Deductive system

# Deductive system

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Encyclopedia
A deductive system consists of the axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

s (or axiom schema
Axiom schema
In mathematical logic, an axiom schema generalizes the notion of axiom.An axiom schema is a formula in the language of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which...

ta) and rules of inference that can be used to derive
Formal proof
A formal proof or derivation is a finite sequence of sentences each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system...

the theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

s of the system.

Such a deductive system is intended to preserve deductive
Deductive reasoning
Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...

qualities in the formulas that are expressed in the system. Usually the quality we are concerned with is truth
Truth
Truth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...

as opposed to falsehood. However, other modalities
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

, such as justification
Theory of justification
Theory of justification is a part of epistemology that attempts to understand the justification of propositions and beliefs. Epistemologists are concerned with various epistemic features of belief, which include the ideas of justification, warrant, rationality, and probability...

or belief
Belief
Belief is the psychological state in which an individual holds a proposition or premise to be true.-Belief, knowledge and epistemology:The terms belief and knowledge are used differently in philosophy....

In order to sustain its deductive integrity, a deductive apparatus must be definable without reference to any intended interpretation
Intended interpretation
One who constructs a syntactical system usually has in mind from the outset some interpretation of this system. While this intended interpretation can have no explicit indication in the syntactical rules - since these rules must be strictly formal - the author's intention respecting...

of the language. The aim is to ensure that each line of a derivation
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

is merely a syntactic consequence of the lines that precede it. There should be no element of any interpretation
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...

of the language that gets involved with the deductive nature of the system.

• Formal grammar
Formal grammar
A formal grammar is a set of formation rules for strings in a formal language. The rules describe how to form strings from the language's alphabet that are valid according to the language's syntax...

• Natural deduction
Natural deduction
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning...

• Axiomatic system
Axiomatic system
In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...

• Proof calculus
Proof calculus
In mathematical logic, a proof calculus corresponds to a family of formal systems that use a common style of formal inference for its inference rules...