2005 | Axiomatic Theories of Truth (2005 ver) |
1 | p.2 | 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) |
1 | p.2 | 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. |
1 | p.2 | 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. |
1 | p.2 | 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. |
1.1 | p.2 | 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. |
1.1 | p.3 | 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. |
1.3 | p.4 | 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'. |
1.3 | p.4 | 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. |
1.3 | p.4 | 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] |
2.1 | p.5 | 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) |
2011 | Axiomatic Theories of Truth |
1 | p.3 | 16293 | Traditional definitions of truth often make it more obscure, rather than less |
Full Idea: A common complaint against traditional definitional theories of truth is that it is far from clear that the definiens is not more in need of clarification than the definiendum (that is, the notion of truth). | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 1) | |||
A reaction: He refers to concepts like 'correspondence', 'facts', 'coherence' or 'utility', which are said to be trickier to understand than 'true'. I suspect that philosophers like Halbach confuse 'clear' with 'precise'. Coherence is quite clear, but imprecise. |
1 | p.3 | 16292 | An explicit definition enables the elimination of what is defined |
Full Idea: Explicit definitions allow for a complete elimination of the defined notion (at least in extensional contexts). | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 1) | |||
A reaction: If the context isn't extensional (concerning the things themselves) then we could define one description of it, but be unable to eliminate it under another description. Elimination is no the aim of an Aristotelian definition. Halbach refers to truth. |
1 | p.4 | 16294 | Axiomatic truth doesn't presuppose a truth-definition, though it could admit it at a later stage |
Full Idea: Choosing an axiomatic approach to truth might well be compatible with the view that truth is definable; the definability of truth is just not presupposed at the outset. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 1) | |||
A reaction: Is it possible that a successful axiomatisation is a successful definition? |
1 | p.6 | 16297 | Semantic theories avoid Tarski's Theorem by sticking to a sublanguage |
Full Idea: In semantic theories (e.g.Tarski's or Kripke's), a definition evades Tarski's Theorem by restricting the possible instances in the schema T[φ]↔φ to sentences of a proper sublanguage of the language formulating the equivalences. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 1) | |||
A reaction: The schema says if it's true it's affirmable, and if it's affirmable it's true. The Liar Paradox is a key reason for imposing this restriction. |
11 | p.148 | 16324 | Any definition of truth requires a metalanguage |
Full Idea: It is plain that the distinction between object and metalanguage is required for the definability of truth. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 11) | |||
A reaction: Halbach's axiomatic approach has given up on definability, and therefore it can seek to abandon the metalanguage and examine 'type-free' theories. |
12 | p.150 | 16325 | Analysis rests on natural language, but its ideal is a framework which revises language |
Full Idea: For me, although the enterprise of philosophical analysis is driven by natural language, its goal is not a linguistic analysis of English but rather an expressively strong framework that may at best be seen as a revision of English. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 12) | |||
A reaction: I agree, but the problem is that there are different ideals for the revision, which may be in conflict. Logicians, mathematicians, metaphysicians, scientists, moralists and aestheticians are queueing up to improve in their own way. |
14 | p.163 | 16326 | The main semantic theories of truth are Kripke's theory, and revisions semantics |
Full Idea: Revision semantics is arguably the main competitor of Kripke's theory of truth among semantic truth theories. …In the former one may hope through revision to arrive at better and better models, ..sorting out unsuitable extensions of the truth predicate. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 14) | |||
A reaction: Halbach notes later that Kripke's theory (believe it or not) is considerably simpler than revision semantics. |
15 | p.195 | 16327 | Friedman-Sheard is type-free Compositional Truth, with two inference rules for truth |
Full Idea: The Friedman-Sheard truth system FS is based on compositional theory CT. The axioms of FS are obtained by relaxing the type restriction on the CT-axioms, and adding rules inferring sentences from their truth, and vice versa. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 15) | |||
A reaction: The rules are called NEC and CONEC by Halbach. The system FSN is FS without the two rules. |
15.1 | p.211 | 16329 | Kripke-Feferman theory KF axiomatises Kripke fixed-points, with Strong Kleene logic with gluts |
Full Idea: The Kripke-Feferman theory KF is an axiomatisation of the fixed points of an operator, that is, of a Kripkean fixed-point semantics with the Strong Kleene evaluation schema with truth-value gluts. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 15.1) |
15.2 | p.212 | 16330 | Truth-value 'gluts' allow two truth values together; 'gaps' give a partial conception of truth |
Full Idea: Truth-value 'gluts' correspond to a so-called dialethic conception of truth; excluding gluts and admitting only 'gaps' leads to a conception of what is usually called 'partial' truth. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 15.2) | |||
