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In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition of new names can also be ...
An F-proof of a formula A is an F-derivation of A from the empty set of axioms (X=∅). F is called a Frege system if F is sound: every F-provable formula is a tautology. F is implicationally complete: for every formula A and a set of formulas X, if X entails A, then there is an F-derivation of A from X.
The satisfiability problem for monadic second-order logic is undecidable in general because this logic subsumes first-order logic. The monadic second-order theory of the infinite complete binary tree, called S2S, is decidable. [8] As a consequence of this result, the following theories are decidable: The monadic second-order theory of trees.
The theory of uninterpreted functions is also sometimes called the free theory, because it is freely generated, and thus a free object, or the empty theory, being the theory having an empty set of sentences (in analogy to an initial algebra). Theories with a non-empty set of equations are known as equational theories.
In mathematical logic, more specifically in the proof theory of first-order theories, extensions by definitions formalize the introduction of new symbols by means of a definition. For example, it is common in naive set theory to introduce a symbol ∅ {\displaystyle \emptyset } for the set that has no member.
Let the non-empty set S be a counter-example to the axiom of regularity; that is, every element of S has a non-empty intersection with S. We define a binary relation R on S by a R b :⇔ b ∈ S ∩ a {\textstyle aRb:\Leftrightarrow b\in S\cap a} , which is entire by assumption.
[1] [3] In the fifth chapter, the same concepts are used to give a non-standard semantics to type theory. After these chapters on other types of logic, the final two chapters introduce many-sorted logic, prove its soundness, completeness, and compactness, and describe how to translate the other forms of logic into it. [3]
Recently, conservative extensions have been used for defining a notion of module for ontologies [citation needed]: if an ontology is formalized as a logical theory, a subtheory is a module if the whole ontology is a conservative extension of the subtheory. An extension which is not conservative may be called a proper extension.