Ideas of Alfred Tarski, by Theme

[Polish, 1902 - 1983, Taught in Warsaw 1925-1939, then University of California at Berkeley from 1942 to 1968.]

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1. Philosophy / E. Nature of Metaphysics / 5. Metaphysics beyond Science
Some say metaphysics is a highly generalised empirical study of objects
     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.
1. Philosophy / F. Analytic Philosophy / 1. Nature of Analysis
Disputes that fail to use precise scientific terminology are all meaningless
     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.
2. Reason / D. Definition / 1. Definitions
For a definition we need the words or concepts used, the rules, and the structure of the language
     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.
3. Truth / A. Truth Problems / 2. Defining Truth
Tarski proved that truth cannot be defined from within a given theory
     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.
Tarski proved that any reasonably expressive language suffers from the liar paradox
     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.
'True sentence' has no use consistent with logic and ordinary language, so definition seems hopeless
     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.
In everyday language, truth seems indefinable, inconsistent, and illogical
     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.
A definition of truth should be materially adequate and formally correct
     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.
Definitions of truth should not introduce a new version of the concept, but capture the old one
     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.
A rigorous definition of truth is only possible in an exactly specified language
     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).
We may eventually need to split the word 'true' into several less ambiguous terms
     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.
3. Truth / C. Correspondence Truth / 3. Correspondence Truth critique
Tarski's Theorem renders any precise version of correspondence impossible
     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.
3. Truth / F. Semantic Truth / 1. Tarski's Truth / a. Tarski's truth definition
Tarski gave up on the essence of truth, and asked how truth is used, or how it functions
     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.
Tarski did not just aim at a definition; he also offered an adequacy criterion for any truth definition
     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.
Tarski enumerates cases of truth, so it can't be applied to new words or languages
     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.
Tarski define truths by giving the extension of the predicate, rather than the meaning
     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?
Tarski made truth relative, by only defining truth within some given artificial language
     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.
Tarski has to avoid stating how truths relate to states of affairs
     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.
Tarskian semantics says that a sentence is true iff it is satisfied by every sequence
     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
'"It is snowing" is true if and only if it is snowing' is a partial definition of the concept of truth
     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 phrase possible in a sentence.
It is convenient to attach 'true' to sentences, and hence the language must be specified
     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.
In the classical concept of truth, 'snow is white' is true if snow is white
     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.
Each interpreted T-sentence is a partial definition of truth; the whole definition is their conjunction
     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.
Use 'true' so that all T-sentences can be asserted, and the definition will then be 'adequate'
     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).
We don't give conditions for asserting 'snow is white'; just that assertion implies 'snow is white' is true
     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.
Scheme (T) is not a definition of truth
     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.
3. Truth / F. Semantic Truth / 1. Tarski's Truth / b. Satisfaction and truth
Truth only applies to closed formulas, but we need satisfaction of open formulas to define it
     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.
Tarski uses sentential functions; truly assigning the objects to variables is what satisfies them
     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.
We can define the truth predicate using 'true of' (satisfaction) for variables and some objects
     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
For physicalism, reduce truth to satisfaction, then define satisfaction as physical-plus-logic
     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.
Insight: don't use truth, use a property which can be compositional in complex quantified sentence
     Full Idea: Tarski's great insight is find another property, since open sentences are not truth. It must be had be 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'),
Tarski gave axioms for satisfaction, then derived its explicit definition, which led to defining truth
     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
The best truth definition involves other semantic notions, like satisfaction (relating terms and objects)
     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.
Specify satisfaction for simple sentences, then compounds; true sentences are satisfied by all objects
     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.
3. Truth / F. Semantic Truth / 1. Tarski's Truth / c. Meta-language for truth
We can't use a semantically closed language, or ditch our logic, so a meta-language is needed
     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.
The metalanguage must contain the object language, logic, and defined semantics
     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?
3. Truth / F. Semantic Truth / 2. Semantic Truth
Physicalists should explain reference nonsemantically, rather than getting rid of it
     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
Tarski defined truth for particular languages, but didn't define it across languages
     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.