A reaction: Talk of 'gaps' and 'gluts' seem to be the neatest way of categorising views of truth. I want a theory with no gaps or gluts. |
15.3 | p.217 | 16331 | The KF is much stronger deductively that FS, which relies on classical truth |
Full Idea: The Kripke-Feferman theory is relatively deductively very strong. In particular, it is much stronger than its competitor FS, which is based on a completely classical notion of truth. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 15.3) |
16 | p.229 | 16332 | The KF theory is useful, but it is not a theory containing its own truth predicate |
Full Idea: KF is useful for explicating Peano arithmetic, but it certainly does not come to close to being a theory that contains its own truth predicate. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 16) | |||
A reaction: Since it is a type-free theory, its main philosophical aspiration was to contain its own truth predicate, so that is bad news (for philosophers). |
16.2 | p.245 | 16333 | The underestimated costs of giving up classical logic are found in mathematical reasoning |
Full Idea: The costs of giving up classical logic are easily underestimated, …the price being paid in terms of mathematical reasoning. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 16.2) | |||
A reaction: No one cares much about such costs, until you say they are 'mathematical'. Presumably this is a message to Graham Priest and his pals. |
18 | p.263 | 16334 | In Weak Kleene logic there are 'gaps', neither true nor false if one component lacks a truth value |
Full Idea: In Weak Kleene Logic, with truth-value gaps, a sentence is neither true nor false if one of its components lacks a truth value. A line of the truth table shows a gap if there is a gap anywhere in the line, and the other lines are classical. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 18) | |||
A reaction: This will presumably apply even if the connective is 'or', so a disjunction won't be true, even if one disjunct is true, when the other disjunct is unknown. 'Either 2+2=4 or Lot's wife was left-handed' sounds true to me. Odd. |
18 | p.263 | 16335 | In Strong Kleene logic a disjunction just needs one disjunct to be true |
Full Idea: In Strong Kleene logic a disjunction of two sentences is true if at least one disjunct is true, even when the other disjunct lacks a truth value. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 18) | |||
A reaction: This sounds fine to me. 'Either I'm typing this or Homer had blue eyes' comes out true in any sensible system. |
19.3 | p.275 | 16336 | The liar paradox applies truth to a negated truth (but the conditional will serve equally) |
Full Idea: An essential feature of the liar paradox is the application of the truth predicate to a sentence with a negated occurrence of the truth predicate, though the negation can be avoided by using the conditional. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 19.3) |
19.5 | p.280 | 16337 | Disquotational truth theories are short of deductive power |
Full Idea: The problem of restricted deductive power has haunted disquotational theories of truth (…because they can't prove generalisations). | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 19.5) |
2 | p.11 | 16298 | We need propositions to ascribe the same beliefs to people with different languages |
Full Idea: Being able to ascribe the same proposition as a belief to persons who do not have a common language seems to be one of the main reasons to employ propositions. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 2) | |||
A reaction: Propositions concern beliefs, as well as sentence meanings. I would want to say that a dog and I could believe the same thing, and that is a non-linguistic reason to believe in propositions. Maybe 'translation' cuts out the proposition middleman? |
2 | p.13 | 16299 | Gödel numbering means a theory of truth can use Peano Arithmetic as its base theory |
Full Idea: Often syntactic objects are identified with their numerical codes. …Expressions of a countable formal language can be coded in the natural numbers. This allows a theory of truth to use Peano Arithmetic (with its results) as a base theory. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 2) | |||
A reaction: The numbering system is the famous device invented by Gödel for his great proof of incompleteness. This idea is a key to understanding modern analytic philosophy. It is the bridge which means philosophical theories can be treated mathematically. |
21 | p.306 | 16338 | Deflationism says truth is a disquotation device to express generalisations, adding no new knowledge |
Full Idea: There are two doctrines at the core of deflationism. The first says truth is a device of disquotation used to express generalisations, and the second says truth is a thin notion that contributes nothing to our knowledge of the world | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 21) |
21.2 | p.314 | 16339 | Truth axioms prove objects exist, so truth doesn't seem to be a logical notion |
Full Idea: Two typed disquotation sentences, truth axioms of TB, suffice for proving that there at least two objects. Hence truth is not a logical notion if one expects logical notions to be ontologically neutral. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 21.2) |
21.2 | p.314 | 16340 | Truth axioms need a base theory, because that is where truth issues arise |
Full Idea: Considering the truth axioms in the absence of a base theory is not very sensible because characteristically truth theoretic reasoning arises from the interplay of the truth axioms with the base theory. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 21.2) | |||
A reaction: The base theory usually seems to be either Peano arithmetic or set theory. We might say that introverted thought (e.g. in infants) has little use for truth; it is when you think about the world that truth becomes a worry. |
