12 ideas
| 18073 | Dummett says classical logic rests on meaning as truth, while intuitionist logic rests on assertability [Dummett, by Kitcher] |
| Full Idea: Dummett argues that classical logic depends on the choice of the concept of truth as central to the theory of meaning, while for the intuitionist the concept of assertability occupies this position. | |
| From: report of Michael Dummett (The philosophical basis of intuitionist logic [1973]) by Philip Kitcher - The Nature of Mathematical Knowledge 06.5 | |
| A reaction: Since I can assert any nonsense I choose, this presumably means 'warranted' assertability, which is tied to the concept of proof in mathematics. You can reason about falsehoods, or about uninterpreted variables. Can you 'assert' 'Fx'? |
| 19057 | Classical quantification is an infinite conjunction or disjunction - but you may not know all the instances [Dummett] |
| Full Idea: Classical quantification represents an infinite conjunction or disjunction, and the truth-value is determined by the infinite sum or product of the instances ....but this presupposes that all the instances already possess determinate truth-values. | |
| From: Michael Dummett (The philosophical basis of intuitionist logic [1973], p.246) | |
| A reaction: In the case of the universal quantifier, Dummett is doing no more than citing the classic empiricism objection to induction - that you can't make the universal claim if you don't know all the instances. The claim is still meaningful, though. |
| 17818 | How many? must first partition an aggregate into sets, and then logic fixes its number [Yourgrau] |
| Full Idea: We want to know How many what? You must first partition an aggregate into parts relevant to the question, where no partition is privileged. How the partitioned set is to be numbered is bound up with its unique members, and follows from logic alone. | |
| From: Palle Yourgrau (Sets, Aggregates and Numbers [1985], 'New Problem') | |
| A reaction: [Compressed wording of Yourgrau's summary of Frege's 'relativity argument'] Concepts do the partitioning. Yourgau says this fails, because the same argument applies to the sets themselves, as well as to the original aggregates. |
| 17822 | Nothing is 'intrinsically' numbered [Yourgrau] |
| Full Idea: Nothing at all is 'intrinsically' numbered. | |
| From: Palle Yourgrau (Sets, Aggregates and Numbers [1985], 'What the') | |
| A reaction: Once you are faced with distinct 'objects' of some sort, they can play the role of 'unit' in counting, so his challenge is that nothing is 'intrinsically' an object, which is the nihilism explored by Unger, Van Inwagen and Merricks. Aristotle disagrees... |
| 17817 | Defining 'three' as the principle of collection or property of threes explains set theory definitions [Yourgrau] |
| Full Idea: The Frege-Maddy definition of number (as the 'property' of being-three) explains why the definitions of Von Neumann, Zermelo and others work, by giving the 'principle of collection' that ties together all threes. | |
| From: Palle Yourgrau (Sets, Aggregates and Numbers [1985], 'A Fregean') | |
| A reaction: [compressed two or three sentences] I am strongly in favour of the best definition being the one which explains the target, rather than just pinning it down. I take this to be Aristotle's view. |
| 17815 | We can't use sets as foundations for mathematics if we must await results from the upper reaches [Yourgrau] |
| Full Idea: Sets could hardly serve as a foundation for number theory if we had to await detailed results in the upper reaches of the edifice before we could make our first move. | |
| From: Palle Yourgrau (Sets, Aggregates and Numbers [1985], 'Two') |
| 17821 | You can ask all sorts of numerical questions about any one given set [Yourgrau] |
| Full Idea: We can address a set with any question at all that admits of a numerical reply. Thus we can ask of {Carter, Reagan} 'How many feet do the members have?'. | |
| From: Palle Yourgrau (Sets, Aggregates and Numbers [1985], 'On Numbering') | |
| A reaction: This is his objection to the Fregean idea that once you have fixed the members of a set, you have thereby fixed the unique number that belongs with the set. |
| 19055 | Stating a sentence's truth-conditions is just paraphrasing the sentence [Dummett] |
| Full Idea: An ability to state the condition for the truth of a sentence is, in effect, no more than an ability to express the content of the sentence in other words. | |
| From: Michael Dummett (The philosophical basis of intuitionist logic [1973], p.224) | |
| A reaction: Alternatively, if you give something other than a paraphrase of the sentence as its meaning (such as a proof of its truth), then you seem to have departed from your target sentence. Can we reduce and eliminate our sentences in this way? |
| 19056 | If a sentence is effectively undecidable, we can never know its truth conditions [Dummett] |
| Full Idea: If a sentence is effectively undecidable, the condition which must obtain for it to be true is not one which we are capable of recognising whenever it obtains, or of getting ourselves in a position to do so. | |
| From: Michael Dummett (The philosophical basis of intuitionist logic [1973], p.225) | |
| A reaction: The instances of 'undecidable' sentences are most clearly seen in mathematics, such as the Continuum Hypothesis or Goldbach's Conjecture, or anything involving vast infinite cardinals. But do you need precise truth-conditions for meaning? |
| 19054 | Meaning as use puts use beyond criticism, and needs a holistic view of language [Dummett] |
| Full Idea: If use constitutes meaning, it might seem that use is beyond criticism. ....But such an attitude can, ultimately, be supported onlly by the adoption of a holistic view of language. | |
| From: Michael Dummett (The philosophical basis of intuitionist logic [1973], p.218) | |
| A reaction: Dummett goes on to say that the rejection of the holistic view of mathematical meaning leads to his preference for intuitionistic logic. |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
| From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
| A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
| From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
| A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |