121 ideas
19199 | Some say metaphysics is a highly generalised empirical study of objects [Tarski] |
Full Idea: For some people metaphysics is a general theory of objects (ontology) - a discipline which is to be developed in a purely empirical way, and which differs from other empirical disciplines in its generality. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 19) | |
A reaction: Tarski says some people despise it, but for him such metaphysics is 'not objectionable'. I subscribe to this view, but the empirical aspect is very remote, because it's too general for detail observation or experiment. Generality is the key to philosophy. |
19193 | Disputes that fail to use precise scientific terminology are all meaningless [Tarski] |
Full Idea: Disputes like the vague one about 'the right conception of truth' occur in all domains where, instead of exact, scientific terminology, common language with its vagueness and ambiguity is used; and they are always meaningless, and therefore in vain. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 14) | |
A reaction: Taski taught a large number of famous philosophers in California in the 1950s, and this approach has had a huge influence. Recently there has been a bit of a rebellion. E.g. Kit Fine doesn't think it can all be done in formal languages. |
19179 | For a definition we need the words or concepts used, the rules, and the structure of the language [Tarski] |
Full Idea: We must specify the words or concepts which we wish to use in defining the notion of truth; and we must also give the formal rules to which the definition should conform. More generally, we must describe the formal structure of the language. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 01) | |
A reaction: This, of course, is a highly formal view of how definition should be achieved, offered in anticipation of one of the most famous definitions in logic (of truth, by Tarski). Normally we assume English and classical logic. |
9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
Full Idea: A contextual definition shows how to analyse an expression in situ, by replacing a complete sentence (of a particular form) in which the expression occurs by another in which it does not. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.2) | |
A reaction: This is a controversial procedure, which (according to Dummett) Frege originally accepted, and later rejected. It might not be the perfect definition that replacing just the expression would give you, but it is a promising step. |
10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
Full Idea: When a definition contains a quantifier whose range includes the very entity being defined, the definition is said to be 'impredicative'. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.2) | |
A reaction: Presumably they are 'impredicative' because they do not predicate a new quality in the definiens, but make use of the qualities already known. |
16295 | Tarski proved that truth cannot be defined from within a given theory [Tarski, by Halbach] |
Full Idea: Tarski's Theorem states that under fairly generally applicable conditions, the assumption that there is a definition of truth within a given theory for the language of that same theory leads to a contradiction. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Volker Halbach - Axiomatic Theories of Truth 1 | |
A reaction: That might leave room for a definition outside the given theory. I take the main motivation for the axiomatic approach to be a desire to get a theory of truth within the given theory, where Tarski's Theorem says traditional approaches are just wrong. |
10153 | In everyday language, truth seems indefinable, inconsistent, and illogical [Tarski] |
Full Idea: In everyday language it seems impossible to define the notion of truth or even to use this notion in a consistent manner and in agreement with the laws of logic. | |
From: Alfred Tarski (works [1936]), quoted by Feferman / Feferman - Alfred Tarski: life and logic Int III | |
A reaction: [1935] See Logic|Theory of Logic|Semantics of Logic for Tarski's approach to truth. |
15342 | Tarski proved that any reasonably expressive language suffers from the liar paradox [Tarski, by Horsten] |
Full Idea: Tarski's Theorem on the undefinability of truth says in a language sufficiently rich to talk about itself (which Gödel proved possible, via coding) the liar paradox can be carried out. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Leon Horsten - The Tarskian Turn 02.2 | |
A reaction: The point is that truth is formally indefinable if it leads inescapably to contradiction, which the liar paradox does. This theorem is the motivation for all modern attempts to give a rigorous account of truth. |
19069 | 'True sentence' has no use consistent with logic and ordinary language, so definition seems hopeless [Tarski] |
Full Idea: The possibility of a consistent use of 'true sentence' which is in harmony with the laws of logic and the spirit of everyday language seems to be very questionable, so the same doubt attaches to the possibility of constructing a correct definition. | |
From: Alfred Tarski (The Concept of Truth for Formalized Languages [1933], §1) | |
A reaction: This is often cited as Tarski having conclusively proved that 'true' cannot be defined from within a language, but his language here is much more circumspect. Modern critics say the claim depends entirely on classical logic. |
19178 | Definitions of truth should not introduce a new version of the concept, but capture the old one [Tarski] |
Full Idea: The desired definition of truth does not aim to specify the meaning of a familiar word used to denote a novel notion; on the contrary, it aims to catch hold of the actual meaning of an old notion. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 01) | |
A reaction: Tarski refers back to Aristotle for an account of the 'old notion'. To many the definition of Tarski looks very weird, so it is important to see that he is trying to capture the original concept. |
19177 | A definition of truth should be materially adequate and formally correct [Tarski] |
Full Idea: The main problem of the notion of truth is to give a satisfactory definition which is materially adequate and formally correct. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 01) | |
A reaction: That is, I take it, that it covers all cases of being true and failing to be true, and it fits in with the logic. The logic is explicitly classical logic, and he is not aiming to give the 'nature' or natural language understanding of the concept. |
19186 | A rigorous definition of truth is only possible in an exactly specified language [Tarski] |
Full Idea: The problem of the definition of truth obtains a precise meaning and can be solved in a rigorous way only for those languages whose structure has been exactly specified. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 06) | |
A reaction: Taski has just stated how to exactly specify the structure of a language. He says definition can only be vague and approximate for natural languages. (The usual criticism of the correspondence theory is its vagueness). |
19194 | We may eventually need to split the word 'true' into several less ambiguous terms [Tarski] |
Full Idea: A time may come when we find ourselves confronted with several incompatible, but equally clear and precise, conceptions of truth. It will then become necessary to abandon the ambiguous usage of the word 'true', and introduce several terms instead. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 14) | |
A reaction: There may be a whiff of the pragmatic attitude to truth here, though that view is not necessarily pluralist. Analytic philosophy needs much more splitting of difficult terms into several more focused terms. |
16296 | Tarski's Theorem renders any precise version of correspondence impossible [Tarski, by Halbach] |
