15 ideas
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) |
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] |
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) |
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? |
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. |
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. |
6011 | There is a remote first god (the Good), and a second god who organises the material world [Numenius, by O'Meara] |
Full Idea: Numenius argues that material reality depends on intelligible being, which depends on a first god - the Good - which is difficult to grasp, but which inspires a second god to imitate it, turning to matter and organizing it as the world. | |
From: report of Numenius (fragments/reports [c.160]) by Dominic J. O'Meara - Numenius | |
A reaction: The interaction problem comes either between the two gods, or between the second god and the world. The argument may have failed to catch on for long when people scented an infinite regress lurking in the middle of it. |