Logic Texts on Compactness - Dictionary of Arguments
Read III 59
Compactness: the classic logical conclusion is compact. To understand this, we must acknowledge that the set of premises can be infinite. Classically, every logical truth (of which there are infinite numbers) is a conclusion from any statement. This can be multiplied, by double negation, the conjunction of itself with its double negation, and so on.
The classical compactness does not mean that a conclusion cannot have an infinite number of premises, it can. But classically it is valid exactly when the conclusion follows from a finite subset of the premises.
Compactness limits the expressiveness of a logic.
Proof: is performed purely syntactically. In itself, the proof has no meaning. Its correctness is defined based on its form and structure.
The counterpart of proof is completeness: there should be a derivation. >Incompleteness/logic texts.
The Omega rule (>Incompleteness/logic texts) is not accepted as a rule of orthodox, classical proof theory. How can I do this? According to classical representation, a rule is valid if the premises are true and the conclusion is false by no interpretation over any range of definition. How can the premises A(0),A(1) etc. was, but be false for each n,A(n)?
The explanation lies in the limitation of the expressiveness.
In non-compact logic, there may be a categorical set of formulas for arithmetic, but the proof methods require compactness.
Difference compact/non compact: classical logic is a 1st order logic. A categorical set of axioms for arithmetic must be a second order logic. ((s) quantifiers also for properties).
For example, Napoleon had all the properties of a great general: "for every quality f, if for every person x, if x was a great general, then x had f, then Napoleon had f".
In reality it is a little more subtle. For syntactically one cannot distinguish whether a formula is like the 1st or 2nd level above. >2nd order logic._____________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. Translations: Dictionary of Arguments The note [Author1]Vs[Author2] or [Author]Vs[term] is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
Me I Albert Menne Folgerichtig Denken Darmstadt 1988
HH II Hoyningen-Huene Formale Logik, Stuttgart 1998
Re III Stephen Read Philosophie der Logik Hamburg 1997
Sal IV Wesley C. Salmon Logic, Englewood Cliffs, New Jersey 1973 - German: Logik Stuttgart 1983
Sai V R.M.Sainsbury Paradoxes, Cambridge/New York/Melbourne 1995 - German: Paradoxien Stuttgart 2001
Thinking About Logic: An Introduction to the Philosophy of Logic. 1995 Oxford University Press
Philosophie der Logik Hamburg 1997