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Formal systems of logic are systematizations of logical truths based on certain principles called axioms. [5] As for formal logic, a central question in the philosophy of logic is what makes a formal system into a system of logic rather than a collection of mere marks together with rules for how they are to be manipulated. [4]
These two definitions of formal logic are not identical, but they are closely related. For example, if the inference from p to q is deductively valid then the claim "if p then q" is a logical truth. [16] Formal logic needs to translate natural language arguments into a formal language, like first-order logic, to assess whether they are valid.
Rudolph Carnap defined the meaning of the adjective formal in 1934 as follows: "A theory, a rule, a definition, or the like is to be called formal when no reference is made in it either to the meaning of the symbols (for example, the words) or to the sense of the expressions (e.g. the sentences), but simply and solely to the kinds and order of the symbols from which the expressions are ...
Formal languages, mathematics, formal logic, programming languages (in principle, they must have zero internal vagueness of interpretation of all language constructs, i.e. they have exact interpretation) can model external vagueness by tools of vagueness and uncertainty representation: fuzzy sets and fuzzy logic, or by stochastic quantities and ...
The definition must not be obscure. The purpose of a definition is to explain the meaning of a term which may be obscure or difficult, by the use of terms that are commonly understood and whose meaning is clear. The violation of this rule is known by the Latin term obscurum per obscurius. However, sometimes scientific and philosophical terms ...
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first understood from context, after which an element ϕ ∈ T {\displaystyle \phi \in T} of a deductively closed theory T {\displaystyle T} is then called a theorem of the theory.
Argument terminology used in logic. In logic, an argument is a set of related statements expressing the premises (which may consists of non-empirical evidence, empirical evidence or may contain some axiomatic truths) and a necessary conclusion based on the relationship of the premises.
This definition is given together with a formal definition of programs (and models of computation), allowing to formally define the notion of implementation, that is when a program implements an algorithm. The notion of algorithm thus obtained avoids some known issues, and is understood as a specification of some kind.