Tarski didn't capture the notion of an adequate truth definition, as Convention T won't prove non-contradiction
     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?
Tarski's 'truth' is a precise relation between the language and its semantics
     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.
Tarskian truth neglects the atomic sentences
     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'.
Tarski says that his semantic theory of truth is completely neutral about all metaphysics
     Full Idea: Tarski says "we may remain naïve 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
A physicalist account must add primitive reference to Tarski's theory
     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.
Tarski made truth respectable, by proving that it could be defined
     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
Tarski had a theory of truth, and a theory of theories of truth
     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.
If listing equivalences is a reduction of truth, witchcraft is just a list of witch-victim pairs
     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
3. Truth / G. Axiomatic Truth / 1. Axiomatic Truth
Tarski's had the first axiomatic theory of truth that was minimally adequate
     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'.
Tarski defined truth, but an axiomatisation can be extracted from his inductive clauses
     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
Tarski thought axiomatic truth was too contingent, and in danger of inconsistencies
     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).
We need an undefined term 'true' in the meta-language, specified by axioms
     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.
3. Truth / H. Deflationary Truth / 1. Redundant Truth
Truth can't be eliminated from universal claims, or from particular unspecified claims
     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.
3. Truth / H. Deflationary Truth / 2. Deflationary Truth
Semantics is a very modest discipline which solves no real problems
     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.
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
Truth tables give prior conditions for logic, but are outside the system, and not definitions
     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.
5. Theory of Logic / A. Overview of Logic / 3. Value of Logic
Set theory and logic are fairy tales, but still worth studying
     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?
5. Theory of Logic / A. Overview of Logic / 4. Pure Logic
There is no clear boundary between the logical and the non-logical
     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.
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
A language: primitive terms, then definition rules, then sentences, then axioms, and finally inference rules
     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.
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
Split out the logical vocabulary, make an assignment to the rest. It's logical if premises and conclusion match
     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'.
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
Logical consequence is when in any model in which the premises are true, the conclusion is true
     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.
Logical consequence: true premises give true conclusions under all interpretations
     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.
X follows from sentences K iff every model of K also models X
     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'.
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
The truth definition proves semantic contradiction and excluded middle laws (not the logic laws)
     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.
5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic
Identity is invariant under arbitrary permutations, so it seems to be a logical term
     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 '=').
5. Theory of Logic / F. Referring in Logic / 1. Naming / c. Names as referential
A name denotes an object if the object satisfies a particular sentential function
     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)
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
Tarski built a compositional semantics for predicate logic, from dependent satisfactions
     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).
Tarksi invented the first semantics for predicate logic, using this conception of truth
     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
Semantics is the concepts of connections of language to reality, such as denotation, definition and truth
     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.
A language containing its own semantics is inconsistent - but we can use a second language
     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.
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
A sentence is satisfied when we can assert the sentence when the variables are assigned
     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.
Satisfaction is the easiest semantical concept to define, and the others will reduce to it
     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'.
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
The object language/ metalanguage distinction is the basis of model theory
     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
A 'model' is a sequence of objects which satisfies a complete set of sentential functions
     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.
5. Theory of Logic / K. Features of Logics / 2. Consistency
Using the definition of truth, we can prove theories consistent within sound logics
     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.
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox
Tarski avoids the Liar Paradox, because truth cannot be asserted within the object language
     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.
The Liar makes us assert a false sentence, so it must be taken seriously
     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?
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
Tarski improved Hilbert's geometry axioms, and without set-theory
     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
6. Mathematics / C. Sources of Mathematics / 7. Formalism
Tarski's theory of truth shifted the approach away from syntax, to set theory and semantics
     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
8. Modes of Existence / E. Nominalism / 1. Nominalism / a. Nominalism
I am a deeply convinced nominalist
     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.
19. Language / E. Analyticity / 1. Analytic Propositions
Sentences are 'analytical' if every sequence of objects models them
     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.
21. Aesthetics / A. Aesthetic Experience / 3. Taste
Taste is the capacity to judge an object or representation which is thought to be beautiful
     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.