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15647 | Truth definitions don't produce a good theory, because they go beyond your current language |
Full Idea: It is far from clear that a definition of truth can lead to a philosophically satisfactory theory of truth. Tarski's theorem on the undefinability of the truth predicate needs resources beyond those of the language for which it is being defined. | |||
From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1) | |||
A reaction: The idea is that you need a 'metalanguage' for the definition. If I say 'p' is a true sentence in language 'L', I am not making that observation from within language L. The dream is a theory confined to the object language. |
15649 | In semantic theories of truth, the predicate is in an object-language, and the definition in a metalanguage |
Full Idea: In semantic theories of truth (Tarski or Kripke), a truth predicate is defined for an object-language. This definition is carried out in a metalanguage, which is typically taken to include set theory or another strong theory or expressive language. | |||
From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1) | |||
A reaction: Presumably the metalanguage includes set theory because that connects it with mathematics, and enables it to be formally rigorous. Tarski showed, in his undefinability theorem, that the meta-language must have increased resources. |
15650 | Axiomatic theories of truth need a weak logical framework, and not a strong metatheory |
Full Idea: Axiomatic theories of truth can be presented within very weak logical frameworks which require very few resources, and avoid the need for a strong metalanguage and metatheory. | |||
From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1) |
15655 | Should axiomatic truth be 'conservative' - not proving anything apart from implications of the axioms? |
Full Idea: If truth is not explanatory, truth axioms should not allow proof of new theorems not involving the truth predicate. It is hence said that axiomatic truth should be 'conservative' - not implying further sentences beyond what the axioms can prove. | |||
From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1.3) | |||
A reaction: [compressed] |
15654 | If truth is defined it can be eliminated, whereas axiomatic truth has various commitments |
Full Idea: If truth can be explicitly defined, it can be eliminated, whereas an axiomatized notion of truth may bring all kinds of commitments. | |||
From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1.3) | |||
A reaction: The general principle that anything which can be defined can be eliminated (in an abstract theory, presumably, not in nature!) raises interesting questions about how many true theories there are which are all equivalent to one another. |
15648 | Instead of a truth definition, add a primitive truth predicate, and axioms for how it works |
Full Idea: The axiomatic approach does not presuppose that truth can be defined. Instead, a formal language is expanded by a new primitive predicate of truth, and axioms for that predicate are then laid down. | |||
From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1) | |||
A reaction: Idea 15647 explains why Halbach thinks the definition route is no good. |
15656 | Deflationists say truth merely serves to express infinite conjunctions |
Full Idea: According to many deflationists, truth serves merely the purpose of expressing infinite conjunctions. | |||
From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1.3) | |||
A reaction: That is, it asserts sentences that are too numerous to express individually. It also seems, on a deflationist view, to serve for anaphoric reference to sentences, such as 'what she just said is true'. |
15657 | To prove the consistency of set theory, we must go beyond set theory |
Full Idea: The consistency of set theory cannot be established without assumptions transcending set theory. | |||
From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 2.1) |
15652 | We can use truth instead of ontologically loaded second-order comprehension assumptions about properties |
Full Idea: The reduction of 2nd-order theories (of properties or sets) to axiomatic theories of truth may be conceived as a form of reductive nominalism, replacing existence assumptions (for comprehension axioms) by ontologically innocent truth assumptions. | |||
From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1.1) | |||
A reaction: I like this very much, as weeding properties out of logic (without weeding them out of the world). So-called properties in logic are too abundant, so there is a misfit with their role in science. |
15651 | Instead of saying x has a property, we can say a formula is true of x - as long as we have 'true' |
Full Idea: Quantification over (certain) properties can be mimicked in a language with a truth predicate by quantifying over formulas. Instead of saying that Tom has the property of being a poor philosopher, we can say 'x is a poor philosopher' is true of Tom. | |||
From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1.1) | |||
A reaction: I love this, and think it is very important. He talks of 'mimicking' properties, but I see it as philosophers mistakenly attributing properties, when actually what they were doing is asserting truths involving certain predicates. |