![Example formalization](https://upload.wikimedia.org/wikipedia/commons/7/71/Example_formalization.svg)
Logic translation is the process of representing a text in the
formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
The alphabet of a formal language consists of symb ...
of a
logical system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A form ...
. If the original text is formulated in
ordinary language
Ordinary language philosophy (OLP) is a philosophical methodology that sees traditional philosophical problems as rooted in misunderstandings philosophers develop by distorting or forgetting how words are ordinarily used to convey meaning in ...
then the term "natural language formalization" is often used. An example is the translation of the English sentence "some men are bald" into
first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
as
. In this regard, the purpose is to reveal the
logical structure of
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
s. This makes it possible to use the precise rules of formal logic to assess whether these arguments are correct. It can also guide reasoning by arriving at new conclusions.
Many of the difficulties associated with the process are caused by
vague
In linguistics and philosophy, a vague predicate is one which gives rise to borderline cases. For example, the English adjective "tall" is vague since it is not clearly true or false for someone of middling height. By contrast, the word "prime" is ...
or
ambiguous
Ambiguity is the type of meaning (linguistics), meaning in which a phrase, statement or resolution is not explicitly defined, making several interpretations wikt:plausible#Adjective, plausible. A common aspect of ambiguity is uncertainty. It ...
expressions in natural language. For example, the English word "is" can mean that something
exists, that it is
identical
Two things are identical if they are the same, see Identity (philosophy).
Identical may also refer to:
* ''Identical'' (Hopkins novel), a 2008 young adult novel by Ellen Hopkins
* ''Identical'' (Turow novel), a 2013 legal drama novel by Scott T ...
to something else, or that it has a certain
property
Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, r ...
. This contrasts with the precise nature of formal logic, which avoids such ambiguities. Natural language formalization is relevant to various fields in the
sciences
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence for ...
and
humanities
Humanities are academic disciplines that study aspects of human society and culture. In the Renaissance, the term contrasted with divinity and referred to what is now called classics, the main area of secular study in universities at the t ...
. Some theorists claim that it plays a key role for
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
in general since it is needed to establish a link between many forms of reasoning and abstract logical systems. The use of
informal logic
Informal logic encompasses the principles of logic and logical thought outside of a formal setting (characterized by the usage of particular statements). However, the precise definition of "informal logic" is a matter of some dispute. Ralph H. ...
is an alternative to formalization since it analyzes the cogency of ordinary language arguments in their original form. Natural language formalization is distinguished from logic translations that convert formulas from one logical system into another. An example is the translation of the
modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
formula
into first-order logic as
. This form of logic translation is specifically relevant for
logic programming
Logic programming is a programming paradigm which is largely based on formal logic. Any program written in a logic programming language is a set of sentences in logical form, expressing facts and rules about some problem domain. Major logic prog ...
and
metalogic
Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.Harry GenslerIntroduction to Logic Routledge, ...
.
Criteria of adequacy specify whether a logic translation is accurate and how to distinguish good from bad translations. An often-cited criterion states that translations should preserve the
inferential relations between sentences. This implies that if an argument is
valid in the original text then the translated argument should also be valid. Other suggested criteria are that the original sentence and the translation have the same
truth condition
In semantics and pragmatics, a truth condition is the condition under which a sentence is true. For example, "It is snowing in Nebraska" is true precisely when it is snowing in Nebraska. Truth conditions of a sentence do not necessarily reflect cu ...
s, that a translation does not include additional or unnecessary symbols, and that its grammatical structure is similar to the original sentence. Various procedures for translating texts have been suggested. Preparatory steps include understanding the meaning of the original text and
paraphrasing
A paraphrase () is a restatement of the meaning of a text or passage using other words. The term itself is derived via Latin ', . The act of paraphrasing is also called ''paraphrasis''.
History
Although paraphrases likely abounded in oral tra ...
it to remove ambiguities and make its logical structure more explicit. As an intermediary step, a translation may happen into a hybrid language that implements a logical formalism while retaining the vocabulary of the original expressions. In the last step, this vocabulary is replaced by logical symbols. Logic translations are often criticized on the grounds that they are unable to accurately represent all the aspects and nuances of the original text.
Definition
A logic translation is a
translation
Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
of a text into a
logical system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A form ...
