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 / 4. Metaphysics beyond Science
Some say metaphysics is a highly generalised empirical study of objects
1. Philosophy / F. Analytic Philosophy / 1. Analysis
Disputes that fail to use precise scientific terminology are all meaningless
2. Reason / D. Definition / 1. Definitions
For a definition we need the words or concepts used, the rules, and the structure of the language
3. Truth / A. Truth Problems / 2. Defining Truth
In everyday language, truth seems indefinable, inconsistent, and illogical
Tarski proved that truth cannot be defined from within a given theory
'True sentence' has no use consistent with logic and ordinary language, so definition seems hopeless
A definition of truth should be materially adequate and formally correct
Definitions of truth should not introduce a new version of the concept, but capture the old one
A rigorous definition of truth is only possible in an exactly specified language
We may eventually need to split the word 'true' into several less ambiguous terms
3. Truth / F. Semantic Truth / 1. Tarski's Truth / a. Tarski's truth definition
Tarski did not just aim at a definition; he also offered an adequacy criterion for any truth definition
Tarski gave up on the essence of truth, and asked how truth is used, or how it functions
'"It is snowing" is true if and only if it is snowing' is a partial definition of the concept of truth
Scheme (T) is not a definition of truth
It is convenient to attach 'true' to sentences, and hence the language must be specified
In the classical concept of truth, 'snow is white' is true if snow is white
Each interpreted T-sentence is a partial definition of truth; the whole definition is their conjunction
Use 'true' so that all T-sentences can be asserted, and the definition will then be 'adequate'
We don't give conditions for asserting 'snow is white'; just that assertion implies 'snow is white' is true
3. Truth / F. Semantic Truth / 1. Tarski's Truth / b. Satisfaction and truth
Tarski uses sentential functions; truly assigning the objects to variables is what satisfies them
We can define the truth predicate using 'true of' (satisfaction) for variables and some objects
The best truth definition involves other semantic notions, like satisfaction (relating terms and objects)
Specify satisfaction for simple sentences, then compounds; true sentences are satisfied by all objects
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
The metalanguage must contain the object language, logic, and defined semantics
3. Truth / F. Semantic Truth / 2. Semantic Truth
Physicalists should explain reference nonsemantically, rather than getting rid of it
Tarski's 'truth' is a precise relation between the language and its semantics
Tarskian truth neglects the atomic sentences
A physicalist account must add primitive reference to Tarski's theory
If listing equivalences is a reduction of truth, witchcraft is just a list of witch-victim pairs
Tarski made truth respectable, by proving that it could be defined
Tarski didn't capture the notion of an adequate truth definition, as Convention T won't prove non-contradiction
Tarski had a theory of truth, and a theory of theories of truth
3. Truth / G. Axiomatic Truth / 1. Axiomatic Truth
Tarski defined truth, but an axiomatisation can be extracted from his inductive clauses
Tarski's had the first axiomatic theory of truth that was minimally adequate
Tarski thought axiomatic truth was too contingent, and in danger of inconsistencies
We need an undefined term 'true' in the meta-language, specified by axioms
3. Truth / H. Deflationary Truth / 1. Redundant Truth
Truth can't be eliminated from universal claims, or from particular unspecified claims
3. Truth / H. Deflationary Truth / 2. Deflationary Truth
Semantics is a very modest discipline which solves no real problems
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
5. Theory of Logic / A. Overview of Logic / 3. Value of Logic
Set theory and logic are fairy tales, but still worth studying
5. Theory of Logic / A. Overview of Logic / 4. Pure Logic
There is no clear boundary between the logical and the non-logical
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
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
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
Logical consequence: true premises give true conclusions under all interpretations
X follows from sentences K iff every model of K also models X
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)
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
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
Tarksi invented the first semantics for predicate logic, using this conception of truth
Semantics is the concepts of connections of language to reality, such as denotation, definition and truth
A language containing its own semantics is inconsistent - but we can use a second language
Tarski built a compositional semantics for predicate logic, from dependent satisfactions
5. Theory of Logic / I. Semantics of Logic / 5. Satisfaction
A sentence is satisfied when we can assert the sentence when the variables are assigned
Satisfaction is the easiest semantical concept to define, and the others will reduce to it
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
A 'model' is a sequence of objects which satisfies a complete set of sentential functions
5. Theory of Logic / K. Features of Logics / 2. Consistency
Using the definition of truth, we can prove theories consistent within sound logics
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox
The Liar makes us assert a false sentence, so it must be taken seriously
6. Mathematics / B. Foundations for Mathematics / 2. Axioms for Geometry
Tarski improved Hilbert's geometry axioms, and without set-theory
8. Modes of Existence / E. Nominalism / 1. Nominalism / a. Nominalism
I am a deeply convinced nominalist
19. Language / F. Analytic/Synthetic / 2. Analytic Propositions
Sentences are 'analytical' if every sequence of objects models them