# lo.GLOSSARY

## GLOSSARY: LOGIC

* [Absoluteness](#absoluteness)
* [Accident](#accident)
* [Analytic-synthetic distinction](#analytic-synthetic-distinction)
* [Argumentation theory](#argumentation-theory)
* [Axiology](#axiology)
* [Axiom](#axiom)
* [Conjecture](#conjecture)
* [Constants symbols](#constants-symbols)
* [Defeasible inference](#defeasible-inference)
* [Defeasible reasoning](#defeasible-reasoning)
* [Essence](#essence)
* [Extensionality](#extensionality)
* [Extremal](#extremal)
* [Fallacy](#fallacy)
* [First-order logic](#first-order-logic)
* [Formal logic](#formal-logic)
* [Fuzzy logic](#fuzzy-logic)
* [iff](#iff)
* [Informal logic](#informal-logic)
* [Laws of Thought](#laws-of-thought)
* [Logical Connectives](#logical-connectives)
* [Logical Consequence](#logical-consequence)
* [Logical Form](#logical-form)
* [Logical Framework](#logical-framework)
* [Logical Truth](#logical-truth)
* [Mathematical logic](#mathematical-logic)
* [Metalogic](#metalogic)
* [Metatheorem](#metatheorem)
* [Naïve theory](#naïve-theory)
* [Necessity and sufficiency](#necessity-and-sufficiency)
* [Non-classical logic](#non-classical-logic)
* [Noumenon](#noumenon)
* [Phenomenon](#phenomenon)
* [Philosophical logic](#philosophical-logic)
* [Philosophy of logic](#philosophy-of-logic)
* [Predicate](#predicate)
* [Predicate symbol](#predicate-symbol)
* [Premise](#premise)
* [Primitive notion](#primitive-notion)
* [Proof by exhaustion](#proof-by-exhaustion)
* [Propositional logic](#propositional-logic)
* [Second-order Logic](#second-order-logic)
* [Statement](#statement)
* [Strict Conditional](#strict-conditional)
* [Substitution](#substitution)
* [Syllogisms](#syllogisms)
* [Symbolic Logic](#symbolic-logic)
* [Theorem](#theorem)
* [Truth Value](#truth-value)
* [Validity](#validity)
* [Variable symbols](#variable-symbols)
* [Weakening](#weakening)

### Absoluteness

\[mathematical-logic], a formula is said to be **absolute** if it has the same truth value in each class of structures/models.

### Accident

\[philosophy] Accident is a property that the entity has contingently, without which it still retains its identity. Accident is contrasted with essence, which is the property that makes an entity fundamentally what it is, which it has by necessity, without which it loses its identity.

### Analytic-synthetic distinction

\[philosophy] The analytic-synthetic dichotomy is a semantic distinction, used primarily in philosophy to distinguish propositions into two types: analytic propositions and synthetic propositions. Analytic propositions are true by virtue of their meaning, while synthetic propositions are true by how their meaning relates to the world.

### Argumentation theory

Argumentation theory is the interdisciplinary study of how conclusions can be reached through logical reasoning i.e. claims based, soundly or not, on premises. It includes the arts and sciences of civil debate, dialogue, conversation, and persuasion.

### Axiology

Axiology is the philosophical study of value. It is either the collective term for ethics and aesthetics, philosophical fields that depend crucially on notions of worth, or the foundation for these fields, and thus similar to value theory and meta-ethics.

### Axiom

An axiom or postulate is a statement that is taken to be true that serves as an initial premise or starting point for further reasoning and arguments. In classical logics, an axiom is an evident and well-established statement that needs no further proof. In modern logics, an axiom is a premise for reasoning.

### Conjecture

An unproved statement that is believed true is called a conjecture. To be considered a conjecture, a statement must usually be proposed publicly, at which point the name of the proponent may be attached to the conjecture, as with Goldbach's conjecture. Other famous conjectures include the Collatz conjecture and the Riemann hypothesis. On the other hand, Fermat's Last Theorem has always been known by that name, even before it was proved; it was never known as "Fermat's conjecture".

### Constants symbols

Constants symbols are strings that are interpreted as representing objects.

### Defeasible inference

Defeasible inferences is a kind of inference in which reasoners draw tentative conclusions, reserving the right to retract their conclusions based on further evidence.

### Defeasible reasoning

Defeasible reasoning is rationally compelling but deductively invalid kind of reasoning.

### Essence

\[philosophy] Essence is the property that makes an entity what it fundamentally is, which it has by necessity, without which it loses its identity. Essence is contrasted with accident, which is a property that the entity has contingently, without which it still retains its identity.

### Extensionality

Extensionality refers to principles that judge objects to be equal if they have the same external properties, as opposed to intensionality, which is concerned with whether the internal definitions of objects are the same. There are many predicates that are intensionally different but extensionally identical. For example, the expressions, 2+4 and 2\*3, are extensionally equal (from outside), but intensionality different (from within).

### Extremal

The clause in a recursive definition that specifies that no items other than those generated by the stated rules fall within the definition; e.g. "0 is an natural number; if n is a natural number, then S(n) is a natural number, and nothing else is a natural number".

### Fallacy

In reasoning to argue a claim, a fallacy is reasoning that is evaluated as logically incorrect and that undermines the logical validity of the argument and permits its recognition as unsound.

### First-order logic

While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates and quantification. Predicate logic uses quantified variables over objects and allows the use of sentences that contain variables, so rather than propositions such as "Socrates is a man" one can have expressions in the form "there exists X such that X is Socrates and X is a man" and "there exists" is a quantifier while "X" is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations. In first-order theories, predicates are often associated with sets.

### Formal logic

Formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule i.e. a rule that is not about any particular thing or property.

### Fuzzy logic

Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or 1.

### iff

Bijection is a logic connective, sometimes read as "if and only if", used to mark two statements as logically equivalent, symbolically denoted as $$p \iff q$$. Two statements are logically equivalent if the former implies the latter, $$p \to q$$, and if the latter implies the former, $$q \to p$$.

### Informal logic

Informal logic is the study of natural language arguments. The study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, informal logic is not logic at all.

### Laws of Thought

Laws of thought are the 3 fundamental laws, _the law of identity_, _the law of non-contradiction_ and _the law of excluded middle_, often considered as the basis of rational discourse itself. These rules have been known and accepted in logic for centuries, until the modern logicians placed them under detailed scrutiny, which resulted in inventions of new kinds of logic that exercised strict control regarding the three fundamental laws.

### Logical Connectives

Logical connectives (logical operators) are symbols or words used to connect sentences in a grammatically valid way, such that the value of the compound sentence produced depends only on that of the original sentences and on the meaning of the connective. Logical connectives include negation (¬), conjunction (∧), inclusive disjunction (∨), exclusive disjunction (⊕), biconditional (⇔), implication (⇒), NAND (↑), NOR (↓), sometimes even parentheses.

### Logical Consequence

Logical consequence (also **entailment**) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises.

### Logical Form

A logical form of a syntactic expression is a precisely-specified semantic version of that expression in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system.

The concept of logical form is central to logic: the validity of an argument is determined by its logical form, not by its content. A logical form of a syntactic expression is a precisely specified semantic version of that expression in a formal system. It is the formalization of possibly ambiguous statements from a natural language into, often symbolic, statements with precise unambiguous interpretation - their logical form.

### Logical Framework

Logical Framework provides a means to define a logic as a signature in a higher-order type theory in such a way that provability of a formula in the original logic reduces to a type inhabitation problem in the framework type theory.

### Logical Truth

Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true, and remains true under all reinterpretations of its components other than its logical constants.

### Mathematical logic

Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of model theory, proof theory, set theory, and computability (recursion) theory.

### 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.

### Metatheorem

A metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory but not the object theory.

### Naïve theory

A theory qualified with the adjective "naïve" is an unformalized theory, presented using natural language.

### Necessity and sufficiency

In logic, necessity and sufficiency are terms used to describe an implicational relationship between statements. The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true.

### Non-classical logic

Non-classical logics (and sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations.

### Noumenon

\[philosophy] Noumenon is a posited object or event that exists independently of human sense or perception. Noumenon is contrasted with phenomenon, which refers to anything that can be apprehended by (or is an object of) the senses.

### Phenomenon

\[philosophy] The term "phenomenon" refers to anything that can be apprehended by (or is an object of) the senses. Phenomenon is contrasted with noumenon, which is a posited object or event that exists independently of human sense or perception.

### Philosophical logic

\[philosophy] Philosophical logic refers to those areas of philosophy in which recognized methods of logic have traditionally been used to solve or advance the discussion of philosophical problems. Among these are the study of argument, meaning, and truth, but also of identity, existence, predication, necessity and truth. Philosophical logic also addresses extensions and alternatives to traditional classical logics, known as non-classical logics.

### Philosophy of logic

Following the developments in formal logic with symbolic logic in the late nineteenth century and mathematical logic in the twentieth, topics traditionally treated by logic not being part of formal logic have tended to be termed either philosophy of logic or philosophical logic if no longer simply logic. Compared to the history of logic the demarcation between philosophy of logic and philosophical logic is of recent coinage and not always entirely clear.

### Predicate

A predicate is a statement that takes argument variables, which range over the elements of the domain of discourse, and maps them to a truth value. In the case there are two such input variables, such a predicate is called a (binary) predicate or a (binary) relation. A unary predicate is often called a property. Predicates of larger arity may also be called relations.

If a predicate `P` marks the property "is divisible by 2", then `P(x)` is a statement of one variable `x`. However, the DOD must be specified so we know what individuals this variable ranges over. For example, if used inside a set builder notation like `𝔼 = { x | ∀x ∈ ℕ. P(x) }`, it indicates that a natural number `x` belongs to the set `𝔼` if `P(x)` holds.

### Predicate symbol

Predicates are represented by a predicate symbol used to denote properties of objects and relations between them. Each predicate has an associated arity. Unary predicates represent positive or negative properties of objects. For example, `φ(x)` could mean that an object `x` is an odd number, i.e. `x` has a property of being odd. Polyadic predicates represent relations between objects, possibly denoted by `R(x,y)`, or `xRy` for binary relations. Common binary relations frequently have own dedicated infix symbol, e.g. `x <= y`.

### Premise

A premise is a statement that an argument claims will induce or justify a conclusion. A premise is an assumption that something is true.

### Primitive notion

In mathematics, a primitive notion is an undefined concept, not defined in terms of previously defined concepts, to be taken for granted. It lacks a proof, and in that regard it's analogous to an axiom of a formal system (axioms don't require proof). Sometimes the primitive notions cannot be avoided because we need to start somewhere lest regress into downward spiral of definitions, forever defining concept in terms of previously defined concept, which also need definition in terms of previous ones, and so on, ad nauseum. Sometimes a concept just doesn't have a formal definition (e.g. "set").

### Proof by exhaustion

Proof by exhaustion (aka proof by cases, proof by case analysis, complete induction, brute force method) is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases and each type of case is checked to see if the proposition in question holds. This is a method of direct proof.

## Propositions

Propositions are declarative sentences that have a truth value.

### Propositional logic

Propositional (sentential) logic is based on propositions and argument flow. Compound propositions are formed by connecting propositions by logical connectives. The propositions without logical connectives are called atomic propositions. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.

### Second-order Logic

Second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies only variables that range over individuals (elements of the domain of discourse), second-order logic additionally quantifies over relations.

### Statement

In logic, the term statement is variously understood to mean either: (a) a meaningful declarative sentence that is true or false, or (b) the assertion that is made by a true or false declarative sentence.In the latter case, a statement is distinct from a sentence in that a sentence is only one formulation of a statement, whereas there may be many other formulations expressing the same statement.

### Strict Conditional

In logic, a strict conditional (strict implication) is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necessity operator from modal logic.

### Substitution

Substitution is a fundamental concept in logic. A substitution is a syntactic transformation on formal expressions.

### Syllogisms

A syllogism is a kind of logical argument where a quantified statement of a specific form (the conclusion) is inferred from two other quantified statements (the premises). It is a logical argument where one starts with premises and reaches a conclusion. A syllogism is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true. In its earliest form, defined by Aristotle, from the combination of a general statement (the major premise) and a specific statement (the minor premise), a conclusion is deduced.

### Symbolic Logic

Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. It is often divided into two main branches, propositional and predicate logic.

### Theorem

A theorem is a statement that has been proven on the basis of previously established statements (other theorems) and generally accepted statements (axioms). It is a logical consequence of the axioms.

### Truth Value

Truth or logical value is a value indicating the relation of a proposition to truth. A proposition has a truth value if it can be evaluated to true or false.

### Validity

In logic, an argument is valid iff it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. It is not required that a valid argument have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion.

### Variable symbols

Variables are used as placeholders for quantifying over objects.

### Weakening

The weakening rule, one of the structural rules that operate directly on the structure of a deduction system, states that the hypotheses or conclusion of a sequent may be extended with additional members and still remain valid.
