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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.
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 ...
The extension of a predicate – a truth-valued function – is the set of tuples of values that, used as arguments, satisfy the predicate. Such a set of tuples is a relation . Examples
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.
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.
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.
The skip-logic consists of a -input AND-gate and one multiplexer. T S K = T A N D ( m ) + T M U X {\displaystyle T_{SK}=T_{AND}(m)+T_{MUX}} As the propagate signals are computed in parallel and are early available, the critical path for the skip logic in a carry-skip adder consists only of the delay imposed by the multiplexer (conditional skip).
In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that – for some coherent meaning of 'logic' – mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. [1]