Full Idea: Tarski's Theorem applies to any sufficient precise version of the correspondence theory of truth, and all the other traditional theories of truth. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Volker Halbach - Axiomatic Theories of Truth 1 | |
A reaction: This is the key reason why modern thinkers have largely dropped talk of the correspondence theory. See Idea 16295. |
19196 | Scheme (T) is not a definition of truth [Tarski] |
Full Idea: It is a mistake to regard scheme (T) as a definition of truth. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 15) | |
A reaction: The point is, I take it, that the definition is the multitude of sentences which are generated by the schema, not the schema itself. |
15339 | Tarski gave up on the essence of truth, and asked how truth is used, or how it functions [Tarski, by Horsten] |
Full Idea: Tarski emancipated truth theory from traditional philosophy, by no longer posing Pilate's question (what is truth? or what is the essence of truth?) but instead 'how is truth used?', 'how does truth function?' and 'how can its functioning be described?'. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Leon Horsten - The Tarskian Turn 02.2 | |
A reaction: Horsten, later in the book, does not give up on the essence of truth, and modern theorists are trying to get back to that question by following Tarski's formal route. Modern analytic philosophy at its best, it seems to me. |
16302 | Tarski did not just aim at a definition; he also offered an adequacy criterion for any truth definition [Tarski, by Halbach] |
Full Idea: Tarski did not settle for a definition of truth, taking its adequacy for granted. Rather he proposed an adequacy criterion for evaluating the adequacy of definitions of truth. The criterion is his famous Convention T. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Volker Halbach - Axiomatic Theories of Truth 3 | |
A reaction: Convention T famously says the sentence is true if and only if a description of the sentence is equivalent to affirming the sentence. 'Snow is white' iff snow is white. |
19135 | Tarski enumerates cases of truth, so it can't be applied to new words or languages [Davidson on Tarski] |
Full Idea: Tarski does not tell us how to apply his concept of truth to a new case, whether the new case is a new language or a word newly added to a language. This is because enumerating cases gives no clue for the next or general case. | |
From: comment on Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Donald Davidson - Truth and Predication 1 | |
A reaction: His account has been compared to a telephone directory. We aim to understand the essence of anything, so that we can fully know it, and explain and predict how it will behave. Either truth is primitive, or I demand to know its essence. |
19138 | Tarski define truths by giving the extension of the predicate, rather than the meaning [Davidson on Tarski] |
Full Idea: Tarski defined the class of true sentences by giving the extension of the truth predicate, but he did not give the meaning. | |
From: comment on Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Donald Davidson - Truth and Predication 1 | |
A reaction: This is analogous to giving an account of the predicate 'red' as the set of red objects. Since I regard that as a hopeless definition of 'red', I am inclined to think the same of Tarski's account of truth. It works in the logic, but so what? |
4699 | Tarski made truth relative, by only defining truth within some given artificial language [Tarski, by O'Grady] |
Full Idea: Tarski's account doesn't hold for natural languages. The general notion of truth is replaced by "true-in-L", where L is a formal language. Hence truth is relativized to each artificial language. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Paul O'Grady - Relativism Ch.2 | |
A reaction: This is a pretty good indication that Tarski's theory is NOT a correspondence theory, even if its structure may sometimes give that impression. |
19324 | Tarski has to avoid stating how truths relate to states of affairs [Kirkham on Tarski] |
Full Idea: Tarski has to define truths so as not to make explicit the relation between a true sentence and an obtaining state of affairs. ...He has to list each sentence separately, and simply assign it a state of affairs. | |
From: comment on Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Richard L. Kirkham - Theories of Truth: a Critical Introduction 5.8 | |
A reaction: He has to avoid semantic concepts like 'reference', because he wants a physicalist theory, according to Kirkham. Thus the hot interest in theories of reference in the 1970s/80s. And also attempts to give a physicalist account of meaning. |
10672 | Tarskian semantics says that a sentence is true iff it is satisfied by every sequence [Tarski, by Hossack] |
Full Idea: Tarskian semantics says that a sentence is true iff it is satisfied by every sequence, where a sequence is a set-theoretic individual, a set of ordered pairs each with a natural number as its first element and an object from the domain for its second. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Keith Hossack - Plurals and Complexes 3 |
13338 | '"It is snowing" is true if and only if it is snowing' is a partial definition of the concept of truth [Tarski] |
Full Idea: Statements of the form '"it is snowing" is true if and only if it is snowing' and '"the world war will begin in 1963" is true if and only if the world war will being in 1963' can be regarded as partial definitions of the concept of truth. | |
From: Alfred Tarski (The Establishment of Scientific Semantics [1936], p.404) | |
A reaction: The key word here is 'partial'. Truth is defined, presumably, when every such translation from the object language has been articulated, which is presumably impossible, given the infinity of concatenated phrases possible in a sentence. |
19180 | It is convenient to attach 'true' to sentences, and hence the language must be specified [Tarski] |
Full Idea: For several reasons it appears most convenient to apply the term 'true' to sentences, and we shall follow this course. Consequently, we must always relate the notion of truth, like that of a sentence, to a specific language. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 02) | |
A reaction: Personally I take truth to attach to propositions, since sentences are ambiguous. In Idea 17308 the one sentence expresses three different truths (in my opinion), even though a single sentence (given in the object language) specifies it. |
19181 | In the classical concept of truth, 'snow is white' is true if snow is white [Tarski] |
Full Idea: If we base ourselves on the classical conception of truth, we shall say that the sentence 'snow is white' is true if snow is white, and it is false if snow is not white. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 04) | |
A reaction: I had not realised, prior to his, how closely Tarski is sticking to Aristotle's famous formulation of truth. The point is that you can only specify 'what is' using a language. Putting 'true' in the metalanguage gives specific content to Aristotle. |
19183 | Each interpreted T-sentence is a partial definition of truth; the whole definition is their conjunction [Tarski] |
Full Idea: In 'X is true iff p' if we replace X by the name of a sentence and p by a particular sentence this can be considered a partial definition of truth. The whole definition has to be ...a logical conjunction of all these partial definitions. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 04) | |
A reaction: This seems an unprecedented and odd way to define something. Define 'red' by '"This tomato is red" iff this tomato is red', etc? Define 'stone' by collecting together all the stones? The complex T-sentences are infinite in number. |
19182 | Use 'true' so that all T-sentences can be asserted, and the definition will then be 'adequate' [Tarski] |