. For example, translating the sentence "all skyscrapers are tall" as
is a logic translation that expresses an English language sentence in the logical system known as
first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
. The aim of logic translations is usually to make the
logical structure of
natural language
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
s explicit. This way, the rules of
formal logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
can be used to assess whether the arguments are
valid.
Understood in a wide sense, a translation is a process that associates expressions belonging to a
source language with expressions belonging to a target language. For example, in a sentence-by-sentence translation of an English text into French, English sentences are associated with their French counterparts. The hallmark of logic translations is that the target language belongs to a logical system.
However, logic translations differ from regular translations in that they are mainly concerned with expressing the logical structure of the original text and less with its concrete content. Regular translations, on the other hand, take various additional factors into account pertaining to the content, meaning, and style of the original expression. For this reason, some theorists have argued that, strictly speaking, it is not a form of translation. They tend to use other terms, such as "formalization", "symbolization", and "explication". However, this opinion is not shared by all logicians and some argue that successful logic translations preserve all the original meaning while making the logical structure explicit.
Discussions on logic translations usually focus on the problem of expressing the logical structure of ordinary language sentences in a formal logical system. However, the term also covers cases where the translation happens from one logical system into another.
Basic concepts
![Modus ponendo ponens](https://upload.wikimedia.org/wikipedia/commons/d/df/Modus_ponendo_ponens.png)
Various basic concepts are employed in the study and analysis of logic translations.
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
as a discipline is the study of correct reasoning. Reasoning happens in the form of
inference
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word '' infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in ...
s or arguments. An argument is a set of
premise
A premise or premiss is a true or false statement that helps form the body of an argument, which logically leads to a true or false conclusion. A premise makes a declarative statement about its subject matter which enables a reader to either agre ...
s together with a conclusion. An argument is
deductively valid if it is impossible for its conclusion to be false if all its premises are true. Valid arguments follow a
rule of inference
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of in ...
, which prescribes how the premises and the conclusion have to be structured. A prominent rule of inference is
modus ponens
In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...
. It states that arguments of the form "(1) ''p''; (2) if ''p'' then ''q''; (3) therefore ''q''" are valid.
An example of an argument following modus ponens is: "(1) today is Sunday; (2) if today is Sunday then I don't have to go to work today; (3) therefore I don't have to go to work today".
A logical system is a theoretical framework for assessing which arguments are valid. Logical systems differ from each other concerning the formal language and the rules of inference they use. In this regard, they are designed to capture different inferential relations between
proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
s. For example,
propositional logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
only focuses on inferences based on
logical connective
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary co ...
s, like "and" or "if...then". First-order logic, on the other hand, also includes additional inferential patterns associated with expressions like "every" or "some". Extended logics cover further inferences, for example, in relation to what is possible and necessary or regarding temporal relations.
This means that logical systems usually do not capture all inferential patterns. This is relevant for logic translation since they may miss patterns for which they were not intended. For example, propositional logic can be used to show that the following ordinary language argument is correct: "(1) John is not a pilot; (2) John is a pilot or Bill is a poet; (3) therefore Bill is a poet". However, it fails to show that the argument "(1) John is a pilot; (2) therefore John can aviate" is correct since it is unable to capture the inferential relation between the terms "Pilot" and "can aviate". If a logical system is applied to cases beyond its limited scope, it is unable to assess the validity of natural language arguments. The advantage of this limitation is that the
vagueness
In linguistics and philosophy, a vague predicate is one which gives rise to borderline cases. For example, the English adjective "tall" is vague since it is not clearly true or false for someone of middling height. By contrast, the word "prime" is ...
and
ambiguity
Ambiguity is the type of meaning in which a phrase, statement or resolution is not explicitly defined, making several interpretations plausible. A common aspect of ambiguity is uncertainty. It is thus an attribute of any idea or statement ...
of natural language arguments are avoided by making some of the inferential patterns very clear.
