Glossary of Terms for Advanced Symbolic Logic

H. Hamner Hill

Southeast Missouri State University


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A

Absolute complement of a set.
See complement.

Absolute consistency.
A system S is absolutely consistent iff at least one wff of the formal language of S is not a theorem.
  • Absolute inconsistency. A system is absolutely inconsistent iff all its wffs are theorems.

Adjectival numerals.
See numeral.

Antecedent.
See implication.

Antitheorem.
A wff whose negation is a theorem. (This is a new term not yet widely used; I include it here to help it catch on.) See theorem.

Argument.
(1) an inference, (2) input to a function, (3) a subject-term for a predicate. See corresponding argument.

Arithmetic, formal system of.
A first-order theory with a finite alphabet, and only finitely long wffs which, on its intended interpretation (1) contains theorems that express truths of number theory, and (2) permits the construction of a term to denote any natural number. See first-order theory.

Arithmetization.
A generalized form of Gödel numbering in which distinct numerals are assigned to distinct symbols in the alphabet of a formal language. As a result every wff of the language can be re-expressed as a numeral (concatenating all the numerals for its component symbols). When done well, there can easily be an effective method for translating wffs into numerals and vice versa. See Gödel numbering.

Associated propositional formula (APF).
A wff A of propositional logic created from a wff B of predicate logic by (1) removing the quantifiers from B, and (2) replacing each predicate symbol (and its arguments) in B with a propositional symbol. Notation: Bprop = p.

Atom.
(1) In propositional logic, a simple proposition as opposed to a compound proposition or molecule. A wff without connectives. (2) In predicate logic, a wff without quantifiers or connectives.

Attribute.
A property of an object; also (at a different level) a monadic predicate symbolizing such a property. See predicate logic; relation.

Axioms.
Wffs that are stipulated as unproved premises for the proof of other wffs inside a formal system.
  • Axiom schema (plural: schemata). A formula containing variables of the metalanguage which becomes an axiom when its variables are instantiated to wffs of the formal language.
  • Logical axiom. An axiom that is a logically valid wff of the language of the system. See logical validity.
  • Proper axiom. An axiom that is not a logically valid wff of the language of the system (but is a closed wff).

Axiom of choice.
In set theory, a controversial axiom asserting that for a non-empty set A of non-empty disjoint sets, there is a set B with exactly one member from each of the disjoint sets comprising A. Sometimes the axiom is written so as to assert that there is a function for choosing the members of the disjoint sets comprising A that will become the members of B. Cantor's generalized continuum hypothesis implies the axiom of choice. Usually abbreviated to AC. Also called the multiplicative axiom. See set theory.


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B

Basis.
See mathematical induction.

Biconditional.
See equivalence.

Bound variable.
In predicate logic, an individual variable at least one of whose occurrences lies within the scope of a quantifier on the same letter. Because other occurrences may be free, a variable may be both free and bound in the same wffs. See closure; free variable; wff.
  • To bind a variable. To add a quantifier on an individual variable, x, to a wff so that one or more previously free occurrences of x lie inside the scope of that quantifier.


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C

c (lower-case "c", often in Gothic type).
The symbol for the cardinality of the continuum; c = 2Ą0, or (assuming the continuum hypothesis) Ą1. See continuum hypothesis.

Cantor's theorem.
The power set of a given set has a greater cardinality than the given set.

Cantorian set theory.
See set theory.

Cardinality (of a set).
The number of elements in the set; intuitively, the set's "size" or "magnitude". Notation: a double bar over the symbol denoting the set; also "|S|" and "card S" when S is the symbol denoting the set.

Categoricity of systems.
(1) A formal system in general is categorical iff all its models are isomorphic. (2) A first-order theory with identity is categorical iff all its normal models are isomorphic. See first-order theory; isomorphism of models; model, normal.
  • Alpha-categoricity. A first order theory is -categorical iff (1) it has a normal model of cardinality and (2) any two normal models of cardinality are isomorphic.

