hcju/proofwikips huggingface dataset:
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:Truth Value
=== Aristotelian Logic === {{:Definition:Truth Value/Aristotelian Logic}} {{expand|Other logic types}}
Definition:Logic
'''Logic''' is the study of the structure of statements and their truth values, divorced from their conceptual content. It has frequently been defined as ''' the science of reasoning'''.
Definition:Atom (Logic)
In a particular branch of logic, certain concepts are at such a basic level of simplicity they can not be broken down into anything simpler. Those concepts are called '''atoms''' or described as '''atomic'''. Different branches of logic admit different '''atoms'''. === Propositional Logic === In propositional logic, the '''atoms''' are statements.
Definition:Subject
The '''subject''' of a simple statement in logic is the part of the statement specifying the object being talked about. The '''subject''' of a simple statement is atomic in predicate logic. The '''subject''' and predicate of a simple statement are referred to as its terms.
Definition:Predicate
The '''predicate''' of a simple statement in logic is the part of the statement which defines ''what is being said'' about the subject. It is a word or phrase which, when combined with one or more names of objects, turns into a meaningful sentence. The predicates of simple statements are atomic in predicate logic. The subject and predicate of a simple statement are referred to as its terms.
Definition:Simple Statement
A '''simple statement''' is a statement which has one subject and one predicate. For example, the statement: :'''London is the capital of England''' is a '''simple statement'''. '''London''' is the subject and '''is the capital of England''' is the predicate.
Definition:Logical Term
Both the subject and the predicate of a simple statement are referred to as its '''(logical) terms'''. Note that this use of the word '''term''' is a more specialized use of the word term as used in algebra.
Definition:Object
An '''object''' is a ''thing''. Everything that can be talked about in mathematics and logic can be referred to as an '''object'''.
Definition:Symbol
In its broadest possible sense: :A '''symbol''' is an object used to represent another object. In a narrower and more "mathematical" sense, a '''symbol''' is a sign of a particular shape to which is assigned a meaning, and is used to represent a concept or identify a particular object. It is generally much more convenient to use a symbol than the plain speaking that it replaces, because it is invariably more compact. One character can replace a large number of words. As definitions become more complex, the symbols tend to convey more information -- but by the same coin, understanding exactly what a symbol means becomes more difficult. Symbols may mean different things in different contexts. A symbol that means something in one context may mean something completely different in another. This is because the number of different concepts is greater than human ingenuity can create symbols for, and some of them naturally have to be used more than once. This does not matter as long as, before we use any symbol, we define exactly what we mean by it. Some symbols are standard and rarely need defining, but in some contexts there are subtle differences to the ''exact'' meaning of a "standard" symbol. Therefore all fields of mathematics generally introduce themselves with a rash of definitions, many of which are symbols.
Definition:Variable/Predicate Logic
In the context of predicate logic, a variable is often called an '''object variable''' or '''arbitrary name'''. As such, it is a symbol which is assigned to an ''arbitrarily selected'' object from a given universe of discourse. The understanding is that (during the scope of the argument to which it is relevant) the arbitrary name could apply equally well to ''any'' of the objects in that universe.
Definition:Operation/Arity
The '''arity''' of an operation is the number of operands it uses. The '''arity''' of an operation may be, in general, any number. It may even be infinite.
Definition:Function
The process which is symbolised by an operation is called a '''function'''. The operand(s) of the operation can be considered to be the input(s). The output of the '''function''' is whatever the operation is defined as doing with the operand(s). A '''function''' is in fact another name for a mapping, but while the latter term is used in the general context of set theory and abstract algebra, the term '''function''' is generally reserved for mappings between sets of numbers.
Definition:Equals
:$x = y$ means '''$x$ is the same object as $y$''', and is read '''$x$ equals $y$''', or '''$x$ is equal to $y$'''. :$x \ne y$ means '''$x$ is not the same object as $y$''', and is read '''$x$ is not equal to $y$'''. The expression: :$a = b$ means: :$a$ and $b$ are names for the same object.
