The term formalism is sometimes used as a rough synonym for formal system, but is also used to refer to a given style of notation, for example, Paul Dirac's bra–ket notation.
Background
Each formal system uses primitive symbols (which collectively form an alphabet) to finitely construct a formal language from a set of axioms through inferential rules of formation.
The system thus consists of valid formulas built up through
finite combinations of the primitive symbols—combinations that are
formed from the axioms in accordance with the stated rules.
More formally, this can be expressed as the following:
- A finite set of symbols, known as the alphabet, which concatenate formulas, so that a formula is just a finite string of symbols taken from the alphabet.
- A grammar consisting of rules to form formulas from simpler formulas. A formula is said to be well-formed if it can be formed using the rules of the formal grammar. It is often required that there be a decision procedure for deciding whether a formula is well-formed.
- A set of axioms, or axiom schemata, consisting of well-formed formulas.
- A set of inference rules. A well-formed formula that can be inferred from the axioms is known as a theorem of the formal system.
Recursive
A formal system is said to be recursive (i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules are decidable sets or semidecidable sets, respectively.
Inference and entailment
The entailment
of the system by its logical foundation is what distinguishes a formal
system from others which may have some basis in an abstract model. Often
the formal system will be the basis for or even identified with a
larger theory or field (e.g. Euclidean geometry) consistent with the usage in modern mathematics such as model theory.
Formal language
A formal language is a language that is defined by a formal system. Like languages in linguistics, formal languages generally have two aspects:
- the syntax of a language is what the language looks like (more formally: the set of possible expressions that are valid utterances in the language) studied in formal language theory
- the semantics of a language are what the utterances of the language mean (which is formalized in various ways, depending on the type of language in question)
In computer science and linguistics a formal grammar is a precise description of a formal language: a set of strings. The two main categories of formal grammar are that of generative grammars, which are sets of rules for how strings in a language can be generated, and that of analytic grammars (or reductive grammar,)
which are sets of rules for how a string can be analyzed to determine
whether it is a member of the language. In short, an analytic grammar
describes how to recognize when strings are members in the set, whereas a generative grammar describes how to write only those strings in the set.
Logical system
A logical system or, for short, a logic, is a formal system together with its semantics. According to model-theoretic interpretation,
the semantics of a logical system describe whether a well-formed
formula is satisfied by a given structure. A structure that satisfies
all the axioms of the formal system is known as a model of the logical
system. A logical system is sound
if each well-formed formula that can be inferred from the axioms is
satisfied by every model of the logical system. Conversely, a logic
system is complete if each well-formed formula that is satisfied by every model of the logical system can be inferred from the axioms.
Deductive inference
A deductive system, also called a deductive apparatus, consists of the axioms (or axiom schemata) and rules of inference that can be used to derive theorems of the system.
Such deductive systems preserve deductive qualities in the formulas that are expressed in the system. Usually the quality we are concerned with is truth as opposed to falsehood. However, other modalities, such as justification or belief may be preserved instead.
In order to sustain its deductive integrity, a deductive apparatus must be definable without reference to any intended interpretation of the language. The aim is to ensure that each line of a derivation is merely a syntactic consequence of the lines that precede it. There should be no element of any interpretation of the language that gets involved with the deductive nature of the system.
History
Early logic systems includes Indian logic of Pāṇini, syllogistic logic of Aristotle, propositional logic of Stoicism, and Chinese logic of Gongsun Long (c. 325–250 BCE) . In recent times, contributors include George Boole, Augustus De Morgan, and Gottlob Frege. Mathematical logic was developed in 19th century Europe.
Formalism
Hilbert's program
David Hilbert instigated a formalist movement that was eventually tempered by Gödel's incompleteness theorems.
QED manifesto
The QED manifesto represented a subsequent unsuccessful effort at formalization of known mathematics.
Examples
Examples of formal systems include:
Variants
The following systems are variations of formal systems.
Proof system
Formal proofs are sequences of well-formed formulas (or wff for short). For a wff to qualify as part of a proof, it might either be an axiom
or be the product of applying an inference rule on previous wffs in the
proof sequence. The last wff in the sequence is recognized as a theorem.
The point of view that generating formal proofs is all there is to mathematics is often called formalism. David Hilbert founded metamathematics as a discipline for discussing formal systems. Any language that one uses to talk about a formal system is called a metalanguage.
The metalanguage may be a natural language, or it may be partially
formalized itself, but it is generally less completely formalized than
the formal language component of the formal system under examination,
which is then called the object language, that is, the object of the discussion in question.
Once a formal system is given, one can define the set of theorems
which can be proved inside the formal system. This set consists of all
wffs for which there is a proof. Thus all axioms are considered
theorems. Unlike the grammar for wffs, there is no guarantee that there
will be a decision procedure for deciding whether a given wff is a theorem or not. The notion of theorem just defined should not be confused with theorems about the formal system, which, in order to avoid confusion, are usually called metatheorems.
