User:Ascii/Sandbox

Logic

 * Definition:Logic
 * Logic is the study of the structure of statements and their truth values, divorced from their conceptual content.
 * Definition:Symbolic Logic
 * Symbolic logic is the study of logic in which the logical form of statements is analyzed by using symbols as tools. Instead of explicit statements, logical formulas are investigated, which are symbolic representations of statements, and compound statements in particular. In symbolic logic, the rules of reasoning and logic are investigated by means of formal systems, which form a good foundation for the symbolic manipulations performed in this field.


 * Definition:Statement
 * A statement is a sentence which has objective and logical meaning.
 * Definition:Proposition
 * A proposition is a statement which is offered up for investigation as to its truth or falsehood. Loosely, a proposition is a statement which is about to be proved (or disproved).
 * Definition:True
 * A statement has a truth value of true what it says matches the way that things are.
 * Definition:False
 * A statement has a truth value of false what it says does not match the way that things are.
 * Definition:Truth Value
 * In Aristotelian logic, a statement can be either true or false, and there is no undefined, in-between value. Whether it is true or false is called its truth value. Note that a statement's truth value may change depending on circumstances.
 * Definition:Proof System
 * Let $\mathcal L$ be a formal language. A proof system $\mathscr P$ for $\mathcal L$ comprises:
 * Axioms and/or axiom schemata;
 * Rules of inference for deriving theorems.
 * It is usual that a proof system does this by declaring certain arguments concerning $\mathcal L$ to be valid. Informally, a proof system amounts to a precise account of what constitutes a (formal) proof.
 * Definition:Axiom
 * In all contexts, the definition of the term axiom is by and large the same. That is, an axiom is a statement which is accepted as being true. A statement that is considered an axiom can be described as being axiomatic.
 * Definition:Assumption
 * An assumption is a statement which is introduced into an argument, whose truth value is (temporarily) accepted as True. In mathematics, the keyword let is often the indicator here that an assumption is going to be introduced. For example: Let $p$ (be true) ... can be interpreted, in natural language, as: Let us assume, for the sake of argument, that $p$ is true ...
 * Definition:Premise
 * A premise is an assumption that is used as a basis from which to start to construct an argument. When the validity or otherwise of a proof is called into question, one may request the arguer to "check your premises".


 * Definition:Valid Argument
 * A valid argument is a logical argument in which the premises provide conclusive reasons for the conclusion.
 * Definition:Sequent
 * A sequent is an expression in the form: $\phi_1, \phi_2, \ldots, \phi_n \vdash \psi$ where $\phi_1, \phi_2, \ldots, \phi_n$ are premises (any number of them), and $\psi$ the conclusion (only one), of an argument.
 * Definition:Logical Language
 * Let $\mathcal L$ be a formal language used in symbolic logic. Then $\mathcal L$ is called a logical language.
 * Definition:Logical Formula
 * Let $\mathcal L$ be a formal language used in the field of symbolic logic. Then the well-formed formulas of $\mathcal L$ are often referred to as logical formulas. They are symbolic representations of statements, and often of compound statements in particular.
 * Definition:Parenthesis
 * Parenthesis is a syntactical technique to disambiguate the meaning of a logical formula. It allows one to specify that a logical formula should (temporarily) be regarded as being a single entity, being on the same level as a statement variable. Such a formula is referred to as being in parenthesis. Typically, a formal language, in defining its formal grammar, ensures by means of parenthesis that all of its well-formed words are uniquely readable. Generally, brackets are used to indicate that certain formulas are in parenthesis. The brackets that are mostly used are round ones, the left (round) bracket $($ and the right (round) bracket $)$.


 * Definition:Propositional Logic
 * Propositional logic is a sub-branch of symbolic logic in which the truth values of propositional formulas are investigated and analysed. The atoms of propositional logic are simple statements. There are various systems of propositional logic for determining the truth values of propositional formulas, for example:
 * Natural deduction is a technique for deducing valid sequents from other valid sequents by applying precisely defined proof rules, each of which themselves are either "self-evident" axioms or themselves derived from other valid sequents.
 * The Method of Truth Tables, which consists of the construction of one or more truth tables which exhaustively list all the possible truth values of all the statement variables with a view to determining the required answer by inspection.
 * Definition:Language of Propositional Logic
 * There are a lot of different formal systems expressing propositional logic. Although they vary wildly in complexity and even disagree (to some extent) on what expressions are valid, generally all of these use a compatible formal language.