Formal logical systems use precise
formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
The alphabet of a formal language consists of symb ...
s to express their formulas and inferences. In the case of propositional logic, letters like
and
are used to represent simple propositions. They can be combined into more complex propositions using propositional connectives like
to express that both propositions are true and
to express that at least one of the propositions is true. So if
stands for "Adam is athletic" and
stands for "Barbara is athletic", then the formula
represents the claim that "Adam is athletic, and also Barbara is athletic". First-order logic also includes propositional connectives but introduces additional symbols. Uppercase letters are used for predicates and lowercase letters stand for individuals. For example, if
stands for the predicate "is angry" and
represents the individual Elsa, then the formula
expresses the proposition "Elsa is angry". Another innovation of first-order logic is the use of quantifiers like
and
to represent the meanings of terms like "some" and "all".
Types
Logic translations can be classified based on the source language of the original text. For many logic translations, the original text belongs to a natural language, like English or French. In this case, the term "natural language formalization" is often used. For example, the sentence "Dana is a logician and Dana is a nice person" can be formalized into propositional logic using the
logical formula
In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be ...
. A further type of logic translation happens between two logical systems. This means that the source text is composed of logical formulas belonging to one logical system and the goal is to associate them with logical formulas belonging to another logical system. For example, the formula
in
modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
can be translated into first-order logic using the formula
.
Natural language formalization
![Natural language formalization](https://upload.wikimedia.org/wikipedia/commons/8/8e/Natural_language_formalization.svg)
Natural language formalization starts with a sentence in natural language and translates it into a logical formula.
Its goal is to make the logical structure of natural language sentences and arguments explicit. It is mainly concerned with their logical form while their specific content is usually ignored. Logical analysis is a closely related term that refers to the process of uncovering the logical form or structure of a sentence. Natural language formalization makes it possible to use formal logic to analyze and evaluate natural language arguments. This is especially relevant for complex arguments, which are often difficult to evaluate without formal tools. Logic translation can also be used to look for new arguments and thereby guide the reasoning process. Closely connected to formalization is the reverse process of translating logical formulas back into natural language sentences. This is sometimes referred to as "verbalization". However, this issue is not as controversial and discussions concerning the relation between natural language and logic usually focus on the problem of formalization.
The success of applications of formal logic to natural language requires that the translation is correct. A formalization is correct if its explicit logical features fit the implicit logical features of the original sentence. However, the logical form of ordinary language sentences is often not obvious and there are many differences between natural languages and the formal languages used by logicians. This poses various difficulties for formalization. For example, ordinary expressions frequently include vague and ambiguous expressions. For this reason, the validity of an argument often depends not just on the expressions themselves but also on how they are interpreted. For example, the sentence "donkeys have ears" could mean that ''all donkeys (without exception) have ears'' or that ''donkeys typically have ears'', which does not exclude the existence of some donkeys without ears. This difference matters for whether a
universal quantifier
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other w ...
can be used to translate the sentence. Such ambiguities are not found in the precise formulations of artificial logical languages and have to be solved before translation is possible.
The problem of natural language formalization has various implications for the
sciences
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence for ...
and
humanities
Humanities are academic disciplines that study aspects of human society and culture. In the Renaissance, the term contrasted with divinity and referred to what is now called classics, the main area of secular study in universities at the t ...
, especially for the fields of
linguistics
Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Linguis ...
,
cognitive science, and
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
. In the field of
formal linguistics Formal linguistics is the branch of linguistics which uses applied mathematics, applied mathematical methods for the analysis of natural languages. Such methods include formal languages, formal grammars and first-order logical expressions. Formal li ...
, for example,
Richard Montague
Richard Merritt Montague (September 20, 1930 – March 7, 1971) was an American mathematician and philosopher who made contributions to mathematical logic and the philosophy of language. He is known for proposing Montague grammar to formalize th ...
provides various suggestions for how to formalize English language expressions in his
theory of universal grammar. Formalization is also discussed in the
philosophy of logic
Philosophy of logic is the area of philosophy that studies the scope and nature of logic. It investigates the philosophical problems raised by logic, such as the presuppositions often implicitly at work in theories of logic and in their application ...
in relation to its role in understanding and applying logic. If logic is understood as the theory of valid inferences in general then formalization plays a central role in it since many of these inferences are formulated in ordinary language. Logic translation is needed to link formal systems of logic to arguments expressed in ordinary language. A related claim is that all logical languages, including highly abstract ones like model logic and
many-valued logic
Many-valued logic (also multi- or multiple-valued logic) refers to a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false" ...