22.1 | p.322 | 16341 | Normally we only endorse a theory if we believe it to be sound |
Full Idea: If one endorses a theory, so one might argue, one should also take it to be sound. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 22.1) |
22.1 | p.322 | 16342 | You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system |
Full Idea: One cannot just accept that all the theorems of Peano arithmetic are true when one accepts Peano arithmetic as the notion of truth is not available in the language of arithmetic. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 22.1) | |||
A reaction: This is given as the reason why Kreisel and Levy (1968) introduced 'reflection principles', which allow you to assert whatever has been proved (with no mention of truth). (I think. The waters are closing over my head). |
22.1 | p.323 | 16343 | The global reflection principle seems to express the soundness of Peano Arithmetic |
Full Idea: The global reflection principle ∀x(Sent(x) ∧ Bew[PA](x) → Tx) …seems to be the full statement of the soundness claim for Peano arithmetic, as it expresses that all theorems of Peano arithmetic are true. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 22.1) | |||
A reaction: That is, an extra principle must be introduced to express the soundness. PA is, of course, not complete. |
22.1 | p.323 | 16344 | Soundness must involve truth; the soundness of PA certainly needs it |
Full Idea: Soundness seems to be a notion essentially involving truth. At least I do not know how to fully express the soundness of Peano arithmetic without invoking a truth predicate. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 22.1) | |||
A reaction: I suppose you could use some alternative locution such as 'assertible' or 'cuddly'. Intuitionists seem a bit vague about the truth end of things. |
23 | p.330 | 16345 | That Peano arithmetic is interpretable in ZF set theory is taken by philosophers as a reduction |
Full Idea: The observation that Peano arithmetic is relatively interpretable in ZF set theory is taken by many philosophers to be a reduction of numbers to sets. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 23) | |||
A reaction: Nice! Being able to express something in a different language is not the same as a reduction. Back to the drawing board. What do you really mean by a reduction? If we model something, we don't 'reduce' it to the model. |
24.2 | p.340 | 16346 | Maybe necessity is a predicate, not the usual operator, to make it more like truth |
Full Idea: Should necessity be treated as a predicate rather than (as in modal logic) as a sentential operator? It is odd to assign different status to necessity and truth, hampering their interaction. That all necessities are true can't be expressed by an operator. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 24.2) | |||
A reaction: [compressed] Halbach and Horsten consistently treat truth as a predicate, but maybe truth is an operator. Making necessity a predicate and not an operator would be a huge upheaval in the world of modal logic. Nice move! |
24.2 | p.341 | 16347 | Many new paradoxes may await us when we study interactions between frameworks |
Full Idea: Paradoxes that arise from interaction of predicates such as truth, necessity, knowledge, future and past truths have receive little attention. There may be many unknown paradoxes lurking when we develop frameworks with these intensional notions. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 24.2) | |||
A reaction: Nice. This is a wonderful pointer to new research in the analytic tradition, in which formal problems will gradually iron out our metaphysical framework. |
3 | p.15 | 16301 | If people have big doubts about truth, a definition might give it more credibility |
Full Idea: If one were wondering whether truth should be considered a legitimate notion at all, a definition might be useful in dispersing doubts about its legitimacy. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 3) | |||
A reaction: Halbach is proposing to skip definitions, and try to give rules for using 'true' instead, but he doesn't rule out definitions. A definition of 'knowledge' or 'virtue' or 'democracy' might equally give those credibility. |
3 | p.23 | 16305 | We know a complete axiomatisation of truth is not feasible |
Full Idea: In the light of incompleteness phenomena, one should not expect a categorical axiomatisation of truth to be feasible, but this should not keep one from studying axiomatic theories of truth (or of arithmetic). | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 3) | |||
A reaction: This, of course, is because of Gödel's famous results. It is important to be aware in this field that there cannot be a dream of a final theory, so we are just seeing what can be learned about truth. |
4 | p.25 | 16307 | Don't trust analogies; they are no more than a guideline |
Full Idea: Arguments from analogy are to be distrusted: at best they can serve as heuristics. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 4) |
4 | p.25 | 16308 | Set theory was liberated early from types, and recently truth-theories are exploring type-free |
Full Idea: While set theory was liberated much earlier from type restrictions, interest in type-free theories of truth only developed more recently. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 4) | |||
A reaction: Tarski's theory of truth involves types (or hierarchies). |
4.1 | p.25 | 16309 | Every attempt at formal rigour uses some set theory |
Full Idea: Almost any subject with any formal rigour employs some set theory. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 4.1) | |||
A reaction: This is partly because mathematics is often seen as founded in set theory, and formal rigour tends to be mathematical in character. |