 * Definition:Logical Connective
 * A logical connective is an object which either modifies a statement, or combines existing statements into a new statement, called a compound statement. It is almost universal to identify a logical connective with the symbol representing it. Thus, logical connective may also, particularly in symbolic logic, be used to refer to that symbol, rather than speaking of a connective symbol separately. In mathematics, logical connectives are considered to be truth-functional. That is, the truth value of a compound statement formed using the connective is assumed to depend only on the truth value of the comprising statements. Thus, as far as the connective is concerned, it does not matter what the comprising statements precisely are. As a consequence of this truth-functionality, a connective has a corresponding truth function, which goes by the same name as the connective itself. The arity of this truth function is the number of statements the logical connective combines into a single compound statement.
 * Definition:Logical Not
 * The logical not or negation operator is a unary connective whose action is to reverse the truth value of the statement on which it operates. $\neg p$ is defined as: $p$ is not true, or It is not the case that $p$ is true.
 * Definition:Conjunction
 * Conjunction is a binary connective written symbolically as $p \land q$ whose behaviour is as follows: $p \land q$ is defined as $p$ is true and $q$ is true. This is called the conjunction of $p$ and $q$. The statements $p$ and $q$ are known as: the conjuncts or the members of the conjunction.
 * Definition:Disjunction
 * Disjunction is a binary connective written symbolically as $p \lor q$ whose behaviour is as follows: $p \lor q$ is defined as: Either $p$ is true or $q$ is true or possibly both. This is called the disjunction of $p$ and $q$. The substatements $p$ and $q$ are known as the disjuncts, or the members of the disjunction.
 * Definition:Conditional
 * The conditional or implication is a binary connective: $p \implies q$ defined as: If $p$ is true, then $q$ is true. This is known as a conditional statement. A conditional statement is also known as a conditional proposition or just a conditional.
 * Definition:Conditional/Necessary Condition
 * Let $p \implies q$ be a conditional statement. Then $q$ is a necessary condition for $p$. That is, if $p \implies q$, then it is necessary that $q$ be true for $p$ to be true. This is because unless $q$ is true, $p$ can not be true.
 * Definition:Converse Statement
 * The converse of the conditional: $p \implies q$ is the statement: $q \implies p$
 * Definition:Biconditional
 * The biconditional is a binary connective: $p \iff q$ defined as: $\left({p \implies q}\right) \land \left({q \implies p}\right)$ That is: If $p$ is true, then $q$ is true, and if $q$ is true, then $p$ is true.
 * Definition:Logical Equivalence
 * If two statements $p$ and $q$ are such that: $p \vdash q$, that is: $p$ therefore $q$; $q \vdash p$, that is: $q$ therefore $p$ then $p$ and $q$ are said to be (logically) equivalent. That is: $p \dashv \vdash q$ means: $p \vdash q$ and $q \vdash p$. Note that because the conclusion of an argument is a single statement, there can be only one statement on either side of the $\dashv \vdash$ sign. In symbolic logic, the notion of logical equivalence occurs in the form of provable equivalence and semantic equivalence.
 * Definition:Iff
 * The logical connective iff is a convenient shorthand for if and only if.
 * Definition:Vacuous Truth
 * Let $P \implies Q$ be a conditional statement. Suppose that $P$ is false. Then the statement $P \implies Q$ is a vacuous truth, or is vacuously true.
 * Definition:Main Connective/Propositional Logic
 * Let $\mathbf C$ be a WFF of propositional logic such that: $\mathbf C = \left({\mathbf A \circ \mathbf B}\right)$ where both $\mathbf A$ and $\mathbf B$ are both WFFs and $\circ$ is a binary connective. Then $\circ$ is the main connective of $\mathbf C$. Alternatively, let $\mathbf A$ be a WFF of propositional logic such that: $\mathbf A = \neg \mathbf B$ where $\mathbf B$ is a WFF. Then $\neg$ is the main connective of $\mathbf A$.
 * Definition:Propositional Function
 * A propositional function $\map P {x_1, x_2, \ldots}$ is an operator which acts on the objects denoted by the object variables $x_1, x_2, \ldots$ in a particular universe to return a truth value which depends on:
 * $(1): \quad$ The values of $x_1, x_2, \ldots$
 * $(2): \quad$ The nature of $P$.
 * Definition:Boolean Interpretation
 * Definition:Tautology
 * A tautology is a statement which is always true, independently of any relevant circumstances that could theoretically influence its truth value. It is epitomised by the form: $p \implies p$ that is: if $p$ is true then $p$ is true. An example of a "relevant circumstance" here is the truth value of $p$. The archetypal tautology is symbolised by $\top$, and referred to as Top.
 * Definition:Contradiction
 * A contradiction is a statement which is always false, independently of any relevant circumstances that could theoretically influence its truth value. This has the form: $p \land \neg p$ or, equivalently: $\neg p \land p$ that is: $p$ is true and, at the same time, $p$ is not true. An example of a "relevant circumstance" here is the truth value of $p$. The archetypal contradiction can be symbolised by $\bot$, and referred to as bottom.
 * Definition:Language of Propositional Logic/Formal Grammar/WFF
 * Let $\mathbf A$ be approved of by the formal grammar of propositional logic. Then $\mathbf A$ is called a well-formed formula of propositional logic. Often, one abbreviates "well-formed formula", speaking of a WFF of propositional logic instead. More informally, a WFF of propositional logic is any sequence of symbols containing statement variables, such that when statements are substituted for the statement variables (the same statement for any given statement variable throughout), the result is a statement.