, have to be "anchored in the structures of natural language". However, one difficulty in this regard is that logic is usually understood as a
formal science
Formal science is a branch of science studying disciplines concerned with abstract structures described by formal systems, such as logic, mathematics, statistics, theoretical computer science, artificial intelligence, information theory, ga ...
. But a theory of its relation to empirical matters pertaining to ordinary languages goes beyond this purely formal conception. For this reason, some theorists identify a pure branch of logic and contrast it with applied logic, which includes the problem of formalization.
Some theorists draw the conclusion from these considerations that informal reasoning takes precedence over formal reasoning and that formal logic can only succeed to the extent that it is based on correct formalizations. For example, Michael Baumgartner and Timm Lampert hold that "there are no
informal fallacies
Informal fallacies are a type of incorrect argument in natural language. The source of the error is not just due to the ''form'' of the argument, as is the case for formal fallacies, but can also be due to their ''content'' and ''context''. Fallaci ...
" but only "misunderstanding of informal arguments expressed by inadequate formalizations". This position is rejected by Jaroslav Peregrin and Vladimír Svoboda, who argue that informal reasoning is not always accurate and may be corrected through the application of formal logic.
An alternative to formalization is to use
informal logic
Informal logic encompasses the principles of logic and logical thought outside of a formal setting (characterized by the usage of particular statements). However, the precise definition of "informal logic" is a matter of some dispute. Ralph H. ...
, which analyzes the cogency of natural language arguments in their original form. This has many advantages by avoiding the difficulties associated with logic translations. However, it also comes with various drawbacks. For example, informal logic lacks the precision found in formal logic for distinguishing between good arguments and
fallacies
A fallacy is the use of invalid or otherwise faulty reasoning, or "wrong moves," in the construction of an argument which may appear stronger than it really is if the fallacy is not spotted. The term in the Western intellectual tradition was intr ...
.
Examples
For propositional logic, the sentence "Tiffany sells jewelry, and Gucci sells cologne" can be translated as
. In this example,
represents the claim "Tiffany sells jewelry",
stands for "Gucci sells cologne", and
is the logical conjunction corresponding to "and". Another example is the sentence "Notre Dame raises tuition if Purdue does", which can be formalized as
.
For predicate logic, the sentence "Ann loves Ben" can be translated as
. In this example,
stands for "loves",
stands for Ann and
stands for Ben. Other examples are "some men are bald" as
, "all animals have a head" as
, "no frogs are birds" as
, and "if Elizabeth is a historian, then some women are historians" as
.
Problematic expressions
For various natural language expressions, it is not clear how they should be translated and the right translation may differ from case to case. The vagueness and ambiguity of ordinary language, in contrast to the precise nature of logic, is often responsible for these problems. For this reason, it has proven difficult to find a general
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
to cover all cases of translation.
For example, the meaning of basic English expressions like "and", "or", and "if...then" can vary from context to context. The corresponding logical operators in symbolic logic (
,
,
), on the other hand, have very precisely defined meanings. In this regard, they only capture certain aspects of the original meaning.
The English word "is" poses another such difficulty since it carries different meanings. It can express
existence
Existence is the ability of an entity to interact with reality. In philosophy, it refers to the ontology, ontological Property (philosophy), property of being.
Etymology
The term ''existence'' comes from Old French ''existence'', from Medieval ...
(as in "there is a Santa Claus"),
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
(as in "Superman is Clark Kent"), and
predication (as in "Venus is a planet"). Each one of these meanings is expressed differently in logical systems like first-order logic.
Another difficulty is that
quantifiers are often not explicitly expressed in ordinary language. For example, the sentence "emeralds are green" does not directly state the universal quantifier "all", i.e. "all emeralds are green". However, some sentences with a similar structure, such as the "children live next door", imply the
existential quantifier
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, w ...
"some", i.e. "some children live next door".
A closely related problem is found in certain valid natural language arguments whose most obvious translations are invalid in formal logic. For example, the argument "(1) Fury is a horse; (2) therefore Fury is an animal" is valid but the corresponding argument in formal logic from
to
is invalid. One solution is to add to the argument an additional premise stating that "all horses are animals". Another is to translate the sentence "Fury is a horse" as
. However, these solutions come with new problems of their own. Further problematic expressions are
definite descriptions
In formal semantics and philosophy of language, a definite description is a denoting phrase in the form of "the X" where X is a noun-phrase or a singular common noun. The definite description is ''proper'' if X applies to a unique individual or o ...
