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3. Truth / F. Semantic Truth / 1. Tarski's Truth / b. Satisfaction and truth

['satisfaction' as a means of defining truth]

18 ideas
An argument 'satisfies' a function φx if φa is true [Russell]
     Full Idea: We say that an argument a 'satisfies' a function φx if φa is true.
     From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XV)
     A reaction: We end up with Tarski defining truth in terms of satisfaction, so we shouldn't get too excited about what he achieved (any more than he got excited).
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.
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.
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.
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.
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
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.
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'),
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
Satisfaction is a sort of reference, so maybe we can define truth in terms of reference? [Davidson]
     Full Idea: That the truth of sentences is defined by appeal to the semantic properties of words suggests that, if we could give an account of the semantic properties of words (essentially, of reference or satisfaction), we would understand the concept of truth.
     From: Donald Davidson (Truth and Predication [2005], 2)
     A reaction: If you thought that words were prior to sentences, this might be the route to go. Davidson gives priority to sentences, and so prefers to work from the other end, which treats truth as primitive, and then defines reference and meaning.
We can explain truth in terms of satisfaction - but also explain satisfaction in terms of truth [Davidson]
     Full Idea: Truth is easily defined in terms of satisfaction (as Tarski showed), but, alternatively, satisfaction can be taken to be whatever relation yields a correct account of truth.
     From: Donald Davidson (Truth and Predication [2005], 2)
     A reaction: Davidson is assessing which is the prior 'primitive' concept, and he votes for truth. A perennial problem in philosophy, and very hard to find reasons for a preference. The axiomatic approach grows from taking truth as primitive. Axioms for satisfaction?
Axioms spell out sentence satisfaction. With no free variables, all sequences satisfy the truths [Davidson]
     Full Idea: Axioms specify how each unstructured predicate is satisfied by a particular sequence. Then recursive axioms characterise complex sentences built from simpler ones. Closed sentences have no free variables, so true sentences are satisfied by all sequences.
     From: Donald Davidson (Truth and Predication [2005], 7)
     A reaction: I take 'all sequences' to mean all combinations of objects in the domain. Thus nothing in domain contradicts the satisfied sentences. Hence Tarski's truth is said to be 'true in a model', where the whole system vouches for the sentence.
Tarski just reduced truth to some other undefined semantic notions [Field,H]
     Full Idea: It is normally claimed that Tarski defined truth using no undefined semantic terms, but I argue that he reduced the notion of truth to certain other semantic notions, but did not in any way explicate these other notions.
     From: Hartry Field (Tarski's Theory of Truth [1972], §0)
A 'sequence' of objects is an order set of them [Kirkham]
     Full Idea: A 'sequence' of objects is like a set of objects, except that, unlike a set, the order of the objects is important when dealing with sequences. ...An infinite sequence satisfies 'x2 is purple' if and only if the second member of the sequence is purple.
     From: Richard L. Kirkham (Theories of Truth: a Critical Introduction [1992], 5.4)
     A reaction: This explains why Tarski needed set theory in his metalanguage.
If one sequence satisfies a sentence, they all do [Kirkham]
     Full Idea: If one sequence satisfies a sentence, they all do. ...Thus it matters not whether we define truth as satisfaction by some sequence or as satisfaction by all sequences.
     From: Richard L. Kirkham (Theories of Truth: a Critical Introduction [1992], 5.4)
     A reaction: So if the striker scores a goal, the team has scored a goal.
Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do [Hart,WD]
     Full Idea: We explain truth for sentences in terms of satisfaction of formulae. The crux here is that for a sentence, either all sequences satisfy it or none do (with no middle ground). For formulae, some sequences may satisfy it and others not.
     From: William D. Hart (The Evolution of Logic [2010], 4)
     A reaction: This is the hardest part of Tarski's theory of truth to grasp.
Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro]
     Full Idea: In a sense, satisfaction is the notion of 'truth in a model', and (as Hodes 1984 elegantly puts it) 'truth in a model' is a model of 'truth'.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1)
     A reaction: So we can say that Tarski doesn't offer a definition of truth itself, but replaces it with a 'model' of truth.
If a language cannot name all objects, then satisfaction must be used, instead of unary truth [Halbach/Leigh]
     Full Idea: If axioms are formulated for a language (such as set theory) that lacks names for all objects, then they require the use of a satisfaction relation rather than a unary truth predicate.
     From: Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 3.3)
     A reaction: I take it this is an important idea for understanding why Tarski developed his account of truth based on satisfaction.