22 ideas
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. |
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) |
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] |
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'. |
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) |
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. |
16657 | Substance, Quantity and Quality are real; other categories depend on those three [Henry of Ghent] |
Full Idea: Among creatures there are only three 'res' belong to the three first categories: Substance, Quantity and Quality. All other are aspects [rationes] and intellectual concepts with respect to them, with reality only as grounded on the res of those three. | |
From: Henry of Ghent (Quodlibeta [1284], VII:1-2), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 12.3 | |
A reaction: Pasnau connects with the 'arrangement of being', giving an 'ontologically innocent' structure to reality. That seems to be what we all want, if only we could work out the ontologically guilty bit. |
16658 | The only reality in the category of Relation is things from another category [Henry of Ghent] |
Full Idea: There is beyond a doubt nothing real in the category of Relation, except what is a thing from another category. | |
From: Henry of Ghent (Quodlibeta [1284], VII:1-2), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 12.3 | |
A reaction: This seems to have been the fairly orthodox scholastic view of relations. |
16645 | Accidents are diminished beings, because they are dispositions of substance (unqualified being) [Henry of Ghent] |
Full Idea: Accidents are beings only in a qualified and diminished sense, because they are not called beings, nor are they beings, except because they are dispositions of an unqualified being, a substance. | |
From: Henry of Ghent (Quodlibeta [1284], XV.5), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 10.4 | |
A reaction: This is aimed to 'half' detach the accidents (as the Eucharist requires). Later scholastics detached them completely. Late scholastics seem to have drifted back to Henry's view. The equivocal use of 'being' here was challenged later. |
15312 | We get the idea of power by abstracting from ropes, magnets and electric shocks [Priestley] |
Full Idea: A rope sustains weight, a magnet attracts iron, a charged electrical jar gives a shock, and from these and other similar observations, we get the idea of power, universally and abstractly considered. | |
From: Joseph Priestley (Theological and other works [1790], p.191), quoted by Harré,R./Madden,E.H. - Causal Powers 9.II.B | |
A reaction: I agree with this, in that we appear to be observing powers directly, and are not observing something which can then be reduced to non-powers. Nature just can't be a set of inert structures, with forces 'imposed' on them. |
22012 | Kant says things-in-themselves cause sensations, but then makes causation transcendental! [Henry of Ghent, by Pinkard] |
Full Idea: Kant claimed that things-in-themselves caused our sensations; but causality was a transcendental condition of experience, not a property of things-in-themselves, so the great Kant had contradicted himself. | |
From: report of Henry of Ghent (Quodlibeta [1284], Supplement) by Terry Pinkard - German Philosophy 1760-1860 04 | |
A reaction: This early objection by the conservative Jacobi (who disliked Enlightenment rational religion) is the key to the dispute over whether Kant is an idealist. Kant denied being an idealist, but how can he be, if this idea is correct? |
15311 | Attraction or repulsion are not imparted to matter, but actually constitute it [Priestley] |
Full Idea: Attraction or repulsion appear to me not to be properly what is imparted to matter, but what really makes it what it is, in so much that, without it, it would be nothing at all. | |
From: Joseph Priestley (Theological and other works [1790], p.237), quoted by Harré,R./Madden,E.H. - Causal Powers 9.II.B | |
A reaction: This is music to the ears of anyone who thinks that powers are the fundamentals of nature (like me). |