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: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: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. There are various formal systems allowing for rigid determination of the theorems of '''predicate logic''':
Definition:Statement Label
A '''statement label''' is a symbol which is assigned to a ''particular'' statement, so that it can be identified without the need to write it out in full. For such conventions to make sense, different statements must always be given different '''statement labels'''. The citing of a statement label can be interpreted as an assertion that the statement represented by that symbol is true. That is: :'''$P$''' means :'''$P \text { is true}$'''
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. === Unary Logical Connective === {{:Definition:Logical Connective/Unary}} === Binary Logical Connective === {{:Definition:Logical Connective/Binary}}
Definition:Natural Language
A '''natural language''' is one of the conventional, everyday languages in which people usually communicate. Although there are many natural languages in the world, we are not generally going to distinguish between them, merely lumping them all into the one concept.  When '''natural language''' is referred to on , it will usually mean English.
Definition:Compound Statement
A '''compound statement''' is a statement which results from the application of one or more logical connectives to a collection of simple statements. === Substatement === {{:Definition:Compound Statement/Substatement}} === Ill-Formed Compound Statement === {{:Definition:Compound Statement/Ill-Formed}}
Definition:Compound Statement/Substatement
A '''substatement''' of a compound statement is one of the statements that comprise it.
Definition:Statement Form
A '''statement form''' is a symbolic representation of a compound statement. It consists of statement variables along with logical connectives joining them. It is traditional, particularly in the field of mathematical logic, to use lowercase Greek letters to stand for general formulas (the usual ones being $\phi, \psi$ and $\chi$), but more modern treatments are starting to use ordinary lowercase letters of the English alphabet, usually $p, q, r$ etc. === Specific Form === {{:Definition:Statement Form/Specific Form}}
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:Logical Connective/Unary
A '''unary logical connective''' (or '''one-place connective''') is a connective whose effect on its compound statement is determined by the truth value of ''one'' substatement. In standard Aristotelian logic, there are four of these. The only non-trivial one is logical not, as shown on Unary Truth Functions.
Definition:Logical Connective/Binary
A '''binary logical connective''' (or '''two-place connective''') is a connective whose effect on its compound statement is determined by the truth value of ''two'' substatements. In standard Aristotelian logic, there are 16 '''binary logical connectives''', cf. Binary Truth Functions. In the field of symbolic logic, the following four (symbols for) '''binary logical connectives''' are commonly used: * Conjunction: the '''And''' connective $p \land q$: '''$p$ is true ''and'' $q$ is true'''. * Disjunction: the '''Or''' connective $p \lor q$: '''$p$ is true ''or'' $q$ is true, ''or possibly both'''''. * The conditional connective $p \implies q$: '''''If'' $p$ is true, ''then'' $q$ is true'''. * The biconditional connective $p \iff q$: '''$p$ is true ''if and only if'' $q$ is true''', or '''$p$ ''is equivalent to'' $q$'''.
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:Logical Argument
A '''logical argument''' (or just '''argument''') is a process of creating a new statement from one or more existing statements. An '''argument''' proceeds from a set of premises to a conclusion, by means of logical implication, via a procedure called logical inference. An '''argument''' may have more than one premise, but only one conclusion. While statements may be classified as either '''true''' or '''false''', an '''argument''' may be classified as either valid or invalid.
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:Conclusion
A '''conclusion''' is a statement that is obtained as the result of the process of an argument.
Definition:Valid Argument
A '''valid argument''' is a logical argument in which the premises provide conclusive reasons for the conclusion. When a proof is valid, we may say one of the following: * The conclusion '''follows from''' the premises; * The premises '''entail''' the conclusion; * The conclusion is true '''on the strength of''' the premises; * The conclusion is '''drawn from''' the premises; * The conclusion is '''deduced from''' the premises; * The conclusion is '''derived from''' the premises.