5.1 | p.29 | 16310 | A theory is some formulae and all of their consequences |
Full Idea: A theory is a set of formulae closed under first-order logical consequence. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 5.1) |
5.2 | p.35 | 16311 | To axiomatise Tarski's truth definition, we need a binary predicate for his 'satisfaction' |
Full Idea: If the clauses of Tarski's definition of truth are turned into axioms (as Davidson proposed) then a primitive binary predicate symbol for satisfaction is needed, as Tarski defined truth in terms of satisfaction. Standard language has a unary predicate. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 5.2) |
6 | p.41 | 16312 | To reduce PA to ZF, we represent the non-negative integers with von Neumann ordinals |
Full Idea: For the reduction of Peano Arithmetic to ZF set theory, usually the set of finite von Neumann ordinals is used to represent the non-negative integers. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 6) | |||
A reaction: Halbach makes it clear that this is just one mode of reduction, relative interpretability. |
6 Df 6.6 | p.44 | 16313 | A theory is 'conservative' if it adds no new theorems to its base theory |
Full Idea: A truth theory is 'conservative' if the addition of the truth predicate does not add any new theorems to the base theory. | |||
From: report of Volker Halbach (Axiomatic Theories of Truth [2011], 6 Df 6.6) by PG - Db (ideas) | |||
A reaction: Halbach presents the definition more formally, and this is my attempt at getting it into plain English. Halbach uses Peano Arithmetic as his base theory, but set theory is also sometimes used. |
7 | p.53 | 16315 | The Tarski Biconditional theory TB is Peano Arithmetic, plus truth, plus all Tarski bi-conditionals |
Full Idea: The truth theory TB (Tarski Biconditional) is all the axioms of Peano Arithmetic, including all instances of the induction schema with the truth predicate, plus all the sentences of the form T[φ] ↔ φ. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 7) | |||
A reaction: The biconditional formula is the famous 'snow is white' iff snow is white. The truth of the named sentence is equivalent to asserting the sentence. This is a typed theory of truth, and it is conservative over PA. |
7 | p.56 | 16316 | Deflationists say truth is just for expressing infinite conjunctions or generalisations |
Full Idea: Deflationists do not hold that truth is completely dispensable. They claim that truth serves the purpose of expressing infinite conjunctions or generalisations. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 7) | |||
A reaction: It is also of obvious value as a shorthand in ordinary conversation, but rigorous accounts can paraphrase that out. 'What he said is true'. 'Pick out the true sentences from p,q,r and s' seems to mean 'affirm some of them'. What does 'affirm' mean? |
7 | p.61 | 16317 | The main problem for deflationists is they can express generalisations, but not prove them |
Full Idea: The main criticism that deflationist theories based on the disquotation sentences or similar axioms have to meet was raised by Tarski: the disquotation sentences do not allow one to prove generalisations. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 7) |
8 | p.66 | 16318 | Compositional Truth CT has the truth of a sentence depending of the semantic values of its constituents |
Full Idea: In the typed Compositional Truth theory CT, it is compositional because the truth of a sentence depends on the semantic values of the constituents of that sentence. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 8) | |||
A reaction: [axioms on p. 65 of Halbach] |
8 | p.67 | 16320 | Some say deflationism is axioms which are conservative over the base theory |
Full Idea: Some authors have tried to understand the deflationist claim that truth is not a substantial notion as the claim that a satisfactory axiomatisation of truth should be conservative over the base theory. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 8) |
8 | p.67 | 16319 | Compositional Truth CT proves generalisations, so is preferred in discussions of deflationism |
Full Idea: Compositional Truth CT and its variants has desirable generalisations among its logical consequences, so they seem to have ousted purely disquotational theories such as TB in the discussion on deflationism. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 8) |
8.3 | p.83 | 16321 | The compactness theorem can prove nonstandard models of PA |
Full Idea: Nonstandard models of Peano arithmetic are models of PA that are not isomorphic to the standard model. Their existence can be established with the compactness theorem or the adequacy theorem of first-order logic. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 8.3) |
8.6 | p.106 | 16322 | CT proves PA consistent, which PA can't do on its own, so CT is not conservative over PA |
Full Idea: Compositional Truth CT proves the consistency of Peano arithmetic, which is not provable in Peano arithmetic by Gödel's second incompleteness theorem. Hence the theory CT is not conservative over Peano arithmetic. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], 8.6) |
II Intro | p.51 | 16314 | Theories of truth are 'typed' (truth can't apply to sentences containing 'true'), or 'type-free' |
Full Idea: I sort theories of truth into the large families of 'typed' and 'type-free'. Roughly, typed theories prohibit a truth predicate's application to sentences with occurrences of that predicate, and one cannot prove the truth of sentences containing 'true'. | |||
From: Volker Halbach (Axiomatic Theories of Truth [2011], II Intro) | |||
A reaction: The problem sentence the typed theories are terrified of is the Liar Sentence. Typing produces a hierarchy of languages, referring down to the languages below them. |