Church's theorem.
If Church's thesis is true, then polyadic predicate logic is undecidable. See Church's thesis.

Church's thesis.
Originally and narrowly, the thesis that all intuitively effective methods are generally recursive (in the sense of this term used in recursive function theory). Due to Alonzo Church, 1935. Currently and more broadly, the thesis that all intuitively effective methods are captured in any one of several formalizations, including recursive function theory, Turing machines, Markov algorithms, the Lambda calculus, and so on. See effective method; recursive function theory.

Closure of a wff.
In predicate logic, the binding of all free variables from a wff by placing them within the scope of suitable quantifiers. A closed wff is considered its own closure. The closure of a closure of a wff A is considered a closure of A. Notation: Ac (a closure of wff A). Closed wffs are also called sentences. See bound variable; free variable; generalization; instantiation; wff.

Complement of a set.
The complement of a set A is the set of elements that are not members of A. Notation: .
  • Absolute complement of a set. The set of all things whatsoever that are not members of the given set. Standard set theory does not recognize absolute complements. See Russell's paradox.
  • Relative complement of a set. The set of all things that are not members of the given set, A, but that are members of some particular "background" set, B. This can be expressed through the notation for set difference: the relative complement of A in B or relative to B (A against the background of B) is the set {x : (x B) · (x A)}. The background set is sometimes called the universe or universe of discourse. Notation: B-A, or B\A. See universe of discourse.

Completeness.
See negation completeness; omega-completeness; semantic completeness; syntactic completeness.

Component.
A proposition that is part of a compound proposition. A component may itself be compound. For example, p is a component in pq, and p®q is a component in (p®q)r.

Composition (of a function).
One of the simple function-building operations of recursive function theory. Given the one-place functions f(x) and g(x), composition allows us to create function h thus: h(x) = f(g(x)). More generally, if f is an m-place, f(x1...xm), and there is a series of n-place functions g, g(x1...xn), then we can create the n-place function h by composition: h(x1...xn) = f(g(x1...xn),...,gm(x1...xn)). Also called substitution. See recursive function theory.

Compound proposition.
A proposition made up of two or more simple propositions (components) joined by a connective. A compound proposition has just one truth-value for a given interpretation. Also called "molecules" (by writers who call simple propositions "atoms").

Computable function.
A total function for which there is an effective method for determining the value (output, member of the range), given the arguments (inputs, members of the domain). See effective method; total function.
  • Incomputable function. A function for which there is no such effective method.

Conclusion.
The result of an argument or inference. The wff derived from or supported by premises. See argument; inference; premise.

Conditional.
See implication.

Conjunction.
A truth-function that is true when both its arguments (called conjuncts) are true. Also the connective denoting this function; also the compound proposition built from this connective. Notation: p · q; sometimes also pq or pŁq.

Conjunctive normal form (CNF).
The form of a of truth-functional compound when it is expressed as a series of conjuncts when each conjunct is either a simple proposition or the disjunction of simple propositions and the negations of simple propositions. See disjunctive normal form.

Connective.
A symbol that functions to join two or more propositions into a compound proposition. Sometimes applied to symbols (like "~" for negation) which apply only to one proposition at a time. Sometimes applied to the function denoted by the symbol, rather than the symbol itself.
  • A truth-functional connective is a truth-function; its components are its arguments and the truth-value of the compound it forms is its value. See truth function; truth-functional compound proposition; truth-functional connective.
  • Connectives that apply to only one proposition at a time are monadic; those that join two propositions are dyadic; those that join three are triadic, and so on. Monadic connectives are also called operators.

Consequence.
See semantic consequence; syntactic consequence.

Consequent.
See implication.

Consistency.
See absolute consistency; model-theoretic consistency; omega-consistency; proof-theoretic consistency; relative consistency proof; simple consistency.