,
conditional sentence
Conditional sentences are natural language sentences that express that one thing is contingent on something else, e.g. "If it rains, the picnic will be cancelled." They are so called because the impact of the main clause of the sentence is ''cond ...
, and
attributive adjectives, as well as
mass nouns
In linguistics, a mass noun, uncountable noun, non-count noun, uncount noun, or just uncountable, is a noun with the syntactic property that any quantity of it is treated as an undifferentiated unit, rather than as something with discrete elemen ...
and
anaphora.
Translation between logics
A different type of logic translation takes place between logical systems. A translation between two logical systems can be defined in a formal sense as a certain type of
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
. This function maps sentences of the first system to sentences of the second system while obeying the
entailment relations between the original sentences. This means that if a sentence entails another sentence in the first logic, then the translation of the first sentence should entail the translation of the second sentence in the second logic. This way, a translation from one logic to another represents the formulas, proofs, and
models
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
of the first logic in terms of the second. This is sometimes referred to as a ''conservative translation'' in contrast to a ''rough translation'', which only maps the sentences of the first logic to sentences of the second logic without regard to their entailment relations.
A preliminary of logic translations is that there is not one logic but many logics. These logics differ from each other concerning the languages they use as well as the rules of inference they see as valid. For example,
intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
differs from
classical logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
Characteristics
Each logical system in this class ...
since it rejects certain rules of inference, such as the
double negation elimination
In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition ''A'' is logically equivalent to ''not (not ...
. This rule states that if a sentence is not not true, then it is true, i.e. that
follows from
.
One way to translate intuitionistic logic into non-intuitionistic logic is by using a
modal operator
A modal connective (or modal operator) is a logical connective for modal logic. It is an operator which forms propositions from propositions. In general, a modal operator has the "formal" property of being non-truth-functional in the following sen ...
. This is based on the idea that intuitionistic logic expresses not just what is true but what is
knowable Fitch's paradox of knowability is one of the fundamental puzzles of epistemic logic. It provides a challenge to the ''knowability thesis'', which states that every truth is, in principle, knowable. The paradox is that this assumption implies the ''o ...
. For example, the formula
in intuitionistic logic can be translated as
where
is a model operator expressing that the following formula is knowable.
Another example is the translation of modal logic to regular predicate logic. Modal logic contains additional symbols for possibility (
) and necessity (
) not found in regular predicate logic. One way to translate them is to introduce new predicates, such as the predicate R, which indicates that one
possible world
A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional logic, intensional and mod ...
is accessible from another possible world. For example, the modal logic expression
(it is possible that ''p'' is true in the actual world) can be translated as
(there exists a possible world that is accessible from the actual world and ''p'' is true in it).
Translations between logics are relevant for
metalogic
Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.Harry GenslerIntroduction to Logic Routledge, ...
and
logic programming
Logic programming is a programming paradigm which is largely based on formal logic. Any program written in a logic programming language is a set of sentences in logical form, expressing facts and rules about some problem domain. Major logic prog ...
. In metalogic, they can be used to study the properties of logical systems and the relations between them. In logic programming, they make it possible for programs limited to one type of logic to be applied to many additional cases. With their help, programs like
Prolog
Prolog is a logic programming language associated with artificial intelligence and computational linguistics.
Prolog has its roots in first-order logic, a formal logic, and unlike many other programming languages, Prolog is intended primarily ...
can be used to solve problems in modal logic and
temporal logic In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time (for example, "I am ''always'' hungry", "I will ''eventually'' be hungry", or "I will be hungry ''until'' I ...
. A closely related issue concerns the question of how to translate a formal language like
Controlled English
Control may refer to:
Basic meanings Economics and business
* Control (management), an element of management
* Control, an element of management accounting
* Comptroller (or controller), a senior financial officer in an organization
* Controlling ...
into a logical system. Controlled English is a
controlled language
Controlled natural languages (CNLs) are subsets of natural languages that are obtained by restricting the grammar and vocabulary in order to reduce or eliminate ambiguity and complexity. Traditionally, controlled languages fall into two major types ...
that limits grammar and vocabulary with the goal of reducing ambiguity and complexity. In this regard, the advantage of Controlled English is that every sentence has a unique interpretation. This makes it possible to use algorithms to translate them into formal logic, which is generally not possible for natural languages.
