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Single Idea 18956

[filed under theme 5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic ]

Full Idea

The natural understanding of first-order logic is that in writing down first-order schemata we are implicitly asserting their validity, that is, making second-order assertions. ...Thus even quantification theory involves reference to classes.

Gist of Idea

Asserting first-order validity implicitly involves second-order reference to classes

Source

Hilary Putnam (Philosophy of Logic [1971], Ch.3)

Book Ref

Putnam,Hilary: 'Philosophy of Logic' [Routledge 1972], p.32

A Reaction

If, as a nominalist, you totally rejected classes, presumably you would get by in first-order logic somehow. To say 'there are no classes so there is no logical validity' sounds bonkers.

Related Idea

Idea 18951 For scientific purposes there is a precise concept of 'true-in-L', using set theory [Putnam]

The 23 ideas with the same theme [logic where variables only refer to objects]:

22873 Liberalism should improve the system, and not just ameliorate it [Dewey]
21495 Theoretical and practical politics are both concerned with the best lives for individuals [Russell]
18956 Asserting first-order validity implicitly involves second-order reference to classes [Putnam]
10767 Elementary logic is complete, but cannot capture mathematics [Tharp]
13840 First-order logic is the strongest complete compact theory with Löwenheim-Skolem [Hacking]
13844 A limitation of first-order logic is that it cannot handle branching quantifiers [Hacking]
18108 First-order logic is not decidable: there is no test of whether any formula is valid [Bostock]
18109 The completeness of first-order logic implies its compactness [Bostock]
10478 Since first-order languages are complete, |= and |- have the same meaning [Hodges,W]
19315 In quantified language the components of complex sentences may not be sentences [Kirkham]
17788 First-order logic only has its main theorems because it is so weak [Mayberry]
13624 The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro]
13660 Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro]
13662 First-order logic was an afterthought in the development of modern logic [Shapiro]
13673 The notion of finitude is actually built into first-order languages [Shapiro]
10588 First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro]
10897 A first-order 'sentence' is a formula with no free variables [Zalabardo]
10986 Not all validity is captured in first-order logic [Read]
13534 In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS]
13535 First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS]
20100 Classical liberalism seeks freedom of opinion, of private life, of expression, and of property [Micklethwait/Wooldridge]
21122 Liberal Nationalism says welfare states and democracy needed a shared sense of nationality [Shorten]
22609 Philosophers accepted first-order logic, because they took science to be descriptive, not explanatory [Ingthorsson]