Constant.
A symbol whose referent has been fixed. An abbreviation or name, as opposed to a place-holder (a variable). See variable.
  • Individual constant. A symbol standing for an individual object from the domain of a system.
  • Predicate constant. A symbol standing for an attribute or relation.
  • Propositional constant. A symbol standing for a proposition.

Constructive proof.
A proof that actually produces an example of that which it proves to exist (which might be a number, wff, function, proof, etc. with certain properties). See existence proof.

Contingency.
In truth-functional propositional logic, any proposition that is neither a tautology nor a contradiction, hence any proposition that is sometimes true, sometimes false, depending on the row of its truth table column or the interpretation. See contradiction; tautology.

Continuum.
The numerical continuum is the series of real numbers; the linear continuum is the series of points on a geometrical line.

Continuum hypothesis.
There is no cardinal, a, such that Ą0 < a < c, where c is the cardinality of the continuum. Proving or disproving the continuum hypothesis was the first problem on Hilbert's famous list of problems in 1900. Gödel (1938) and Cohen (1963) have proved that it is neither provable nor disprovable from standard set theory. Usually abbreviated to CH. See set theory.
  • Generalized continuum hypothesis. For every transfinite cardinal, a, there is no cardinal b such that a < b < 2a. Usually abbreviated to GCH.

Contradiction.
(1) The conjunction of any proposition and its negation, (2) in truth-functional propositional logic, the negation of any tautology, hence any proposition that is false in every row of its truth table or in every interpretation. See contingency; tautology.

Corresponding argument or derivation.
Every conditional statement, A®B, can be reexpressed as a derivation, AB, called the corresponding argument or derivation of the conditional.

Corresponding conditional.
Every derivation, A1, A2,...An B, can be reexpressed as a conditional statement, (A1 · A2 · ...· An)®B, called the corresponding conditional of the argument.

Countable set.
A set whose cardinality is either finite or Ą0. See uncountable set.


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D

Dagger function.
The dyadic connective or truth function "neither/nor". One of only two dyadic connectives capable of expressing all truth functions by itself. Notation: p q. Also called joint denial. See stroke function.

Decidable set.
A set for which there is an effective method to determine whether any given object is a member. See also effective method, recursive set.
  • Undecidable set. A set for which there is no such effective method.

Decidable system.
A formal system in which there is an effective method for determining whether any given wff is a theorem. A system in which the set of theorems is a decidable set. The question whether a system is decidable is often called the Entscheidungsproblem, or decision problem. See decidable set, effective method, effective proof procedure.
  • Undecidable system. A system for which there is no such effective method.

Decidable wff.
A wff that is either a theorem or the negation of a theorem. Either the wff or its negation is a theorem. Jargon: if wff A is decidable in system S, we often say that "S decides A".
  • Undecidable wff. A wff that is neither a theorem nor the negation of a theorem.

Deduction.
An inference in which (when valid) the conclusion contains no information that was not already present in the premises, or whose corresponding conditional is a tautology. See corresponding conditional; induction; tautology; validity.

Deduction theorem.
When Γ is a set of wffs, and A and B are wffs, then if Γ, AB, then Γ(A®B). Also called the rule of conditional proof.

Deductive apparatus.
The axioms and rules of inference of a formal system. Formal systems may lack axioms or rules of inference but not both. See axioms; rules of inference.

Definability of a function.
A function f of one argument is definable in a system iff it is strongly represented in the system, say by wff Axy, and (x)(y)(z)[Axy®(Axz ® y=z)]. See representation of a function.

Denumerable set.
A set whose cardinality is exactly Ą0, for example the set of natural numbers.

Derivation.
A finite, non-empty sequence of wffs in which the last member is the wff derived, and each of the others (the premises) is either an axiom, a member of a set of accepted premises, or the result of applying a rule of inference to wffs preceding it in the sequence. See corresponding argument; proof. Notation: ΓA (the wff A can be derived from the set of wffs Γ).