Full Idea: We wish to use the term 'true' in such a way that all the equivalences of the form (T) [i.e. X is true iff p] can be asserted, and we shall call a definition of truth 'adequate' if all these equivalences follow from it. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 04) | |
A reaction: The interpretation of Tarski's theory is difficult. From this I'm thinking that 'true' is simply being defined as 'assertible'. This is the status of each line in a logical proof, if there is a semantic dimension to the proof (and not mere syntax). |
19198 | We don't give conditions for asserting 'snow is white'; just that assertion implies 'snow is white' is true [Tarski] |
Full Idea: Semantic truth implies nothing regarding the conditions under which 'snow is white' can be asserted. It implies only that, whenever we assert or reject this sentence, we must be ready to assert or reject the correlated sentence '"snow is white" is true'. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 18) | |
A reaction: This appears to identify truth with assertibility, which is pretty much what modern pragmatists say. How do you distinguish 'genuine' assertion from rhetorical, teasing or lying assertions? Genuine assertion implies truth? Hm. |
15410 | Truth only applies to closed formulas, but we need satisfaction of open formulas to define it [Burgess on Tarski] |
Full Idea: In Tarski's theory of truth, although the notion of truth is applicable only to closed formulas, to define it we must define a more general notion of satisfaction applicable to open formulas. | |
From: comment on Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by John P. Burgess - Philosophical Logic 1.8 | |
A reaction: This is a helpful pointer to what is going on in the Tarski definition. It culminates in the 'satisfaction of all sequences', which presumable delivers the required closed formula. |
18811 | Tarski uses sentential functions; truly assigning the objects to variables is what satisfies them [Tarski, by Rumfitt] |
Full Idea: Tarski invoked the notion of a sentential function, where components are replaced by appropriate variables. A function is then satisfied by assigning objects to variables. An assignment satisfies if the function is true of the things assigned. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Ian Rumfitt - The Boundary Stones of Thought 3.2 | |
A reaction: [very compressed] This use of sentential functions, rather than sentences, looks like the key to Tarski's definition of truth. |
15365 | We can define the truth predicate using 'true of' (satisfaction) for variables and some objects [Tarski, by Horsten] |
Full Idea: The truth predicate, says Tarski, should be defined in terms of the more primitive satisfaction relation: the relation of being 'true of'. The fundamental notion is a formula (containing the free variables) being true of a sequence of objects as values. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Leon Horsten - The Tarskian Turn 06.3 |
19314 | For physicalism, reduce truth to satisfaction, then define satisfaction as physical-plus-logic [Tarski, by Kirkham] |
Full Idea: Tarski, a physicalist, reduced semantics to physical and/or logicomathematical concepts. He defined all semantic concepts, save satisfaction, in terms of truth. Then truth is defined in terms of satisfaction, and satisfaction is given non-semantically. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Richard L. Kirkham - Theories of Truth: a Critical Introduction 5.1 | |
A reaction: The term 'logicomathematical' is intended to cover set theory. Kirkham says you can remove these restrictions from Tarski's theory, and the result is a version of the correspondence theory. |
19316 | Insight: don't use truth, use a property which can be compositional in complex quantified sentence [Tarski, by Kirkham] |
Full Idea: Tarski's great insight is find another property, since open sentences are not truth. It must be had by open and genuine sentences. Clauses having it must generate it for the whole sentence. Truth can be defined for sentences by using it. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Richard L. Kirkham - Theories of Truth: a Critical Introduction 5.4 | |
A reaction: The proposed property is 'satisfaction', which can (unlike truth) be a feature open sentences (such as 'x is green', which is satisfied by x='grass'), |
19175 | Tarski gave axioms for satisfaction, then derived its explicit definition, which led to defining truth [Tarski, by Davidson] |
Full Idea: Tarski turned his axiomatic characterisation of satisfaction into an explicit definition of the satisfaction-predicate using some fancy set theoretical apparatus, and this in turn leads to the explicit definition of the truth predicate. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Donald Davidson - Truth and Predication 7 |
19184 | The best truth definition involves other semantic notions, like satisfaction (relating terms and objects) [Tarski] |
Full Idea: It turns out that the simplest and most natural way of obtaining an exact definition of truth is one which involves the use of other semantic notions, e.g. the notion of satisfaction (...which expresses relations between expressions and objects). | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 05) | |
A reaction: While the T-sentences appear to be 'minimal' and 'deflationary', it seems important to remember that 'satisfaction', which is basic to his theory, is a very robust notion. He actually mentions 'objects'. But see Idea 19185. |
19191 | Specify satisfaction for simple sentences, then compounds; true sentences are satisfied by all objects [Tarski] |
Full Idea: To define satisfaction we indicate which objects satisfy the simplest sentential functions, then state the conditions for compound functions. This applies automatically to sentences (with no free variables) so a true sentence is satisfied by all objects. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 11) | |
A reaction: I presume nothing in the domain of objects can conflict with a sentence that has been satisfied by some of them, so 'all' the objects satisfy the sentence. Tarski doesn't use the word 'domain'. Basic satisfaction seems to be stipulated. |
19188 | We can't use a semantically closed language, or ditch our logic, so a meta-language is needed [Tarski] |
Full Idea: In a 'semantically closed' language all sentences which determine the adequate usage of 'true' can be asserted in the language. ...We can't change our logic, so we reject such languages. ...So must use two different languages to discuss truth. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 08-09) | |
A reaction: This section explains why a meta-language is required. It rests entirely on the existence of the Liar paradox is a semantically closed language. |
19189 | The metalanguage must contain the object language, logic, and defined semantics [Tarski] |
Full Idea: Every sentence which occurs in the object language must also occur in the metalanguage, or can be translated into the metalanguage. There must also be logical terms, ...and semantic terms can only be introduced in the metalanguage by definition. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 09) | |
A reaction: He suggest that if the languages are 'typed', the meta-languag, to be 'richer', must contain variables of a higher logica type. Does this mean second-order logic? |
10969 | Tarski had a theory of truth, and a theory of theories of truth [Tarski, by Read] |
Full Idea: Besides a theory of truth of his own, Tarski developed a theory of theories of truth. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Stephen Read - Thinking About Logic Ch.1 | |
A reaction: The famous snow biconditional is the latter, and the recursive account based on satisfaction is the former. |
19134 | Tarski defined truth for particular languages, but didn't define it across languages [Davidson on Tarski] |
