Ideas from 'Higher-Order Logic' by Stewart Shapiro [2001], by Theme Structure

[found in 'Blackwell Guide to Philosophical Logic' (ed/tr Goble,Lou) [Blackwell 2001,0-631-20693-0]].

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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX

10301

The axiom of choice is controversial, but it could be replaced

			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)

5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic

10588

First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems

			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)

5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic

10298

Some say that second-order logic is mathematics, not logic

			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

			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'.

5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence

10300

Logical consequence can be defined in terms of the logical terminology

			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.

5. Theory of Logic / G. Quantification / 5. Second-Order Quantification

10290

Second-order variables also range over properties, sets, relations or functions

			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)

5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems

10296

The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics

			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)

10590

Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them

			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)

10292

Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model

			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)

10297

The Löwenheim-Skolem theorem seems to be a defect of first-order logic

			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]

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order

10294

Second-order logic has the expressive power for mathematics, but an unworkable model theory

			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?

8. Modes of Existence / B. Properties / 11. Properties as Sets

10591

Logicians use 'property' and 'set' interchangeably, with little hanging on it

			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.