Difference of sets.
The difference of set B from set A is the set of all members of A that are not also members of B. Notation: A-B, or A\B. A-B =df {x : (xA)·(xB)}

Disjoint sets.
Two sets are disjoint iff they share no members, i.e. iff their intersection is the null set.

Disjunction.
A truth function that is true when one or the other of its components (called disjuncts) is true, and false otherwise. Also the connective denoting this function; also the compound proposition built from this connective.
  • Exclusive disjunction. One or the other of the disjuncts is true, but not both. Notation: no standard symbol, but the concept is accurately captured thus: p q (negation of material equivalence).
  • Inclusive disjunction. One or the other or both of the disjuncts is true. Notation: pq.

Disjunctive normal form (DNF).
The form of a of truth-functional compound when it is expressed as a series of disjuncts when each disjunct is either a simple proposition or the conjunction of simple propositions and the negations of simple propositions.

Domain.
(1) Of a function, the set of objects or sequences of objects that may serve as the arguments (inputs) of the function. (2) Of an interpretation of a formal language of predicate logic, the set of objects that may serve as the assigned referents of the constants of the language, the arguments of functions, and the arguments of predicates.
  • Cardinality of a domain. The cardinality of the set of objects comprising the domain.


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E

Effective enumeration.
See enumerable set.

Effective method (for a class of problems).
A method for solving problems in the class when the method (1) is logically bound as opposed to physically bound (2) to give some answer, as opposed to no answer, (3) that is correct, as opposed to incorrect, (4) in a finite number of steps, as opposed to an infinite number, (5) every time, or for all inputs, or for all problems in the class, as opposed to selectively, (6) if the method is followed carefully, as opposed to carelessly, (7) as far as necessary, as opposed to only as far as our resources permit, (8) when each step in the process is "dumb" or "mechanical". The eighth requirement introduces an irreducibly intuitive element into the definition. Some add (9) when given a problem from outside the the class for which the method is effective, the method may halt or loop forever without halting, but must not return a value as if it were the answer to the problem. (The wording of this definition was influenced by Geoffrey Hunter.) Also called algorithm; decision procedure. See Church's Thesis.

Effective proof procedure.
An effective method for generating the proof of any theorem in a formal system. A system for which there exists an effective proof procedure is decidable; but not all decidable systems have effective proof procedures. See decidable system.

Enumerable set.
Roughly, a set that can be translated into a sequence. More precisely, a set such that every one of its members has at least one counterpart in a certain sequence (though they may have more than one counterpart), and every term in the sequence has a counterpart in the set. The resulting sequence is called an enumeration of the set. The set {a, b, c} is enumerated by the sequence <a, b, c>, but also by the sequence <a, a, c, b>; it is not enumerated by the sequence <a, a, c, c>.
  • Effectively enumerable set. An enumerable set for which there is an effective method for ascertaining the nth term of the sequence for every positive integer n.
  • Recursively enumerable set. A set that is effectively enumerable by some recursive function. Under Church's thesis, a set is recursively enumerable iff it is effectively enumerable. See Church's thesis; recursive function.

Epimenides paradox.
See liar paradox.

Equivalence.
A truth function that returns truth when its two arguments have the same truth-value, and false otherwise. Also the connective denoting this function; also the compound proposition built from this connective. Syntactically: the two propositions imply one another. Semantically: they have the same models. Also called a biconditional, or biconditional statement.
  • Logical equivalence. A tautologous statement of material equivalence (next).
  • Material equivalence. A truth function that is true when its two arguments have the same truth-value (not necessarily the same meaning). Notation: pq, or p iff q.

Equivalent sets.
Two sets are equivalent iff they have the same cardinality, that is, if they can be put into one-to-one correspondence. Also called equinumerous sets. Notation: AB; sometimes A~B.

Exclusive disjunction.
See disjunction.