Full Idea: Tarski defined various predicates of the form 's is true in L', each applicable to a single language, but he failed to define a predicate of the form 's is true in L' for variable 'L'. | |
From: comment on Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Donald Davidson - Truth and Predication 1 | |
A reaction: You might say that no one defines 'tree' to be just 'in English', but we might define 'multiplies' to be in Peano Arithmetic. This indicates the limited and formal nature of what Tarski was trying to achieve. |
16303 | Tarski made truth respectable, by proving that it could be defined [Tarski, by Halbach] |
Full Idea: Tarski's proof of the definability of truth allowed him to establish truth as a respectable notion by his standards. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Volker Halbach - Axiomatic Theories of Truth 3 |
16304 | Tarski didn't capture the notion of an adequate truth definition, as Convention T won't prove non-contradiction [Halbach on Tarski] |
Full Idea: Every really adequate theory of truth should also prove the law of non-contradiction. Therefore Tarski's notion of adequacy in Convention T fails to capture the intuitive notion of adequacy he is after. | |
From: comment on Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Volker Halbach - Axiomatic Theories of Truth 3 | |
A reaction: Tarski points out this weakness, in a passage quoted by Halbach. This obviously raises the question of what truth theories should prove, and this is explored by Halbach. If they start to prove arithmetic, we get nervous. Non-contradiction and x-middle? |
17746 | Tarski's 'truth' is a precise relation between the language and its semantics [Tarski, by Walicki] |
Full Idea: Tarski's analysis of the concept of 'truth' ...is given a precise treatment as a particular relation between syntax (language) and semantics (the world). | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Michal Walicki - Introduction to Mathematical Logic History E.1 | |
A reaction: My problem is that the concept of truth seems to apply to animal minds, which are capable of making right or wrong judgements, and of realising their errors. Tarski didn't make universal claims for his account. |
10904 | Tarskian truth neglects the atomic sentences [Mulligan/Simons/Smith on Tarski] |
Full Idea: The Tarskian account of truth neglects the atomic sentences. | |
From: comment on Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Mulligan/Simons/Smith - Truth-makers §1 | |
A reaction: Yes! The whole Tarskian edifice is built on a foundation which it is taboo even to mention. If truth is just the assignment of 'T' and 'F', that isn't even the beginnings of a theory of 'truth'. |
2571 | Tarski says that his semantic theory of truth is completely neutral about all metaphysics [Tarski, by Haack] |
Full Idea: Tarski says "we may remain naďve realists or idealists, empiricists or metaphysicians The semantic conception is completely neutral toward all these issues." | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Susan Haack - Philosophy of Logics 7.5 |
10821 | Physicalists should explain reference nonsemantically, rather than getting rid of it [Tarski, by Field,H] |
Full Idea: Tarski work was to persuade physicalist that eliminating semantics was on the wrong track, and that we should explicate notions in the theory of reference nonsemantically rather than simply get rid of them. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Hartry Field - Tarski's Theory of Truth §3 |
10822 | A physicalist account must add primitive reference to Tarski's theory [Field,H on Tarski] |
Full Idea: We need to add theories of primitive reference to Tarski's account if we are to establish the notion of truth as a physicalistically acceptable notion. | |
From: comment on Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Hartry Field - Tarski's Theory of Truth §4 | |
A reaction: This is the main point of Field's paper, and sounds very plausible to me. There is something major missing from Tarski, and at some point there needs to be a 'primitive' notion of thought and language making contact with the world, as it can't be proved. |
10824 | If listing equivalences is a reduction of truth, witchcraft is just a list of witch-victim pairs [Field,H on Tarski] |
Full Idea: By similar standards of reduction to Tarski's, one might prove witchcraft compatible with physicalism, as long as witches cast only a finite number of spells. We merely list witch-and-victim pairs, with no mention of the terms of witchcraft theory. | |
From: comment on Alfred Tarski (The Semantic Conception of Truth [1944], 04) by Hartry Field - Tarski's Theory of Truth §4 |
19141 | Tarski thought axiomatic truth was too contingent, and in danger of inconsistencies [Tarski, by Davidson] |
Full Idea: Tarski preferred an explicit definition of truth to axioms. He says axioms have a rather accidental character, only a definition can guarantee the continued consistency of the system, and it keeps truth in harmony with physical science and physicalism. | |
From: report of Alfred Tarski (works [1936]) by Donald Davidson - Truth and Predication 2 n2 | |
A reaction: Davidson's summary, gleaned from various sources in Tarski. A big challenge for modern axiom systems is to avoid inconsistency, which is extremely hard to do (given that set theory is not sure of having achieved it). |
19190 | We need an undefined term 'true' in the meta-language, specified by axioms [Tarski] |
Full Idea: We have to include the term 'true', or some other semantic term, in the list of undefined terms of the meta-language, and to express fundamental properties of the notion of truth in a series of axioms. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 10) | |
A reaction: It sounds as if Tarski semantic theory gives truth for the object language, but then an axiomatic theory of truth is also needed for the metalanguage. Halbch and Horsten seem to want an axiomatic theory in the object language. |
15322 | Tarski's had the first axiomatic theory of truth that was minimally adequate [Tarski, by Horsten] |
Full Idea: Tarski's work is the earliest axiomatic theory of truth that meets minimal adequacy conditions. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Leon Horsten - The Tarskian Turn 01.1 | |
A reaction: This shows a way in which Tarski gave a new direction to the study of truth. Subsequent theories have been 'stronger'. |
16306 | Tarski defined truth, but an axiomatisation can be extracted from his inductive clauses [Tarski, by Halbach] |
Full Idea: Tarski preferred a definition of truth, but from that an axiomatisation can be extracted. His induction clauses can be turned into axioms. Hence he opened the way to axiomatic theories of truth. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Volker Halbach - Axiomatic Theories of Truth 3 |
19197 | Truth can't be eliminated from universal claims, or from particular unspecified claims [Tarski] |
Full Idea: Truth can't be eliminated from universal statements saying all sentences of a certain type are true, or from the proof that 'all consequences of true sentences are true'. It is also needed if we can't name the sentence ('Plato's first sentence is true'). | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 16) | |
A reaction: This points to the deflationary view of truth, if its only role is in talking about other sentences in this way. Tarski gives the standard reason for rejecting the Redundancy view. |
19185 | Semantics is a very modest discipline which solves no real problems [Tarski] |
Full Idea: Semantics as it is conceived in this paper is a sober and modest discipline which has no pretensions to being a universal patent-medicine for all the ills and diseases of mankind, whether imaginary or real. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 05) | |