Definition:Proof
A '''proof''' is another name for a valid argument, but in this context the assumption is made that the premises are all true. That is, a valid argument that has one or more false premises is not a proof.
Definition:Invalid Argument
An '''invalid argument''' is a argument in which the premises do ''not'' provide conclusive reasons for the conclusion.
Definition:Depend
A statement $q$ '''depends upon''' another statement $p$ if the truth value of $q$ is ''influenced'' (in some way) by that of $p$, but is ''not necessarily'' implied by that of $p$. Some authors prefer '''rests on''' for '''depends upon'''.
Definition:Therefore
If statement $p$ logically implies statement $q$, then we may say: :'''$p$, therefore $q$'''. The symbology: :$p, q \vdash r$ means: :'''Given as premises $p$ and $q$, we may validly conclude $r$''' So the symbol $\vdash$ is interpreted to mean '''therefore'''. Thus, $p, q \vdash r$ reads as: :'''$p$ and $q$, therefore $r$.''' A fallacy may be indicated by $p, q \not \vdash r$, which can be interpreted as: :'''Given as premises $p$ and $q$, we may ''not'' validly conclude $r$.'''
Definition:Because
If statement $p$ logically implies statement $q$, then we may say: : '''$q$, because $p$.''' The symbol $\dashv$ is interpreted to mean '''because'''. Thus: : $r \dashv p, q$ means: :'''Given as premises $p$ and $q$, we may validly conclude $r$''' or :'''$r$, because $p$ and $q$.'''
Definition:Fallacy
A '''(logical) fallacy''' is a mistake caused by an application of an invalid argument. '''Logical fallacies''' abound. Most of the documented '''fallacies''' that can be found in the literature arise from linguistic imprecision or deliberately misleading statements, or both. Frequently the commission of '''fallacies''' is as deliberate as the telling of lies.
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:Zermelo-Fraenkel Set Theory
'''Zermelo-Fraenkel Set Theory''' is a system of axiomatic set theory upon which the whole of (at least conventional) mathematics can be based. Its basis consists of a system of Aristotelian logic, appropriately axiomatised, together with the Zermelo-Fraenkel axioms of set theory. These are as follows: {{:Definition:Zermelo-Fraenkel Axioms}}
Definition:Axiomatic Set Theory
'''Axiomatic set theory''' is a system of set theory which differs from so-called naive set theory in that the sets which are allowed to be generated are strictly constrained by the axioms.
Definition:ZFC
'''Zermelo-Fraenkel Set Theory with the Axiom of Choice''' is a system of axiomatic set theory upon which the whole of (at least conventional) mathematics can be based. Its basis consists of a system of Aristotelian logic, appropriately axiomatised, together with the Zermelo-Fraenkel axioms of Set Theory and the (controversial) Axiom of Choice. These are as follows: {{:Definition:Zermelo-Fraenkel Axioms}} === The Axiom of Choice === {{:Axiom:Axiom of Choice}}
Definition:Set Theory
'''Set Theory''' is the branch of mathematics which studies sets.
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'''. Thus the statement $\neg p$ is called the '''negation''' of $p$. $\neg p$ is voiced '''not $p$'''.
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''' : the '''members of the conjunction'''. $p \land q$ is voiced: :'''$p$ and $q$'''.
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 $q$. $p \lor q$ is voiced: :'''$p$ or $q$'''
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'''. $p \implies q$ is voiced: :'''if $p$ then $q$''' or: :'''$p$ implies $q$'''
Definition:Biconditional
The '''biconditional''' is a binary connective: :$p \iff q$ defined as: :$\paren {p \implies q} \land \paren {q \implies p}$ That is: :'''''If'' $p$ is true, ''then'' $q$ is true, ''and if'' $q$ is true, ''then'' $p$ is true'''.
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:Binding Priority
The '''binding priority''' is the convention defining the order of '''binding strength''' of the individual connectives in a logical formula. '''Binding priorities''' can be overridden by using parenthesis in appropriate places. Parenthesis always takes priority over conventional '''binding priorities'''.