Existence proof.
A proof that something exists (e.g. a number, wff, proof, etc. with certain properties) but that does not produce an example. See constructive proof.

Existential import.
Quantified statements have existential import iff (in the standard interpretation) they are taken to assert the existence of their subjects. Aristotle held that all quantified propositions have existential import. The modern view, due to George Boole, is that existentially quantified statements do and that universally quantified statements do not. Hence in the modern view, (x)(Ax Bx) ("All A's are B's") is non-committal on the existence of any A's; it may be true even for an interpretation whose domain contains no objects to instantiate x, or none that happen to be A's. By contrast, (x)(Ax·Bx) ("Some A's are B's") asserts the existence of at least one A, and it would be false for any interpretation whose domain contained no such values for x. See predicate logic, inclusive; quantifier.

Existential quantifier.
See quantifier.

Extension of a system.
A system S' is an extension of S iff every theorem of S is a theorem of S'. It follows that every model of S' is a model of S.
  • Finite extension of a system. System S' is a finite extension of S iff S' has all of the axioms of S, and differs from S only by adding a finite number of additional axioms that are wffs but not axioms of S.


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F

Finitary formal system.
A formal system in which (1) there are countably many symbols in the formal language, (2) wffs are finite in length, and (3) every rule of inference takes only a finite number of premises.

First-order logic.
See predicate logic.

First-order theory (FOT).
A formal system of first-order predicate logic in which (1) there may be countably many new individual constants in the formal language, provided they are effectively enumerable, and (2) there may be countably many proper axioms to supplement the logical axioms. See proper axiom.
  • First-order theory with identity. A first-order theory with (x)(x=x) as an axiom, and the following axiom schema, [(x=y) (AA')]c, when Bc is an arbitrary closure of B, and when A' differs from A only in that y may replace any free occurrence of x in A so long as y is free wherever it replaces x (y need not replace every occurrence of x in A). See identity; predicate logic with identity.

Formal language.
An alphabet and grammar. The alphabet is a set of uninterpreted symbols. The grammar is a set of rules that determine which strings of symbols from the alphabet will be acceptable (grammatically correct or well-formed) in that language. The grammar may also be conceived as a set of functions taking strings of symbols as input and returning either "yes" or "no" as output. The rules of the grammar are also called formation rules. See decidable system; finitary formal system; wff.

Formal system.
A formal language (alphabet and grammar) and a deductive apparatus (axioms and rules of inference). See arithmetic, formal system of; categoricity of systems; closure of a system; decidable system; formal language; deductive apparatus.

Formation rules.
See formal language.

Formula.
A string of symbols from the alphabet of a formal language. It may or may not conform to the grammar of the formal language; if it does it is also called a well-formed formula or wff. See wff.

Free variable.
In predicate logic, an individual variable at least one of whose occurrences in a wff does not lie within the scope of a quantifier on the same letter. Because other occurrences may be bound, a variable may be both free and bound in the same wff. See bound variable; closure; wff.
  • Free occurrence of a variable. Any occurrence of an individual variable not within the scope of a quantifier on the same letter.

Function.
A rule for associating a member or a sequence of members of one set (the domain) with a member of another set (the range). See composition; computable function; definability of a function; minimization; n-adic function; partial function; primitive recursion; propositional function; recursive function; recursive function theory; representation of a function; total function; truth function.


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G

Generalization.
To add a quantifier to a wff so that it either binds previously free variables, or binds new variables substituted for constants. See bound variables; free variables; instantiation; quantifier.
  • Existential generalization. To generalize using the existential quantifier. For example to move from propositional functions like Px or propositions like Pa to (x)Px; from "x is purple" or "alabaster is purple" to "something is purple". Valid without restriction.
  • Universal generalization. To generalize using the universal quantifier. For example to move from propositional functions like Px or propositions like Pa to (x)Px; from "x is purple" or "alabaster is purple" to "everything is purple". Valid only under several restrictions.