A reaction: Written in 1944. This remark encourages the minimal or deflationary interpretation of his theory of truth, but see the robust use of 'satisfaction' in Idea 19184. |
19195 | Truth tables give prior conditions for logic, but are outside the system, and not definitions [Tarski] |
Full Idea: Logical sentences are often assigned preliminary conditions under which they are true or false (often given as truth tables). However, these are outside the system of logic, and should not be regarded as definitions of the terms involved. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 15) | |
A reaction: Hence, presumably, the connectives are primitives (with no nature or meaning), and the truth tables are axioms for their use? This opinion of Tarski's may have helped shift the preference towards natural deduction introduction and elimination rules. |
10098 | The 'power set' of A is all the subsets of A [George/Velleman] |
Full Idea: The 'power set' of A is all the subsets of A. P(A) = {B : B ⊆ A}. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) |
10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman] |
Full Idea: The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}}. The existence of this set is guaranteed by three applications of the Axiom of Pairing. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: See Idea 10100 for the Axiom of Pairing. |
10101 | Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B [George/Velleman] |
Full Idea: The 'Cartesian Product' of any two sets A and B is the set of all ordered pairs <a, b> in which a ∈ A and b ∈ B, and it is denoted as A x B. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) |
10103 | Grouping by property is common in mathematics, usually using equivalence [George/Velleman] |
Full Idea: The idea of grouping together objects that share some property is a common one in mathematics, ...and the technique most often involves the use of equivalence relations. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) |
10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman] |
Full Idea: A relation is an equivalence relation if it is reflexive, symmetric and transitive. The 'same first letter' is an equivalence relation on the set of English words. Any relation that puts a partition into clusters will be equivalence - and vice versa. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: This is a key concept in the Fregean strategy for defining numbers. |
10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman] |
Full Idea: ZFC is a theory concerned only with sets. Even the elements of all of the sets studied in ZFC are also sets (whose elements are also sets, and so on). This rests on one clearly pure set, the empty set Φ. ..Mathematics only needs pure sets. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: This makes ZFC a much more metaphysically comfortable way to think about sets, because it can be viewed entirely formally. It is rather hard to disentangle a chair from the singleton set of that chair. |
10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman] |
Full Idea: The Axiom of Extensionality says that for all sets x and y, if x and y have the same elements then x = y. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: This seems fine in pure set theory, but hits the problem of renates and cordates in the real world. The elements coincide, but the axiom can't tell you why they coincide. |
10100 | Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman] |
Full Idea: The Axiom of Pairing says that for all sets x and y, there is a set z containing x and y, and nothing else. In symbols: ∀x∀y∃z∀w(w ∈ z ↔ (w = x ∨ w = y)). | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: See Idea 10099 for an application of this axiom. |
17900 | The Axiom of Reducibility made impredicative definitions possible [George/Velleman] |
Full Idea: The Axiom of Reducibility ...had the effect of making impredicative definitions possible. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) |
10109 | ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman] |
Full Idea: Sets, unlike extensions, fail to correspond to all concepts. We can prove in ZFC that there is no set corresponding to the concept 'set' - that is, there is no set of all sets. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4) | |
A reaction: This is rather an important point for Frege. However, all concepts have extensions, but they may be proper classes, rather than precisely defined sets. |
10108 | As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman] |
Full Idea: The problem with reducing arithmetic to ZFC is not that this theory is inconsistent (as far as we know it is not), but rather that is not completely general, and for this reason not logical. For example, it asserts the existence of sets. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4) | |
A reaction: Note that ZFC has not been proved consistent. |
10152 | Set theory and logic are fairy tales, but still worth studying [Tarski] |
Full Idea: People have asked me, 'How can you, a nominalist, do work in set theory and in logic, which are theories about things you do not believe in?' ...I believe that there is a value even in fairy tales and the study of fairy tales. | |
From: Alfred Tarski (talk [1965]), quoted by Feferman / Feferman - Alfred Tarski: life and logic | |
A reaction: This is obviously an oversimplification. I don't think for a moment that Tarski literally believed that the study of fairy tales had as much value as the study of logic. Why do we have this particular logic, and not some other? |
10048 | There is no clear boundary between the logical and the non-logical [Tarski] |
Full Idea: No objective grounds are known to me which permit us to draw a sharp boundary between the two groups of terms, the logical and the non-logical. | |
From: Alfred Tarski (works [1936]), quoted by Alan Musgrave - Logicism Revisited §3 | |
A reaction: Musgrave is pointing out that this is bad news if you want to 'reduce' something like arithmetic to logic. 'Logic' is a vague object. |
13337 | A language: primitive terms, then definition rules, then sentences, then axioms, and finally inference rules [Tarski] |
Full Idea: For a language, we must enumerate the primitive terms, and the rules of definition for new terms. Then we must distinguish the sentences, and separate out the axioms from amng them, and finally add rules of inference. | |
From: Alfred Tarski (The Establishment of Scientific Semantics [1936], p.402) | |
A reaction: [compressed] This lays down the standard modern procedure for defining a logical language. Once all of this is in place, we then add a semantics and we are in business. Natural deduction tries to do without the axioms. |
18812 | Split out the logical vocabulary, make an assignment to the rest. It's logical if premises and conclusion match [Tarski, by Rumfitt] |
Full Idea: Tarski made a division of logical and non-logical vocabulary. He then defined a model as a non-logical assignment satisfying the corresponding sentential function. Then a conclusion follows logically if every model of the premises models the conclusion. | |
From: report of Alfred Tarski (The Concept of Logical Consequence [1936]) by Ian Rumfitt - The Boundary Stones of Thought 3.2 | |
A reaction: [compressed] This is Tarski's account of logical consequence, which follows on from his account of truth. 'Logical validity' is then 'true in every model'. Rumfitt doubts whether Tarski has given the meaning of 'logical consequence'. |
10694 | Logical consequence is when in any model in which the premises are true, the conclusion is true [Tarski, by Beall/Restall] |
Full Idea: Tarski's 1936 definition of logical consequence is that in any model in which the premises are true, the conclusion is true too (so that no model can make the conclusion false). | |
From: report of Alfred Tarski (works [1936]) by JC Beall / G Restall - Logical Consequence 3 | |
