30 ideas
| 15134 | The truthmaker principle requires some specific named thing to make the difference [Williamson] |
| Full Idea: The truthmaker principle seems compelling, because if a proposition is true, something must be different from a world in which it is false. The principle makes this specific, by treating 'something' as a quantifier binding a variable in name position. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §2) | |
| A reaction: See Williamson for an examination of the logical implications of this. The point is that the principle seems to require some very specific 'thing', which may be asking too much. For a start, it might be the absence of a thing. |
| 15141 | Truthmaker is incompatible with modal semantics of varying domains [Williamson] |
| Full Idea: Friends of the truthmaker principle should reject the Kripke semantics of varying domains. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §3) | |
| A reaction: See other ideas from this paper to get a sense of what that is about. |
| 15140 | The converse Barcan formula will not allow contingent truths to have truthmakers [Williamson] |
| Full Idea: The converse Barcan formula does not allow any contingent truths at all to have a truthmaker. Once cannot combine the converse Barcan formula with any truthmaker principle worth having. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §3) | |
| A reaction: One might reply, so much the worse for the converse Barcan formula, but Williamson doesn't think that. |
| 15131 | If metaphysical possibility is not a contingent matter, then S5 seems to suit it best [Williamson] |
| Full Idea: In S5, necessity and possibility are not themselves contingent matters. This is plausible for metaphysical modality, since metaphysical possibility, unlike practical possibility, does not depend on the contingencies of one's situation. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §1) | |
| A reaction: This is the clearest statement I have found of why S5 might be preferable for metaphysics. See Nathan Salmon for the rival view. Williamson's point sounds pretty persuasive to me. |
| 15135 | If the domain of propositional quantification is constant, the Barcan formulas hold [Williamson] |
| Full Idea: If the domain of propositional quantification is constant across worlds, the Barcan formula and its converse hold. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §2) | |
| A reaction: So the issue is whether we should take metaphysics to be dealing with a constant or varying domains. Williamson seems to favour the former, but my instincts incline towards the latter. |
| 15139 | Converse Barcan: could something fail to meet a condition, if everything meets that condition? [Williamson] |
| Full Idea: The converse Barcan is at least plausible, since its denial says there is something that could fail to meet a condition when everything met that condition; but how could everything meet that condition if that thing did not? | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §3) | |
| A reaction: Presumably the response involves a discussion of domains, since everything in a given domain might meet a condition, but something in a different domain might fail it. |
| 10775 | The axiom of choice now seems acceptable and obvious (if it is meaningful) [Tharp] |
| Full Idea: The main objection to the axiom of choice was that it had to be given by some law or definition, but since sets are arbitrary this seems irrelevant. Formalists consider it meaningless, but set-theorists consider it as true, and practically obvious. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §3) |
| 10766 | Logic is either for demonstration, or for characterizing structures [Tharp] |
| Full Idea: One can distinguish at least two quite different senses of logic: as an instrument of demonstration, and perhaps as an instrument for the characterization of structures. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) | |
| A reaction: This is trying to capture the proof-theory and semantic aspects, but merely 'characterizing' something sounds like a rather feeble aspiration for the semantic side of things. Isn't it to do with truth, rather than just rule-following? |
| 10767 | Elementary logic is complete, but cannot capture mathematics [Tharp] |
| Full Idea: Elementary logic cannot characterize the usual mathematical structures, but seems to be distinguished by its completeness. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) |
| 10769 | Second-order logic isn't provable, but will express set-theory and classic problems [Tharp] |
| Full Idea: The expressive power of second-order logic is too great to admit a proof procedure, but is adequate to express set-theoretical statements, and open questions such as the continuum hypothesis or the existence of big cardinals are easily stated. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) |
| 10762 | In sentential logic there is a simple proof that all truth functions can be reduced to 'not' and 'and' [Tharp] |
| Full Idea: In sentential logic there is a simple proof that all truth functions, of any number of arguments, are definable from (say) 'not' and 'and'. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §0) | |
| A reaction: The point of 'say' is that it can be got down to two connectives, and these are just the usual preferred pair. |
| 18492 | Not all quantification is either objectual or substitutional [Williamson] |
| Full Idea: We should not assume that all quantification is either objectual or substitutional. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], p.262) | |
| A reaction: [see Prior 1971:31-4] He talks of quantifying into sentence position. |
| 10776 | The main quantifiers extend 'and' and 'or' to infinite domains [Tharp] |
| Full Idea: The symbols ∀ and ∃ may, to start with, be regarded as extrapolations of the truth functional connectives ∧ ('and') and ∨ ('or') to infinite domains. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §5) |
| 15136 | Substitutional quantification is metaphysical neutral, and equivalent to a disjunction of instances [Williamson] |
