|Satisfaction, logic: a formula is satisfied when their variables are interpreted in a way that the formula as a whole is a true statement. The interpretation is a substitution of the variables of the formula by appropriate constants (e.g. names). When the interpreted formula is true, we call it a model. See also satisfiability, models, model theory.|
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|Satisfaction||K.Glüer Davidson zur Einführung Hamburg 1993 S 24ff
> Recursive Method: but fails with 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. Therefore, Tarski introduces the term "fulfillment":
Definition fulfillment: 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.
Clause 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. - For true statements fulfillment is simple: because whether an ordered sequence of objects satisfies a sentence depends only on the free variables it contains.
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."
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.
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 - for whole sentences, there is also fulfillment: then a sentence is either satisfied by no object or by all - fulfillment: 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 fulfillment 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.
Horwich I 139
Fulfillment/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.
Skirbekk I 146
semantic: refers to statements - fulfillment, designation: refers to objects.
Truth/Tarski: we get the truth definition simply because of the definition of fulfillment: Definition fulfillment/Tarski: fulfillment 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: fulfillment itself must be defined recursively - if we have the fulfillemt, it relates by itself on the statements themselves - a statement is either satisfied by all objects, or by none.
Logic, Semantics, Metamathematics: Papers from 1923-38 Indianapolis 1983
K. Berka/L. Kreiser
Logik Texte Berlin 1983
P. Horwich (Ed.)
Theories of Truth Aldershot 1994
G. Skirbekk (Hg)
Wahrheitstheorien Frankfurt 1977