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Single Idea 10594

[filed under theme 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic ]

Full Idea

Henkin-style semantics seem to me more plausible for plural logic than for second-order logic.

Gist of Idea

Henkin semantics is more plausible for plural logic than for second-order logic

Source

Penelope Maddy (Second Philosophy [2007], III.8 n1)

Book Ref

Maddy,Penelope: 'Second Philosophy: naturalistic method' [OUP 2007], p.299

A Reaction

Henkin-style semantics are presented by Shapiro as the standard semantics for second-order logic.

The 57 ideas from Penelope Maddy

13011 New axioms are being sought, to determine the size of the continuum [Maddy]
13013 The Axiom of Extensionality seems to be analytic [Maddy]
13014 Extensional sets are clearer, simpler, unique and expressive [Maddy]
13019 The Iterative Conception says everything appears at a stage, derived from the preceding appearances [Maddy]
13018 Limitation of Size is a vague intuition that over-large sets may generate paradoxes [Maddy]
13021 The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics [Maddy]
13022 Infinite sets are essential for giving an account of the real numbers [Maddy]
13023 The Power Set Axiom is needed for, and supported by, accounts of the continuum [Maddy]
13024 Efforts to prove the Axiom of Choice have failed [Maddy]
13025 Modern views say the Choice set exists, even if it can't be constructed [Maddy]
13026 A large array of theorems depend on the Axiom of Choice [Maddy]
17610 The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy]
17605 Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy]
17614 The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy]
17615 Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy]
17620 Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy]
17618 Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy]
17625 If two mathematical themes coincide, that suggest a single deep truth [Maddy]
18182 The extension of concepts is not important to me [Maddy]
18163 Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy]
18168 'Propositional functions' are propositions with a variable as subject or predicate [Maddy]
18172 Infinity has degrees, and large cardinals are the heart of set theory [Maddy]
18175 For any cardinal there is always a larger one (so there is no set of all sets) [Maddy]
18169 Axiom of Reducibility: propositional functions are extensionally predicative [Maddy]
18167 We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy]
18164 Frege solves the Caesar problem by explicitly defining each number [Maddy]
18171 Cantor and Dedekind brought completed infinities into mathematics [Maddy]
18177 In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy]
18187 Theorems about limits could only be proved once the real numbers were understood [Maddy]
18186 Identifying geometric points with real numbers revealed the power of set theory [Maddy]
18188 The line of rationals has gaps, but set theory provided an ordered continuum [Maddy]
18184 Making set theory foundational to mathematics leads to very fruitful axioms [Maddy]
18185 Unified set theory gives a final court of appeal for mathematics [Maddy]
18183 Set theory brings mathematics into one arena, where interrelations become clearer [Maddy]
18190 Completed infinities resulted from giving foundations to calculus [Maddy]
18193 The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy]
18191 Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy]
18194 'Forcing' can produce new models of ZFC from old models [Maddy]
18195 A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy [Maddy]
18196 An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy]
18204 Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy]
18205 The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy]
18206 Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy]
18207 Maybe applications of continuum mathematics are all idealisations [Maddy]
8755 Maddy replaces pure sets with just objects and perceived sets of objects [Maddy, by Shapiro]
8756 Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro]
17733 We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins]
10718 A natural number is a property of sets [Maddy, by Oliver]
10594 Henkin semantics is more plausible for plural logic than for second-order logic [Maddy]
17823 If mathematical objects exist, how can we know them, and which objects are they? [Maddy]
17825 Set theory (unlike the Peano postulates) can explain why multiplication is commutative [Maddy]
17824 The master science is physical objects divided into sets [Maddy]
17826 Standardly, numbers are said to be sets, which is neat ontology and epistemology [Maddy]
17828 Numbers are properties of sets, just as lengths are properties of physical objects [Maddy]
17827 Sets exist where their elements are, but numbers are more like universals [Maddy]
17829 Number words are unusual as adjectives; we don't say 'is five', and numbers always come first [Maddy]
17830 Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy]