Metalogic in the context of Instance (computer science)

The type–token distinction is the difference between a type of objects (analogous to a class) and the individual tokens of that type (analogous to instances). Since each type may be instantiated by multiple tokens, there are generally more tokens than types of an object.

For example, the sentence "A rose is a rose is a rose" contains three word types: three word tokens of the type a, two word tokens of the type is, and three word tokens of the type rose. The distinction is important in disciplines such as logic, linguistics, metalogic, typography, and computer programming.

Logic translation is the process of representing a text in the formal language of a logical system. If the original text is formulated in ordinary language 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 as ${\displaystyle \exists x(M(x)\land B(x))}$. The purpose is to reveal the logical structure of arguments. 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 of the process are caused by vague or ambiguous expressions in natural language. For example, the English word "is" can mean that something exists, that it is identical to something else, or that it has a certain property. This contrasts with the precise nature of formal logic, which avoids such ambiguities. Natural language formalization is relevant to various fields in the sciences and humanities. It may play a key role for logic in general since it is needed to establish a link between many forms of reasoning and abstract logical systems. The use of informal logic 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, for example, from modal logic to first-order logic. This form of logic translation is specifically relevant for logic programming and metalogic.

In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete.The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantic validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true.

In metalogic, mathematical logic, and computability theory, an effective method or effective procedure is a finite-time, deterministic procedure for solving a problem from a specific class. An effective method is sometimes also called a mechanical method or procedure.

Philosophy of logic is the branch 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. This involves questions about how logic is to be defined and how different logical systems are connected to each other. It includes the study of the nature of the fundamental concepts used by logic and the relation of logic to other disciplines. According to a common characterisation, philosophical logic is the part of the philosophy of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. But other theorists draw the distinction between the philosophy of logic and philosophical logic differently or not at all. Metalogic is closely related to the philosophy of logic as the discipline investigating the properties of formal logical systems, like consistency and completeness.

Various characterizations of the nature of logic are found in the academic literature. Logic is often seen as the study of the laws of thought, correct reasoning, valid inference, or logical truth. It is a formal science that investigates how conclusions follow from premises in a topic-neutral manner, i.e. independent of the specific subject matter discussed. One form of inquiring into the nature of logic focuses on the commonalities between various logical formal systems and on how they differ from non-logical formal systems. Important considerations in this respect are whether the formal system in question is compatible with fundamental logical intuitions and whether it is complete. Different conceptions of logic can be distinguished according to whether they define logic as the study of valid inference or logical truth. A further distinction among conceptions of logic is based on whether the criteria of valid inference and logical truth are specified in terms of syntax or semantics.

In computer science, in particular in knowledge representation and reasoning and metalogic, the area of automated reasoning is dedicated to understanding different aspects of reasoning. The study of automated reasoning helps produce computer programs that allow computers to reason completely, or nearly completely, automatically. Although automated reasoning is considered a sub-field of artificial intelligence, it also has connections with theoretical computer science and philosophy.

The most developed subareas of automated reasoning are automated theorem proving (and the less automated but more pragmatic subfield of interactive theorem proving) and automated proof checking (viewed as guaranteed correct reasoning under fixed assumptions). Extensive work has also been done in reasoning by analogy using induction and abduction.

where "${\displaystyle \Leftrightarrow }$" is a metalogical symbol representing "can be replaced in a proof with", P and Q are any given logical statements, and ${\displaystyle \neg P\lor Q}$ can be read as "(not P) or Q". To illustrate this, consider the following statements:

Exportation is a valid rule of replacement in propositional logic. The rule allows conditional statements having conjunctive antecedents to be replaced by statements having conditional consequents and vice versa in logical proofs. It is the rule that:

Where "${\displaystyle \Leftrightarrow }$" is a metalogical symbol representing "can be replaced in a proof with." In strict terminology, ${\displaystyle ((P\land Q)\to R)\Rightarrow (P\to (Q\to R))}$ is the law of exportation, for it "exports" a proposition from the antecedent of ${\displaystyle (P\land Q)\to R}$ to its consequent. Its converse, the law of importation, ${\displaystyle (P\to (Q\to R))\Rightarrow ((P\land Q)\to R)}$, "imports" a proposition from the consequent of ${\displaystyle P\to (Q\to R)}$ to its antecedent.

In propositional logic and Boolean algebra, there is a duality between conjunction and disjunction, also called the duality principle. It is the most widely known example of duality in logic. The duality consists in these metalogical theorems:

• In classical propositional logic, the connectives for conjunction and disjunction can be defined in terms of each other, and consequently, only one of them needs to be taken as primitive.
• If ${\displaystyle \varphi ^{D}}$ is used as notation to designate the result of replacing every instance of conjunction with disjunction, and every instance of disjunction with conjunction (e.g. ${\displaystyle p\land q}$ with ${\displaystyle q\lor p}$, or vice-versa), in a given formula ${\displaystyle \varphi }$, and if ${\displaystyle {\overline {\varphi }}}$ is used as notation for replacing every sentence-letter in ${\displaystyle \varphi }$ with its negation (e.g., ${\displaystyle p}$ with ${\displaystyle \neg p}$), and if the symbol ${\displaystyle \models }$ is used for semantic consequence and ⟚ for semantical equivalence between logical formulas, then it is demonstrable that ${\displaystyle \varphi ^{D}}$ ⟚ ${\displaystyle \neg {\overline {\varphi }}}$, and also that ${\displaystyle \varphi \models \psi }$ if, and only if, ${\displaystyle \psi ^{D}\models \varphi ^{D}}$, and furthermore that if ${\displaystyle \varphi }$ ⟚ ${\displaystyle \psi }$ then ${\displaystyle \varphi ^{D}}$ ⟚ ${\displaystyle \psi ^{D}}$. (In this context, ${\displaystyle {\overline {\varphi }}^{D}}$ is called the dual of a formula ${\displaystyle \varphi }$.)