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:Contingent Statement
A '''contingent statement''' is a statement form which is neither a tautology, nor unsatisfiable, but whose truth value depends upon the truth value of its component substatements. <!-- == Formal Definition == Let $\mathfrak M$ be a collection of models for a particular formal language $\LL$. A well-formed word $\mathbf A$ of $\LL$ is said to be '''contingent (for $\mathfrak M$)''' : :$\exists \MM_1, \MM_2 \in \mathfrak M: \MM_1 \models \mathbf A \land \MM_2 \not \models \mathbf A$ that is,  some models in $\mathfrak M$ approve, and others disapprove of $\mathbf A$. == Also known as == {{refactor|Seems to be replaceable by Also see entries. But, will need source review}} In the context of propositional formulas the term '''satisfiable''' is usually used: A propositional formula is '''satisfiable''' if its value is True in at least ''one'' boolean interpretation. A propositional formula is '''not-valid''' or '''falsifiable''' if its value is False in at least ''one'' boolean interpretation. === Set of Logical Formulas === Let $U = \set {P_1, P_2, \ldots, P_n}$ be a set of propositional formulas. Let $U' = \set {p_1, p_2, \ldots, p_m}$ be the set of all the atoms of all the propositional formulas in $U$. (Some of these atoms, and indeed this will most likely be the case, may be in more than one logical formula.) Then $U$ is '''(mutually) satisfiable''' if there exists a boolean interpretation $v$ for all the atoms in $U'$ such that: :$\map v {P_1} = \map v {P_2} = \cdots = \map v {P_n} = \T$. -->
Definition:Natural Deduction
'''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, by a technique called logical inference.
Definition:Natural Deduction/Proof Rule
A '''proof rule''' is a rule in natural deduction which allows one to infer the validity of propositional formulas from other propositional formulas. === Rule of Substitution === {{:Rule of Substitution}} === Rule of Sequent Introduction === {{:Rule of Sequent Introduction}} === Rule of Theorem Introduction === {{:Rule of Theorem Introduction}}
Definition:Logical Inference
'''Logical inference''' is the process used in natural deduction to deduce the validity of statement forms from other statement forms by use of proof rules. Given a set of logical formulae and the proof rules, we '''(logically) infer''' other formulas.
Definition:Pool of Assumptions
The '''pool of assumptions''', for a formula deduced in a particular proof by natural deduction, is the collection of all the assumptions upon which the formula depends. These consist of premises and possibly some intermediate assumptions. The assumptions in this pool are called, naturally enough, the '''pooled assumptions'''. The '''pooled assumptions''' at the end of a proof (its conclusion) are thus seen to be the premises.
Definition:Aristotelian Logic
'''Aristotelian logic''' is a system of logic which is based upon the philosophy of {{AuthorRef|Aristotle}}. It forms the cornerstone of the entirety of classical logic. The school of '''Aristotelian logic''' consists almost entirely of the relationships between the various categorical syllogisms. This school of philosophy forms the basis of mainstream mathematics, although, for example, mathematicians of the intuitionistic school do not accept the Law of the Excluded middle value.
Definition:Hypothesis
A '''hypothesis''' is a statement that is made whose truth value has not been established, but which is ''believed'' to be true. The underlying idea is that its truth or falsehood is '''about to be investigated'''. A hypothesis is considered by some branches of philosophy as being synonymous with the words '''wild guess'''[http://www.experiencefestival.com/a/Hypothesis/id/183571]. Possibly the most famous still-unproved hypothesis is the Riemann Hypothesis.
Definition:Discharged Assumption
The context of this definition is in the process of deducing the validity of sequents using logical inference in the framework of natural deduction. Assumptions that are made during the course of using certain proof rules remain in force only so long as the use of those rules requires them. Once the appropriate rules have been completed, these are known as '''discharged assumptions''', and are not included in the pool of assumptions on which the conclusion of the rule depends.