16 ideas
6782 | Realism is the only philosophy of science that doesn't make the success of science a miracle [Putnam] |
Full Idea: Realism….is the only philosophy science which does not make the success of science a miracle. | |
From: Hilary Putnam (works [1980]), quoted by Alexander Bird - Philosophy of Science Ch.4 | |
A reaction: This was from his earlier work; he became more pragmatist and anti-realist later. Personally I approve of the remark. The philosophy of science must certainly offer an explanation for its success. Truth seems the obvious explanation. |
10301 | The axiom of choice is controversial, but it could be replaced [Shapiro] |
Full Idea: The axiom of choice has a troubled history, but is now standard in mathematics. It could be replaced with a principle of comprehension for functions), or one could omit the variables ranging over functions. | |
From: Stewart Shapiro (Higher-Order Logic [2001], n 3) |
10588 | First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro] |
Full Idea: Early study of first-order logic revealed a number of important features. Gödel showed that there is a complete, sound and effective deductive system. It follows that it is Compact, and there are also the downward and upward Löwenheim-Skolem Theorems. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
10298 | Some say that second-order logic is mathematics, not logic [Shapiro] |
Full Idea: Some authors argue that second-order logic (with standard semantics) is not logic at all, but is a rather obscure form of mathematics. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) |
10299 | If the aim of logic is to codify inferences, second-order logic is useless [Shapiro] |
Full Idea: If the goal of logical study is to present a canon of inference, a calculus which codifies correct inference patterns, then second-order logic is a non-starter. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |
A reaction: This seems to be because it is not 'complete'. However, moves like plural quantification seem aimed at capturing ordinary language inferences, so the difficulty is only that there isn't a precise 'calculus'. |
10300 | Logical consequence can be defined in terms of the logical terminology [Shapiro] |
Full Idea: Informally, logical consequence is sometimes defined in terms of the meanings of a certain collection of terms, the so-called 'logical terminology'. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |
A reaction: This seems to be a compositional account, where we build a full account from an account of the atomic bits, perhaps presented as truth-tables. |
10290 | Second-order variables also range over properties, sets, relations or functions [Shapiro] |
Full Idea: Second-order variables can range over properties, sets, or relations on the items in the domain-of-discourse, or over functions from the domain itself. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
10292 | Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro] |
Full Idea: Downward Löwenheim-Skolem: a finite or denumerable set of first-order formulas that is satisfied by a model whose domain is infinite is satisfied in a model whose domain is the natural numbers | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
10590 | Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro] |
Full Idea: Upward Löwenheim-Skolem: if a set of first-order formulas is satisfied by a domain of at least the natural numbers, then it is satisfied by a model of at least some infinite cardinal. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
10296 | The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro] |
Full Idea: Both of the Löwenheim-Skolem Theorems fail for second-order languages with a standard semantics | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.3.2) |
10297 | The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro] |
Full Idea: The Löwenheim-Skolem theorem is usually taken as a sort of defect (often thought to be inevitable) of the first-order logic. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |
A reaction: [He is quoting Wang 1974 p.154] |
10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro] |
Full Idea: Full second-order logic has all the expressive power needed to do mathematics, but has an unworkable model theory. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) | |
A reaction: [he credits Cowles for this remark] Having an unworkable model theory sounds pretty serious to me, as I'm not inclined to be interested in languages which don't produce models of some sort. Surely models are the whole point? |
22181 | Putnam says anti-realism is a bad explanation of accurate predictions [Putnam, by Okasha] |
Full Idea: Putnam's 'no miracle' argument says that being an anti-realist is akin to believing in miracles (because of the accurate predictons). …It is a plausibility argument - an inference to the best explanation. | |
From: report of Hilary Putnam (works [1980]) by Samir Okasha - Philosophy of Science: Very Short Intro (2nd ed) 4 | |
A reaction: [not sure of ref] Putnam later backs off from this argument, but my personal realism rests on best explanation. Does anyone want to prefer an inferior explanation? The objection is that successful theories can turn out to be false. Phlogiston, ether. |
10591 | Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro] |
Full Idea: In studying second-order logic one can think of relations and functions as extensional or intensional, or one can leave it open. Little turns on this here, and so words like 'property', 'class', and 'set' are used interchangeably. | |
From: Stewart Shapiro (Higher-Order Logic [2001], 2.2.1) | |
A reaction: Important. Students of the metaphysics of properties, who arrive with limited experience of logic, are bewildered by this attitude. Note that the metaphysics is left wide open, so never let logicians hijack the metaphysical problem of properties. |
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. |