Criteria of adequate translations
Criteria of adequate translations specify how to distinguish good from bad translations. They determine whether a logical formula accurately represents the logical structure of the sentence it translates and help logicians decide between competing translations of the same sentence. Various criteria are discussed in the academic literature. According to various theorists, like Peregrin and Svoboda, the most basic criterion is that translations should preserve the inferential relations between sentences. This principle is sometimes called the criterion of syntactic correctness or the criterion of reliability. It stipulates that if an argument is valid in the original text then the translated argument is also valid. One difficulty in this regard is that the same sentence may form part of several arguments, sometimes as a premise and sometimes as a conclusion. A translation of a sentence is only correct if in all or nearly all these cases, the inferential relations are preserved. Some theorists argue that this implies that translations are
holistic
Holism () is the idea that various systems (e.g. physical, biological, social) should be viewed as wholes, not merely as a collection of parts. The term "holism" was coined by Jan Smuts in his 1926 book ''Holism and Evolution''."holism, n." OED Onl ...
. This means that one cannot evaluate sentence translations individually. Instead, the correctness of a translation of one sentence depends on how other sentences are translated to ensure correspondence in the inferential relations.
A closely related criterion focuses on the truth conditions of sentences. A truth condition of a sentence is what the world must be like for that sentence to be true. This criterion states that for adequate translations, the truth conditions of the original sentence are identical to the truth conditions of the translated sentence. The mere fact that the sentence and its translation have the same truth value is not sufficient. Instead, it implies that whenever one is true, the other is also true, i.e. they have to have the same truth value in all possible circumstances. This criterion is not universally accepted. It has been criticized based on the claim that logical formulas do not have truth conditions. According to this view, the symbols they use are meaningless by themselves and only have the purpose of expressing the logical form of a sentence without implying any concrete content. Another problem with this approach is that all
tautologies have the same truth conditions: they are true independently of the circumstances. This would imply that any tautology is a correct translation of any other tautology.
Besides these core criteria, various additional criteria are often discussed in the academic literature. Their goal is usually to exclude bad translations that nonetheless comply with the other criteria. For example, according to the first two criteria, the sentence "it rains" could be formalized as
or as
since both formulas have the same truth conditions and the same inferential patterns even though the second formula is a bad translation. One additional criterion is that translations should not include symbols that do not correspond to expressions in the original sentence. According to it, the translation of "it rains" should not include the symbol for logical negation (
) since a corresponding expression is not found in the original sentence. Another criterion holds that the order of symbols in the translation should reflect the order of the expressions in the original sentence. For example, the sentence "Pete went up the hill and Quinn went up the hill" should be translated as
and not as
. A closely related criterion is the principle of transparency, which states that translations should aim to be similar to the original expression. This concerns, for example, that a translation reflects the grammatical structure of the original sentence as closely as possible. The principle of parsimony states that simple translations (i.e. logical formulas that use as few symbols as possible) are to be preferred. One way to test whether a formalization is correct is to translate it back into natural language and see if this second translation matches the original.
The problem of the criteria of adequate translations is often not discussed in detail in introductions to logic. One reason for this is that some theorists see logic translation as an
art
Art is a diverse range of human activity, and resulting product, that involves creative or imaginative talent expressive of technical proficiency, beauty, emotional power, or conceptual ideas.
There is no generally agreed definition of wha ...
or an intuitive practice. According to this view, it is based on a practical skill learned from
experience
Experience refers to conscious events in general, more specifically to perceptions, or to the practical knowledge and familiarity that is produced by these conscious processes. Understood as a conscious event in the widest sense, experience involv ...
with many examples and guided by some rough rules of thumb. This outlook implies that there are no strict rules of adequate formalization. Against this position, it has been argued that, without clear criteria of adequacy, it becomes very difficult to decide between competing formalizations of the same sentence.
Translation procedures
Various logicians have proposed translation procedures employing several steps to arrive at correct translations. Some only constitute rough guidelines to help translators in the process while others consist of detailed and effective procedures covering all the steps needed to arrive at a translation. In either case, they are usually not exact algorithms that could be blindly followed but rather tools to simplify the process.