A reaction: So the general idea is that a logical consequence is distinguished by being unstoppable. Sounds good. But then we have monotonic and non-monotonic logics, which (I'm guessing) embody different notions of consequence. |
10479 | Logical consequence: true premises give true conclusions under all interpretations [Tarski, by Hodges,W] |
Full Idea: Tarski's definition of logical consequence (1936) is that in a fully interpreted formal language an argument is valid iff under any allowed interpretation of its nonlogical symbols, if the premises are true then so is the conclusion. | |
From: report of Alfred Tarski (works [1936]) by Wilfrid Hodges - Model Theory 3 | |
A reaction: The idea that you can only make these claims 'under an interpretation' seems to have had a huge influence on later philosophical thinking. |
13344 | X follows from sentences K iff every model of K also models X [Tarski] |
Full Idea: The sentence X follows logically from the sentences of the class K if and only if every model of the class K is also a model of the sentence X. | |
From: Alfred Tarski (The Concept of Logical Consequence [1936], p.417) | |
A reaction: [see Idea 13343 for his account of a 'model'] He is offering to define logical consequence in general, but this definition fits what we now call 'semantic consequence', written |=. This it is standard practice to read |= as 'models'. |
19192 | The truth definition proves semantic contradiction and excluded middle laws (not the logic laws) [Tarski] |
Full Idea: With our definition of truth we can prove the laws of contradiction and excluded middle. These semantic laws should not be identified with the related logical laws, which belong to the sentential calculus, and do not involve 'true' at all. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 12) | |
A reaction: Very illuminating. I wish modern thinkers could be so clear about this matter. The logic contains 'P or not-P'. The semantics contains 'P is either true or false'. Critics say Tarski has presupposed 'classical' logic. |
10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
Full Idea: A hallmark of our realist stance towards the natural world is that we are prepared to assert the Law of Excluded Middle for all statements about it. For all statements S, either S is true, or not-S is true. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4) | |
A reaction: Personally I firmly subscribe to realism, so I suppose I must subscribe to Excluded Middle. ...Provided the statement is properly formulated. Or does liking excluded middle lead me to realism? |
18759 | Identity is invariant under arbitrary permutations, so it seems to be a logical term [Tarski, by McGee] |
Full Idea: Tarski showed that the only binary relations invariant under arbitrary permutations are the universal relation, the empty relation, identity and non-identity, thus giving us a reason to include '=' among the logical terms. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Vann McGee - Logical Consequence 6 | |
A reaction: Tarski was looking for a criterion to distinguish logical from non-logical terms, since his account of logical validity depended on it. This idea lies behind whether a logic is or is not specified to be 'with identity' (i.e. using '='). |
10823 | A name denotes an object if the object satisfies a particular sentential function [Tarski] |
Full Idea: To say that the name x denotes a given object a is the same as to stipulate that the object a ... satisfies a sentential function of a particular type. | |
From: Alfred Tarski (The Concept of Truth for Formalized Languages [1933], p.194) |
18756 | Tarski built a compositional semantics for predicate logic, from dependent satisfactions [Tarski, by McGee] |
Full Idea: Tarski discovered how to give a compositional semantics for predicate calculus, defining truth in terms of satisfaction, and showing how satisfaction for a complicated formula depends on satisfaction of the simple subformulas. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Vann McGee - Logical Consequence 4 | |
A reaction: The problem was that the subformulas may contain free variables, and thus not be sentences with truth values. 'Satisfaction' can handle this, where 'truth' cannot (I think). |
19313 | Tarksi invented the first semantics for predicate logic, using this conception of truth [Tarski, by Kirkham] |
Full Idea: Tarski invented a formal semantics for quantified predicate logic, the logic of reasoning about mathematics. The heart of this great accomplishment is his theory of truth. It has been called semantic 'theory' of truth, but Tarski preferred 'conception'. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Richard L. Kirkham - Theories of Truth: a Critical Introduction 5.1 |
13335 | Semantics is the concepts of connections of language to reality, such as denotation, definition and truth [Tarski] |
Full Idea: Semantics is the totality of considerations concerning concepts which express connections between expressions of a language and objects and states of affairs referred to by these expressions. Examples are denotation, satisfaction, definition and truth. | |
From: Alfred Tarski (The Establishment of Scientific Semantics [1936], p.401) | |
A reaction: Interestingly, he notes that it 'is not commonly recognised' that truth is part of semantics. Nowadays truth seems to be the central concept in most semantics. |
13336 | A language containing its own semantics is inconsistent - but we can use a second language [Tarski] |
Full Idea: People have not been aware that the language about which we speak need by no means coincide with the language in which we speak. ..But the language which contains its own semantics must inevitably be inconsistent. | |
From: Alfred Tarski (The Establishment of Scientific Semantics [1936], p.402) | |
A reaction: It seems that Tarski was driven to propose the metalanguage approach mainly by the Liar Paradox. |
13339 | A sentence is satisfied when we can assert the sentence when the variables are assigned [Tarski] |
Full Idea: Here is a partial definition of the concept of satisfaction: John and Peter satisfy the sentential function 'X and Y are brothers' if and only if John and Peter are brothers. | |
From: Alfred Tarski (The Establishment of Scientific Semantics [1936], p.405) | |
A reaction: Satisfaction applies to open sentences and truth to closed sentences (with named objects). He uses the notion of total satisfaction to define truth. The example is a partial definition, not just an illustration. |
13340 | Satisfaction is the easiest semantical concept to define, and the others will reduce to it [Tarski] |
Full Idea: It has been found useful in defining semantical concepts to deal first with the concept of satisfaction; both because the definition of this concept presents relatively few difficulties, and because the other semantical concepts are easily reduced to it. | |
From: Alfred Tarski (The Establishment of Scientific Semantics [1936], p.406) | |
A reaction: See Idea 13339 for his explanation of satisfaction. We just say that a open sentence is 'acceptable' or 'assertible' (or even 'true') when particular values are assigned to the variables. Then sentence is then 'satisfied'. |
13343 | A 'model' is a sequence of objects which satisfies a complete set of sentential functions [Tarski] |
Full Idea: An arbitrary sequence of objects which satisfies every sentential function of the sentences L' will be called a 'model' or realization of the class L of sentences. There can also be a model of a single sentence is this way. | |
From: Alfred Tarski (The Concept of Logical Consequence [1936], p.417) | |
A reaction: [L' is L with the constants replaced by variables] Tarski is the originator of model theory, which is central to modern logic. The word 'realization' is a helpful indicator of what he has in mind. A model begins to look like a possible world. |
