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Single Idea 10973

[filed under theme 5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic ]

Full Idea

A 'theory' is any logically closed set of propositions, ..and since any proposition has infinitely many consequences, including all the logical truths, so that theories have infinitely many premisses.

Gist of Idea

A theory is logically closed, which means infinite premisses

Source

Stephen Read (Thinking About Logic [1995], Ch.2)

Book Ref

Read,Stephen: 'Thinking About Logic' [OUP 1995], p.43

A Reaction

Read is introducing this as the essential preliminary to an account of the Compactness Theorem, which relates these infinite premisses to the finite.

The 42 ideas from 'Thinking About Logic'

10966 A proposition objectifies what a sentence says, as indicative, with secure references [Read]
10983 Knowledge of possible worlds is not causal, but is an ontology entailed by semantics [Read]
10982 How can modal Platonists know the truth of a modal proposition? [Read]
10987 Three traditional names of rules are 'Simplification', 'Addition' and 'Disjunctive Syllogism' [Read]
10985 We should exclude second-order logic, precisely because it captures arithmetic [Read]
10970 A theory of logical consequence is a conceptual analysis, and a set of validity techniques [Read]
10984 Logical consequence isn't just a matter of form; it depends on connections like round-square [Read]
10973 A theory is logically closed, which means infinite premisses [Read]
10978 In second-order logic the higher-order variables range over all the properties of the objects [Read]
10971 A logical truth is the conclusion of a valid inference with no premisses [Read]
10988 Any first-order theory of sets is inadequate [Read]
10974 Compactness is when any consequence of infinite propositions is the consequence of a finite subset [Read]
10975 Compactness does not deny that an inference can have infinitely many premisses [Read]
10977 Compactness blocks the proof of 'for every n, A(n)' (as the proof would be infinite) [Read]
10976 Compactness makes consequence manageable, but restricts expressive power [Read]
10972 The non-emptiness of the domain is characteristic of classical logic [Read]
10986 Not all validity is captured in first-order logic [Read]
10979 Although second-order arithmetic is incomplete, it can fully model normal arithmetic [Read]
10980 Second-order arithmetic covers all properties, ensuring categoricity [Read]
10981 A possible world is a determination of the truth-values of all propositions of a domain [Read]
10989 The standard view of conditionals is that they are truth-functional [Read]
10992 The point of conditionals is to show that one will accept modus ponens [Read]
10997 Von Neumann numbers are helpful, but don't correctly describe numbers [Read]
11001 Equating necessity with truth in every possible world is the S5 conception of necessity [Read]
11000 If worlds are concrete, objects can't be present in more than one, and can only have counterparts [Read]
10998 The mind abstracts ways things might be, which are nonetheless real [Read]
11004 Necessity is provability in S4, and true in all worlds in S5 [Read]
10996 Actualism is reductionist (to parts of actuality), or moderate realist (accepting real abstractions) [Read]
10995 A haecceity is a set of individual properties, essential to each thing [Read]
11011 Same say there are positive, negative and neuter free logics [Read]
11007 Quantifiers are second-order predicates [Read]
11005 Negative existentials with compositionality make the whole sentence meaningless [Read]
11012 A 'supervaluation' gives a proposition consistent truth-value for classical assignments [Read]
11013 Identities and the Indiscernibility of Identicals don't work with supervaluations [Read]
11014 Self-reference paradoxes seem to arise only when falsity is involved [Read]
11018 There are fuzzy predicates (and sets), and fuzzy quantifiers and modifiers [Read]
11016 Would a language without vagueness be usable at all? [Read]
11019 Supervaluations say there is a cut-off somewhere, but at no particular place [Read]
11017 Some people even claim that conditionals do not express propositions [Read]
11025 Infinite cuts and successors seems to suggest an actual infinity there waiting for us [Read]
11020 Realisms like the full Comprehension Principle, that all good concepts determine sets [Read]
11024 Semantics must precede proof in higher-order logics, since they are incomplete [Read]