false. | |
| From: Frank Jackson (Conditionals [2006], 'Equiv') | |
| A reaction: This is his first step in showing how the truth functional account of A→B acquires its truth table. If you are giving up the truth functional view of conditionals, presumably you are not also going to give up modus ponens? |
| 14354 | When A and B have the same truth value, A→B is true, because A→A is a logical truth [Jackson] |
| Full Idea: (A→A) is a logical truth, so some conditionals with antecedent and consequent the same truth value are true. But if '→' is a truth function, that will be true for all cases. Hence whenever A and B are alike in truth value, (A→B) is true. | |
| From: Frank Jackson (Conditionals [2006], 'Equiv') | |
| A reaction: His second step in demonstrating the truth table for →, assuming it is truth functional. |
| 14355 | (A&B)→A is a logical truth, even if antecedent false and consequent true, so it is T if A is F and B is T [Jackson] |
| Full Idea: (A&B)→A is a logical truth, but A can be true and B false, so that (A&B) is false. So some conditionals with false antecedent and true consequent are true. If → is a truth function, then whenever A is false and B is true (A→B) is true. | |
| From: Frank Jackson (Conditionals [2006], 'Equiv') | |
| A reaction: This is his third and final step in showing the truth table of → if it is truth functional. |
| 14358 | In the possible worlds account of conditionals, modus ponens and modus tollens are validated [Jackson] |
| Full Idea: In the possible worlds account modus ponens is validated (the closest world, the actual, is a B-world just if B is true), and modus tollens is validated (if B is false, the actual world is not an A-world, so A is false). | |
| From: Frank Jackson (Conditionals [2006], 'Famous') | |
| A reaction: [see Jackson for slightly fuller versions] This looks like a minimal requirement for a decent theory of conditionals, so Jackson explains the attractions of the possible worlds view very persuasively. |
| 14359 | Only assertions have truth-values, and conditionals are not proper assertions [Jackson] |
| Full Idea: In the no-truth theory of conditionals they have justified assertion or acceptability conditions but not truth conditions. ...The motivation is that only assertions have truth values, and conditionals are arguments, not proper assertions. | |
| From: Frank Jackson (Conditionals [2006], 'No-truth') | |
| A reaction: Once I trim this idea down to its basics, it suddenly looks very persuasive. Except that I am inclined to think that conditional truths do state facts about the world - perhaps as facts about how more basic truths are related to each other. |
| 14357 | Possible worlds account, unlike A⊃B, says nothing about when A is false [Jackson] |
| Full Idea: In the possible worlds account of conditionals A⊃B is not sufficient for A→B. If A is false then A⊃B is true, but here nothing is implied about whether the world most like the actual world except that A is true is or is not a B-world. | |
| From: Frank Jackson (Conditionals [2006], 'Possible') | |
| A reaction: The possible worlds account seems to be built on Ramsey's idea of just holding A true and seeing what you get. Being committed to B being automatically true if A is false seems highly counterintuitive. |
| 14356 | We can't insist that A is relevant to B, as conditionals can express lack of relevance [Jackson] |
| Full Idea: One addition to the truth functional account of conditionals is that A be somehow relevant to B. However, sometimes we use conditionals to express lack of relevance, as in 'If Fred works he will fail, and if Fred doesn't work he will fail'. | |
| From: Frank Jackson (Conditionals [2006], 'Possible') | |
| A reaction: This certainly seems to put paid to an attractive instant solution to the problem. |
| 15863 | The principles of my treatise are designed to fit with a belief in God [Newton] |
| Full Idea: When I wrote my treatise about our system, I had an eye upon such principles as might work with considering men, for the belief of a deity. | |
| From: Isaac Newton (Letters to Bentley [1692], 1692.12.10) | |
| A reaction: Harré quotes this, and it shows that the rather passive view of nature Newton developed was to be supplemented by the active power of God. Without God, we need a more active view of nature. |
| 8340 | I do not pretend to know the cause of gravity [Newton] |
| Full Idea: You sometimes speak of gravity as essential and inherent in matter. Pray do no ascribe that notion to me; for the cause of gravity is what I do not pretend to know. | |
| From: Isaac Newton (Letters to Bentley [1692], 1693.01.17) | |
| A reaction: I take science to be a two-stage operation - first we discern the regularities, and then we explain them. Evolution was spotted, then explained by Darwin. Cancer from cigarettes was spotted, but hasn't been explained. Regularity is the beginning. |
| 13150 | The motions of the planets could only derive from an intelligent agent [Newton] |
| Full Idea: The motions which the planets now have could not spring from any natural cause alone, but were impressed by an intelligent agent. | |
| From: Isaac Newton (Letters to Bentley [1692], 1692.12.10) | |
| A reaction: He is writing to a cleric, but seems to be quite sincere about this. Elsewhere he just says he doesn't know what causes gravity. |
| 12178 | That gravity should be innate and essential to matter is absurd [Newton] |
| Full Idea: That gravity should be innate, inherent and essential to matter ...is to me so great an absurdity that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it. | |
| From: Isaac Newton (Letters to Bentley [1692], 1693.02.25) | |
| A reaction: He is replying to some sermons, and he pays vague lip service to a possible divine force. Nevertheless, this is thoroughgoing anti-essentialism, and he talks of external 'laws' in the next sentence. Newton still sought the cause of gravity. |