 * Definition:Categorical Statement
 * Let $S$ and $P$ be predicates. A categorical statement is a statement that can be expressed in one of the following ways in natural language:
 * $(A)$: Universal Affirmative: Every $S$ is $P$
 * $(E)$: Universal Negative: No $S$ is $P$
 * $(I)$: Particular Affirmative: Some $S$ is $P$
 * $(O)$: Particular Negative: Some $S$ is not $P$
 * Definition:Categorical Syllogism
 * A categorical syllogism is a logical argument which is structured as follows:
 * $(1):$ It has exactly two premises and one conclusion. The first premise is usually referred to as the major premise. The second premise is usually referred to as the minor premise.
 * $(2):$ It concerns exactly three terms, which are usually denoted:
 * $P$ the primary term
 * $M$ the middle term
 * $S$ the secondary term
 * $(3):$ Each of the premises and conclusion is a categorical statement.


 * Definition:Predicate Logic
 * Predicate logic is a sub-branch of symbolic logic. It is an extension of propositional logic in which the internal structure of simple statements is analyzed. Thus in predicate logic, simple statements are no longer atomic. The atoms of predicate logic are subjects and predicates of simple statements.
 * Definition:Universal Quantifier
 * The symbol $\forall$ is called the universal quantifier. It expresses the fact that, in a particular universe of discourse, all objects have a particular property. That is: $\forall x:$ means: For all objects $x$, it is true that ... In the language of set theory, this can be formally defined: $\forall x \in S: \map P x := \set {x \in S: \map P x} = S$ where $S$ is some set and $\map P x$ is a propositional function on $S$.
 * Definition:Existential Quantifier
 * The symbol $\exists$ is called the existential quantifier. It expresses the fact that, in a particular universe of discourse, there exists (at least one) object having a particular property. That is: $\exists x:$ means: There exists at least one object $x$ such that ... In the language of set theory, this can be formally defined: $\exists x \in S: \map P x := \set {x \in S: \map P x} \ne \O$ where $S$ is some set and $\map P x$ is a propositional function on $S$.
 * Definition:Free Variable
 * Let $x$ be a variable in an expression $E$. $x$ is a free variable in $E$ it is not a bound variable. In the context of predicate logic, $x$ is a free variable in $E$  it has not been introduced by a quantifier, either: the universal quantifier $\forall$ or the existential quantifier $\exists$.
 * Definition:Language of Predicate Logic
 * Definition:Language of Predicate Logic/Formal Grammar
 * Definition:Structure for Predicate Logic


 * Definition:By Hypothesis
 * Definition:Mutatis Mutandis
 * Definition:WLOG


 * Definition:Basis for the Induction
 * Definition:Induction Hypothesis
 * Definition:Induction Step


 * Definition:RHS