Certain preparatory steps may be taken within natural language before the actual translation starts. An initial step is often to understand the meaning of the original text, for example, by analyzing the different claims made in it. This includes identifying which arguments are made and whether a certain claim acts as a premise or as a conclusion.
At this stage, some logicians recommend paraphrasing the sentences to make the claims more explicit, remove ambiguities, and highlight their logical structure. For example, the sentence "John Paul II is infallible" could be paraphrased as "it is not the case that John Paul II is fallible".
This can involve identifying
truth-functional
In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly one ...
connectives, like "and", "if...then", or "not", and decomposing the text accordingly. Each of the units analyzed this way is an individual claim that is either true or false. A closely related step is to group the individual expressions into logical units and classify them according to their logical role. In the sentence above, for example, "is fallible" is a predicate and the expression "it is not the case that" corresponds to the logical connective for
negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
.
Once these preparations are done, some theorists recommend translation into a hybrid language. Such hybrid expressions already contain a logical formalism but retain regular names for predicates and proper names. For example, the sentence "All animals have heads" could be translated as
. The idea behind this step is that the regular terms still carry their original meaning and thereby make it easier to understand the formulas and to see how they relate to the original text. However, the natural language vocabulary is usually not precisely defined and therefore lacks the exactness demanded by formal logic. As a last step, these regular terms are then replaced by logical symbols. For the expression above, this would result in the formula
. This way, the connection to the ordinary language meanings is cut and the formulas become a purely formal expression of the logical structure of the original text while abstracting from its specific content.
The formalization of a full argument consists in several steps since the argument is made up of several propositions. Once the translation is complete, the formal tools of the logical system, such as its rules of inference, can be employed to assess whether the argument is valid.
Criticism
Logic translations are frequently criticized based on the claim that they are unable to accurately represent all the aspects and nuances of the original text. For example, logical vocabulary is usually unable to capture things like
sarcasm
Sarcasm is the caustic use of words, often in a humorous way, to mock someone or something. Sarcasm may employ ambivalence, although it is not necessarily ironic. Most noticeable in spoken word, sarcasm is mainly distinguished by the inflection ...
, indirect
insinuation
An innuendo is a wikt:hint, hint, wikt:insinuation, insinuation or wikt:intimation, intimation about a person or thing, especially of a denigrating or derogatory nature. It can also be a remark or question, typically disparaging (also called in ...
, or
emphasis. In this regard, many aspects of the meaning of the original expression that go beyond truth value, validity, and logical structure are frequently ignored. On the level of informal inferences, there are various expressions that cannot easily be represented using the precise but limited languages of formal logic.
For these reasons, it is often controversial whether a logic translation is correct. When a logic translation is used to defend the conclusion of a natural language argument, a common way to undermine such a defense is to claim that the logic translation is incorrect and that insights gained from the formal logical analysis do not carry any weight for the original argument.
Another type of criticism is not directed at logic translations themselves but at how they are discussed in many standard works and courses of logic. In this regard, theorists like Georg Brun, Peregrin, and Svoboda argue such works do not provide a proper discussion of the role and limitations of logic translations. Instead, it is claimed that they merely treat this topic as a side note by providing a few examples while having their main focus on the formal systems themselves without an in-depth discussion of how these systems are to be applied to ordinary arguments.
See also
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Common Logic
Common Logic (CL) is a framework for a family of logic languages, based on first-order logic, intended to facilitate the exchange and transmission of knowledge in computer-based systems.
The CL definition permits and encourages the development o ...
*
Double-negation translation In proof theory, a discipline within mathematical logic, double-negation translation, sometimes called negative translation, is a general approach for embedding classical logic into intuitionistic logic, typically by translating formulas to formulas ...
*
Standard translation {{short description, Algorithm in modal logic
In modal logic, standard translation is a logic translation that transforms formulas of modal logic into formulas of first-order logic which capture the meaning of the modal formulas. Standard translati ...
References
Sources
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*{{cite thesis , last1=Rabe , first1=Florian , title=Representing Logics and Logic Translations , date=2008 , publisher=Jacobs University Bremen , location=Bremen , url=https://kwarc.info/people/frabe/Research/phdthesis.pdf , type=PhD dissertation
Logic
Semantics
Translation