| Full Idea: If quantification into sentence position is substitutional, then it is metaphysically neutral. A substitutionally interpreted 'existential' quantification is semantically equivalent to the disjunction (possibly infinite) of its substitution instances. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §2) | |
| A reaction: Is it not committed to the disjunction, just as the objectual reading commits to objects? Something must make the disjunction true. Or is it too verbal to be about reality? |
| 10774 | There are at least five unorthodox quantifiers that could be used [Tharp] |
| Full Idea: One might add to one's logic an 'uncountable quantifier', or a 'Chang quantifier', or a 'two-argument quantifier', or 'Shelah's quantifier', or 'branching quantifiers'. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §3) | |
| A reaction: [compressed - just listed for reference, if you collect quantifiers, like collecting butterflies] |
| 15138 | Not all quantification is objectual or substitutional [Williamson] |
| Full Idea: We should not assume that all quantification is objectual or substitutional. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §2) |
| 10773 | The Löwenheim-Skolem property is a limitation (e.g. can't say there are uncountably many reals) [Tharp] |
| Full Idea: The Löwenheim-Skolem property seems to be undesirable, in that it states a limitation concerning the distinctions the logic is capable of making, such as saying there are uncountably many reals ('Skolem's Paradox'). | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) |
| 10777 | Skolem mistakenly inferred that Cantor's conceptions were illusory [Tharp] |
| Full Idea: Skolem deduced from the Löwenheim-Skolem theorem that 'the absolutist conceptions of Cantor's theory' are 'illusory'. I think it is clear that this conclusion would not follow even if elementary logic were in some sense the true logic, as Skolem assumed. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §7) | |
| A reaction: [Tharp cites Skolem 1962 p.47] Kit Fine refers to accepters of this scepticism about the arithmetic of infinities as 'Skolemites'. |
| 10765 | Soundness would seem to be an essential requirement of a proof procedure [Tharp] |
| Full Idea: Soundness would seem to be an essential requirement of a proof procedure, since there is little point in proving formulas which may turn out to be false under some interpretation. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) |
| 10763 | Completeness and compactness together give axiomatizability [Tharp] |
| Full Idea: Putting completeness and compactness together, one has axiomatizability. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §1) |
| 10770 | If completeness fails there is no algorithm to list the valid formulas [Tharp] |
| Full Idea: In general, if completeness fails there is no algorithm to list the valid formulas. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) | |
| A reaction: I.e. the theory is not effectively enumerable. |
| 10771 | Compactness is important for major theories which have infinitely many axioms [Tharp] |
| Full Idea: It is strange that compactness is often ignored in discussions of philosophy of logic, since the most important theories have infinitely many axioms. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) | |
| A reaction: An example of infinite axioms is the induction schema in first-order Peano Arithmetic. |
| 10772 | Compactness blocks infinite expansion, and admits non-standard models [Tharp] |
| Full Idea: The compactness condition seems to state some weakness of the logic (as if it were futile to add infinitely many hypotheses). To look at it another way, formalizations of (say) arithmetic will admit of non-standard models. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) |
| 10764 | A complete logic has an effective enumeration of the valid formulas [Tharp] |
| Full Idea: A complete logic has an effective enumeration of the valid formulas. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) |
| 10768 | Effective enumeration might be proved but not specified, so it won't guarantee knowledge [Tharp] |
| Full Idea: Despite completeness, the mere existence of an effective enumeration of the valid formulas will not, by itself, provide knowledge. For example, one might be able to prove that there is an effective enumeration, without being able to specify one. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) | |
| A reaction: The point is that completeness is supposed to ensure knowledge (of what is valid but unprovable), and completeness entails effective enumerability, but more than the latter is needed to do the key job. |
| 15137 | If 'fact' is a noun, can we name the fact that dogs bark 'Mary'? [Williamson] |
| Full Idea: If one uses 'fact' as a noun, the question arises why one cannot name the fact that dogs bark 'Mary'. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §2 n10) | |
| A reaction: What an intriguing thought! Must all nouns pass this test? 'The courage of the regiment was called Alfred'? |
| 15142 | Our ability to count objects across possibilities favours the Barcan formulas [Williamson] |
| Full Idea: Consideration of our ability to count objects across possibilities strongly favour both the Barcan formula and its converse. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §3) | |
| A reaction: I'm not sure that I can understand counting objects across possibilities. The objects themselves are possibilia, and possibilia seem to include unknowns. The unexpected is highly possible. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |
| 15133 | A thing can't be the only necessary existent, because its singleton set would be as well [Williamson] |
| Full Idea: That there is just one necessary existent is surely false, for if x is a necessary, {x} is a distinct necessary existent. | |
| From: Timothy Williamson (Truthmakers and Converse Barcan Formula [1999], §1) | |
| A reaction: You would have to believe that sets actually 'exist' to accept this, but it is a very neat point. |