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10017 | Truth in a model is more tractable than the general notion of truth |
Full Idea: Truth in a model is interesting because it provides a transparent and mathematically tractable model - in the 'ordinary' rather than formal sense of the term 'model' - of the less tractable notion of truth. | |||
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131) | |||
A reaction: This is an important warning to those who wish to build their entire account of truth on Tarski's rigorously formal account of the term. Personally I think we should start by deciding whether 'true' can refer to the mental state of a dog. I say it can. |
10018 | Truth is quite different in interpreted set theory and in the skeleton of its language |
Full Idea: There is an enormous difference between the truth of sentences in the interpreted language of set theory and truth in some model for the disinterpreted skeleton of that language. | |||
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.132) | |||
A reaction: This is a warning to me, because I thought truth and semantics only entered theories at the stage of 'interpretation'. I must go back and get the hang of 'skeletal' truth, which sounds rather charming. [He refers to set theory, not to logic.] |
10015 | Higher-order logic may be unintelligible, but it isn't set theory |
Full Idea: Brand higher-order logic as unintelligible if you will, but don't conflate it with set theory. | |||
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131) | |||
A reaction: [he gives Boolos 1975 as a further reference] This is simply a corrective, because the conflation of second-order logic with set theory is an idea floating around in the literature. |
10011 | Identity is a level one relation with a second-order definition |
Full Idea: Identity should he considered a logical notion only because it is the tip of a second-order iceberg - a level 1 relation with a pure second-order definition. | |||
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984]) |
10016 | When an 'interpretation' creates a model based on truth, this doesn't include Fregean 'sense' |
Full Idea: A model is created when a language is 'interpreted', by assigning non-logical terms to objects in a set, according to a 'true-in' relation, but we must bear in mind that this 'interpretation' does not associate anything like Fregean senses with terms. | |||
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131) | |||
A reaction: This seems like a key point (also made by Hofweber) that formal accounts of numbers, as required by logic, will not give an adequate account of the semantics of number-terms in natural languages. |
10027 | Mathematics is higher-order modal logic |
Full Idea: I take the view that (agreeing with Aristotle) mathematics only requires the notion of a potential infinity, ...and that mathematics is higher-order modal logic. | |||
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984]) | |||
A reaction: Modern 'modal' accounts of mathematics I take to be heirs of 'if-thenism', which seems to have been Russell's development of Frege's original logicism. I'm beginning to think it is right. But what is the subject-matter of arithmetic? |
10026 | Arithmetic must allow for the possibility of only a finite total of objects |
Full Idea: Arithmetic should be able to face boldly the dreadful chance that in the actual world there are only finitely many objects. | |||
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.148) | |||
A reaction: This seems to be a basic requirement for any account of arithmetic, but it was famously a difficulty for early logicism, evaded by making the existence of an infinity of objects into an axiom of the system. |
10021 | It is claimed that numbers are objects which essentially represent cardinality quantifiers |
Full Idea: The mathematical object-theorist says a number is an object that represents a cardinality quantifier, with the representation relation as the entire essence of the nature of such objects as cardinal numbers like 4. | |||
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984]) | |||
A reaction: [compressed] This a classic case of a theory beginning to look dubious once you spell it our precisely. The obvious thought is to make do with the numerical quantifiers, and dispense with the objects. Do other quantifiers need objects to support them? |
10022 | Numerical terms can't really stand for quantifiers, because that would make them first-level |
Full Idea: The dogmatic Frege is more right than wrong in denying that numerical terms can stand for numerical quantifiers, for there cannot be a language in which object-quantifiers and objects are simultaneously viewed as level zero. | |||
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.142) | |||
A reaction: Subtle. We see why Frege goes on to say that numbers are level zero (i.e. they are objects). We are free, it seems, to rewrite sentences containing number terms to suit whatever logical form appeals. Numbers are just quantifiers? |
10023 | Talk of mirror images is 'encoded fictions' about real facts |
Full Idea: Talk about mirror images is a sort of fictional discourse. Statements 'about' such fictions are not made true or false by our whims; rather they 'encode' facts about the things reflected in mirrors. | |||
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.146) | |||
A reaction: Hodes's proposal for how we should view abstract objects (c.f. Frege and Dummett on 'the equator'). The facts involved are concrete, but Hodes is offering 'encoding fictionalism' as a linguistic account of such abstractions. He applies it to numbers. |