|General validity: within a calculus a formula that is satisfied by any interpretation (variable assignment with expressions for objects) is valid. See also satisfaction, satisfiability, interpretation._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments.|
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Validity/Quine: even validity and extension of predicates can be eliminated in favor of truth value tables - validity in the quantifier theory can be eliminated by proof theory.
Validity/Quine: sentences that are valid for a universe, are also valid for a small universe - except for an empty universe. - Therefore, laws for large universes also should consider possible smaller universes. - Test, whether theorems are also valid for empty universes: put all universal quantifiers as true and all existential quantifiers as false.
Validity/valid/Quine: There are two definitions of validity,
a) (so far) as a property of schemes that refer to insertion.
b) uses the set theory: therefore two auxiliary terms:
1. Auxiliary term "set-theoretic analogue": a logical scheme, open sentence of set theory: instead of predications "Fx", "Fy", "Gx" etc., so we write
"X ε a" y ε α "x ε β" etc. the values of the variable "α", "β" etc. are amounts.
Two-digit predicate letters. For "Hxy" we use ordered pairs
Existential quantification: E.g. (Ex)(Fx.Gx): Set-theoretic analogue: the open sentence "Ex(x ε α. x ε β)".
N.B.: This sentence talks about quantities and allows quantification about them. E.g. "(α)".
Scheme letters "F" etc. on the other hand, only predicates represent and are not variables that take values.
Set-theoretic analogue/s.a.: while the scheme is only the logical form of sentences, the set-theoretic analogue is actually a sentence of this form.
2. Auxiliary term for the new definition of validity: Model._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
Wort und Gegenstand Stuttgart 1980
Theorien und Dinge Frankfurt 1985
Grundzüge der Logik Frankfurt 1978
Mengenlehre und ihre Logik Wiesbaden 1967
Die Wurzeln der Referenz Frankfurt 1989
Unterwegs zur Wahrheit Paderborn 1995
From a logical point of view Cambridge, Mass. 1953
Bezeichnung und Referenz
Zur Philosophie der idealen Sprache, J. Sinnreich (Hg), München 1982
Philosophie der Logik Bamberg 2005
Ontologische Relativität Frankfurt 2003