10129 | A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman] |
Full Idea: A 'model' of a theory is an assignment of meanings to the symbols of its language which makes all of its axioms come out true. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7) | |
A reaction: If the axioms are all true, and the theory is sound, then all of the theorems will also come out true. |
16323 | The object language/ metalanguage distinction is the basis of model theory [Tarski, by Halbach] |
Full Idea: Tarski's distinction between object and metalanguage forms the basis of model theory. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Volker Halbach - Axiomatic Theories of Truth 11 |
10105 | Differences between isomorphic structures seem unimportant [George/Velleman] |
Full Idea: Mathematicians tend to regard the differences between isomorphic mathematical structures as unimportant. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: This seems to be a pointer towards Structuralism as the underlying story in mathematics. The intrinsic character of so-called 'objects' seems unimportant. How theories map onto one another (and onto the world?) is all that matters? |
13341 | Using the definition of truth, we can prove theories consistent within sound logics [Tarski] |
Full Idea: Using the definition of truth we are in a position to carry out the proof of consistency for deductive theories in which only (materially) true sentences are (formally) provable. | |
From: Alfred Tarski (The Establishment of Scientific Semantics [1936], p.407) | |
A reaction: This is evidently what Tarski saw as the most important first fruit of his new semantic theory of truth. |
10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman] |
Full Idea: Consistency is a purely syntactic property, unlike the semantic property of soundness. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6) |
10126 | A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman] |
Full Idea: If there is a sentence such that both the sentence and its negation are theorems of a theory, then the theory is 'inconsistent'. Otherwise it is 'consistent'. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7) |
10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman] |
Full Idea: Soundness is a semantic property, unlike the purely syntactic property of consistency. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6) |
10127 | A 'complete' theory contains either any sentence or its negation [George/Velleman] |
Full Idea: If there is a sentence such that neither the sentence nor its negation are theorems of a theory, then the theory is 'incomplete'. Otherwise it is 'complete'. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7) | |
A reaction: Interesting questions are raised about undecidable sentences, irrelevant sentences, unknown sentences.... |
8940 | Tarski avoids the Liar Paradox, because truth cannot be asserted within the object language [Tarski, by Fisher] |
Full Idea: In Tarski's account of truth, self-reference (as found in the Liar Paradox) is prevented because the truth predicate for any given object language is never a part of that object language, and so a sentence can never predicate truth of itself. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Jennifer Fisher - On the Philosophy of Logic 03.I | |
A reaction: Thus we solve the Liar Paradox by ruling that 'you are not allowed to say that'. Hm. The slightly odd result is that in any conversation about whether p is true, we end up using (logically speaking) two different languages simultaneously. Hm. |
19187 | The Liar makes us assert a false sentence, so it must be taken seriously [Tarski] |
Full Idea: In my judgement, it would be quite wrong and dangerous from the point of view of scientific progress to depreciate the importance of nhtinomies like the Liar Paradox, and treat them as jokes. The fact is we have been compelled to assert a false sentence. | |
From: Alfred Tarski (The Semantic Conception of Truth [1944], 07) | |
A reaction: This is the heartfelt cry of the perfectionist, who wants everything under control. It was the dream of the age of Frege to Hilbert, which gradually eroded after Gödel's Incompleteness proof. Short ordinary folk panic about the Liar? |
10106 | Rational numbers give answers to division problems with integers [George/Velleman] |
Full Idea: We can think of rational numbers as providing answers to division problems involving integers. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: Cf. Idea 10102. |
10102 | The integers are answers to subtraction problems involving natural numbers [George/Velleman] |
Full Idea: In defining the integers in set theory, our definition will be motivated by thinking of the integers as answers to subtraction problems involving natural numbers. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: Typical of how all of the families of numbers came into existence; they are 'invented' so that we can have answers to problems, even if we can't interpret the answers. It it is money, we may say the minus-number is a 'debt', but is it? Cf Idea 10106. |
10107 | Real numbers provide answers to square root problems [George/Velleman] |
Full Idea: One reason for introducing the real numbers is to provide answers to square root problems. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: Presumably the other main reasons is to deal with problems of exact measurement. It is interesting that there seem to be two quite distinct reasons for introducing the reals. Cf. Ideas 10102 and 10106. |
9946 | Logicists say mathematics is applicable because it is totally general [George/Velleman] |
Full Idea: The logicist idea is that if mathematics is logic, and logic is the most general of disciplines, one that applies to all rational thought regardless of its content, then it is not surprising that mathematics is widely applicable. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.2) | |
A reaction: Frege was keen to emphasise this. You are left wondering why pure logic is applicable to the physical world. The only account I can give is big-time Platonism, or Pythagoreanism. Logic reveals the engine-room of nature, where the design is done. |
10125 | The classical mathematician believes the real numbers form an actual set [George/Velleman] |
Full Idea: Unlike the intuitionist, the classical mathematician believes in an actual set that contains all the real numbers. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6) |
10157 | Tarski improved Hilbert's geometry axioms, and without set-theory [Tarski, by Feferman/Feferman] |
Full Idea: Tarski found an elegant new axiom system for Euclidean geometry that improved Hilbert's earlier version - and he formulated it without the use of set-theoretical notions. | |
From: report of Alfred Tarski (works [1936]) by Feferman / Feferman - Alfred Tarski: life and logic Ch.9 |
17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman] |
Full Idea: The first-order version of the induction axiom is weaker than the second-order, because the latter applies to all concepts, but the first-order applies only to concepts definable by a formula in the first-order language of number theory. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7 n7) |
10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman] |
Full Idea: The idea behind the proofs of the Incompleteness Theorems is to use the language of Peano Arithmetic to talk about the formal system of Peano Arithmetic itself. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7) | |
A reaction: The mechanism used is to assign a Gödel Number to every possible formula, so that all reasonings become instances of arithmetic. |
17902 | A successor is the union of a set with its singleton [George/Velleman] |
Full Idea: For any set x, we define the 'successor' of x to be the set S(x) = x U {x}. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: This is the Fregean approach to successor, where the Dedekind approach takes 'successor' to be a primitive. Frege 1884:§76. |
