Glüer II 24f
Recursive method/recursion: fails with quantifiers.
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Recursive Method, >
Quantifiers.
E.g.
"No tree is large and small"
cannot be analyzed as two complete elementary propositions.
Most complex sentences formed with variables, connectives, predicates, must be interpreted as links of open sentences.
But open sentences have no truth value.
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Open sentence, >
Truth value.
Therefore, Tarski introduces the term "satisfaction":
Def satisfaction: relation between (ordered) sequences of objects and open sentences.
Here works the recursive method: for elementary sentences it is defined which objects 2 they satisfy, and there are rules specified for all compositions of open sentences by which can be determined which objects they satisfy.
Statements are determined as a special case of open sentences. Either they do not contain free variables, or they have been closed by means of quantifiers.
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Free variables, >
Open sentence.
For true statements satisfaction is simple: because whether an ordered sequence of objects satisfies a sentence depends only on the free variables it contains.
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Sequences/Tarski.
E.g. "The moon is round" contains no free variables. Thus, the nature of the objects of the respective sequence is irrelevant and it can be determined by definition, whether such a proposition is true when it is satisfied by all the consequences - or by none.
It is slightly more complicated for quantified statements: E.g. "All stars are around" or "There is at least one star, which is round." Here, too, the fulfillment is defined such that either all sequences satisfy a sentence, or none.
So it is clear that it would be absurd to associate truth of closed sentences with fulfillment by any sequence of objects. A sentence like "All stars are round" is true if there are certain objects that satisfy "X is round": all stars. Tarski: a statement is true if it is satisfied by all objects, otherwise false."
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Berka I 399
Part definition/satisfy/Tarski. E.g. Johann and Peter satisfy the propositional function "X and Y are brothers" if they are brothers.
(1)
1. A.Tarski, „Grundlegung der wissenschaftlichen Semantik“, in: Actes du Congrès International de Philosophie Scientifique, Paris 1935, Bd. III, ASI 390, Paris 1936, pp. 1-8
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Horwich I 119
Fulfillment/Tarski: here we replace the free variables of propositional functions by the names of objects and see if we can get true statements - but that does not work when we use fulfillment to define truth.
Solution: recursive procedure: rules for the conditions under which objects satisfy a composite propositional function.
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Propositional functions.
For whole sentences, there is also fulfillment: then a sentence is either satisfied by no object or by all.
Satisfaction: has as a relation always one more spot - E.g. "is greater than": is a function between a relation and pairs of objects.
Therefore, there are many fulfillment terms.
Solution: "infinite sequence". Then satisfaction is a binary relation between functions and sequences of objects.
The reason for this indirect truth definition is that compound sentences are composed of several propositional functions, not always of complete sentences. So there is no recursive definition.
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Recursive definition, >
Recursion.
Horwich I 139
Satisfaction/antinomy/Tarski: for the fulfillment, we can also construct an antinomy: E.g. the statements function X does not satisfy X. - Now we look at the question of whether this term, which is surely a propositional function satisfied itself or not.
(2)
2. A. Tarski, The semantic Conceptions of Truth, Philosophy and Phenomenological Research 4, pp. 341-75
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Skirbekk I 146
Semantics: refers to statements,
Satisfaction, designation: refers to objects.
Skirbekk I 156
Truth/Tarski: we get the truth definition simply because of the definition of satisfaction:
Def satisfaction/Tarski: satisfaction is a relationship between any object and propositional functions.
An object satisfies a function when the function becomes a true statement, when the free variables replace with the name of object.
Snow satisfies the propositional function "x is white".
Vs: that is circular, because "true" occurs in the definition of fulfillment.
Solution: satisfaction itself must be defined recursively. If we have the satisfaction, it relates by itself on the statements themselves. A statement is either satisfied by all objects, or by none.
(3)
3. A.Tarski, „Die semantische Konzeption der Wahrheit und die Grundlagen der Semantik“ (1944) in. G: Skirbekk (Hg.) Wahrheitstheorien, Frankfurt 1996