10133 | Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman] |
Full Idea: The derivability of Peano's Postulates from Hume's Principle in second-order logic has been dubbed 'Frege's Theorem', (though Frege would not have been interested, because he didn't think Hume's Principle gave an adequate definition of numebrs). | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.8 n1) | |
A reaction: Frege said the numbers were the sets which were the extensions of the sets created by Hume's Principle. |
10130 | Set theory can prove the Peano Postulates [George/Velleman] |
Full Idea: The Peano Postulates can be proven in ZFC. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7) |
10089 | Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman] |
Full Idea: One might well wonder whether talk of abstract entities is less a solution to the empiricist's problem of how a priori knowledge is possible than it is a label for the problem. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Intro) | |
A reaction: This pinpoints my view nicely. What the platonist postulates is remote, bewildering, implausible and useless! |
10131 | If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman] |
Full Idea: As, in the logicist view, mathematics is about nothing particular, it is little wonder that nothing in particular needs to be observed in order to acquire mathematical knowledge. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002]) | |
A reaction: At the very least we can say that no one would have even dreamt of the general system of arithmetic is they hadn't had experience of the particulars. Frege thought generality ensured applicability, but extreme generality might entail irrelevance. |
17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman] |
Full Idea: If a is an individual and b is a set of individuals, then in the theory of types we cannot talk about the set {a,b}, since it is not an individual or a set of individuals, ...but it is hard to see what harm can come from it. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) |
10092 | In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman] |
Full Idea: In the unramified theory of types, all objects are classified into a hierarchy of types. The lowest level has individual objects that are not sets. Next come sets whose elements are individuals, then sets of sets, etc. Variables are confined to types. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: The objects are Type 0, the basic sets Type 1, etc. |
10094 | The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman] |
Full Idea: The theory of types seems to rule out harmless sets as well as paradoxical ones. If a is an individual and b is a set of individuals, then in type theory we cannot talk about the set {a,b}. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: Since we cheerfully talk about 'Cicero and other Romans', this sounds like a rather disasterous weakness. |
10095 | Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman] |
Full Idea: A problem with type theory is that there are only finitely many individuals, and finitely many sets of individuals, and so on. The hierarchy may be infinite, but each level is finite. Mathematics required an axiom asserting infinitely many individuals. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: Most accounts of mathematics founder when it comes to infinities. Perhaps we should just reject them? |
10154 | Tarski's theory of truth shifted the approach away from syntax, to set theory and semantics [Feferman/Feferman on Tarski] |
Full Idea: Tarski's theory of truth has been most influential in eventually creating a shift from the entirely syntactic way of doing things in metamathematics (promoted by Hilbert in the 1920s, in his theory of proofs), towards a set-theoretical, semantic approach. | |
From: comment on Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Feferman / Feferman - Alfred Tarski: life and logic Int III |
10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman] |
Full Idea: In the first instance all bounded quantifications are finitary, for they can be viewed as abbreviations for conjunctions and disjunctions. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6) | |
A reaction: This strikes me as quite good support for finitism. The origin of a concept gives a good guide to what it really means (not a popular view, I admit). When Aristotle started quantifying, I suspect of he thought of lists, not totalities. |
10134 | Much infinite mathematics can still be justified finitely [George/Velleman] |
Full Idea: It is possible to use finitary reasoning to justify a significant part of infinitary mathematics. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.8) | |
A reaction: This might save Hilbert's project, by gradually accepting into the fold all the parts which have been giving a finitist justification. |
10123 | The intuitionists are the idealists of mathematics [George/Velleman] |
Full Idea: The intuitionists are the idealists of mathematics. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6) |
10124 | Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman] |
Full Idea: For intuitionists, truth is not independent of proof, but this independence is precisely what seems to be suggested by Gödel's First Incompleteness Theorem. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.8) | |
A reaction: Thus Gödel was worse news for the Intuitionists than he was for Hilbert's Programme. Gödel himself responded by becoming a platonist about his unprovable truths. |
9558 | All scientific tests will verify mathematics, so it is a background, not something being tested [Sober] |
Full Idea: If mathematical statements are part of every competing hypothesis, then no matter which hypothesis comes out best in the light of observations, they will be part of the best hypothesis. They are not tested, but are a background assumption. | |
From: Elliott Sober (Mathematics and Indispensibility [1993], 45), quoted by Charles Chihara - A Structural Account of Mathematics | |
A reaction: This is a very nice objection to the Quine-Putnam thesis that mathematics is confirmed by the ongoing successes of science. |
10151 | I am a deeply convinced nominalist [Tarski] |
Full Idea: I am a nominalist. This is a very deep conviction of mine. ...I am a tortured nominalist. | |
From: Alfred Tarski (talk [1965]), quoted by Feferman / Feferman - Alfred Tarski: life and logic Int I | |
A reaction: I too am of the nominalist persuasion, but I don't feel justified in such a strong commitment. |
10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
Full Idea: Corresponding to every concept there is a class (some classes will be sets, the others proper classes). | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4) |
13345 | Sentences are 'analytical' if every sequence of objects models them [Tarski] |
Full Idea: A class of sentences can be called 'analytical' if every sequence of objects is a model of it. | |
From: Alfred Tarski (The Concept of Logical Consequence [1936], p.418) | |
A reaction: See Idea 13344 and Idea 13343 for the context of this assertion. |
20407 | Taste is the capacity to judge an object or representation which is thought to be beautiful [Tarski, by Schellekens] |
Full Idea: Taste is the faculty for judging an object or a kind of representation through a satisfaction or a dissatisfaction, ...where the object of such a satisfaction is called beautiful. | |
From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Elizabeth Schellekens - Immanuel Kant (aesthetics) 1 | |
A reaction: We usually avoid the word 'faculty' nowadays, because it implies a specific mechanism, but 'capacity' will do. Kant is said to focus specifically on beauty, whereas modern aestheticians have a broader view of the type of subject matter. |