Gödel numbering.
A code in which distinct numerals are assigned to the expressions of a language such that we can tell from the numeral whether it is assigned to a symbol, a sequence of symbols (potential wff), or a sequence of wffs (potential proof). There must be an effective method for translating symbols, formulas, or sequences of formulas into their Gödel numbers, and vice versa. See arithmetization.

Gödel's theorems.
Any of many theorems proved by Kurt Gödel. The following three are normally named after Gödel. When one hears simply "Gödel's theorem" it usually refers to the first incompleteness theorem.
  • Gödel's completeness theorem (1930). First-order polyadic predicate logic is semantically complete, that is, all logically valid wffs are theorems.
  • Gödel's first incompleteness theorem (1931). Roughly, any consistent or omega-consistent formal system of arithmetic of "sufficient strength" is incomplete (negation incomplete and omega-incomplete). To be of sufficient strength, the system must (1) have decidable sets of wffs and proofs, and (2) represent every decidable set of natural numbers. See arithmetic, formal systems of; representation of a set.
  • Gödel's second incompleteness theorem (1931). The consistency of a system of "sufficient strength" (same as for the first incompleteness theorem) is not provable in the system, unless the system is inconsistent. The second incompleteness theorem is a corollary of the first. See Hilbert's program.

Grammar.
See formal language; wff.

Grelling's paradox.
If an adjective truly describes itself, call it "autological", otherwise call it "heterological". For example, "polysyllabic" and "English" are autological, while "monosyllabic" and "pulchritudinous" are heterological. Is "heterological" heterological? If it is, then it isn't; if it isn't, then it is. Grelling's paradox cannot be expressed in first-order predicate logic, and is difficult to prevent in higher-order predicate logics.


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H

Heredity.
A property possessed by all the wffs in a set is logically hereditary iff the accepted rules of inference pass it on (transmit it) to all the conclusions derivable from that set by those rules.

Heterological paradox.
See Grelling's paradox.

Higher-order logic.
See predicate logic.

Hilbert's program.
An attempt to avoid both relativity and vicious circularity in the proof of the consistency of formal systems of arithmetic, by using only a small set of extremely intuitive operations to prove the consistency of the system containing that set. (A second phase of the program was to build all of mathematics on the system thus certified to be consistent.) Hopes of accomplishing Hilbert's program were dashed by Gödel's second incompleteness theorem. See Gödel's theorems; relative consistency proof.


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I

Identity.
A 2-adic predicate, say Ixy, asserting that its two arguments are identical. Customarily symbolized by "=" and written in infix notation, "x=y". While all systems of polyadic predicate logic can express identity as easily as any other 2-adic relation, a system is said to be "with identity" iff it also contains axioms, axiom schemata, and/or rules of inference determining how "=" is to be used. Note that an axiom like "(x)(x=x)" or "(x)Ixx" is not logically valid because there are interpretations of "=" or "I" that do not take the meaning of identity. See first-order theory with identity; predicate logic with identity; interpretation, normal.

Iff.
Abbreviation of "if and only if", which designates material equivalence. See equivalence, material.

Implication.
A statement of the form, "if A, then B," when A and B stand for wffs or propositions. The wff in the if-clause is called the antecedent (also the implicans and protasis). The wff in the then-clause is called the consequent (also the implicate and apodosis). As a truth function, see material implication, below. Also called a conditional, or a conditional statement. See corresponding conditional.
  • Logical implication. A tautologous statement of material implication (next).
  • Material implication. A truth function that is false when its antecedent is true and its consequent false, and true otherwise. Also the connective that denotes this function; also the compound proposition built from this connective. This truth function is rarely what implication or "if...then" means in English, but it captures the logical core of that usage and is truth-functional. Notation: pq.
  • Paradoxes of material implication. Two consequences of the formal definition of material implication that violate informal intuitions about implication: (1) that a material implication is true whenever its antecedent is false, and (2) that a material implication is true whenever its consequent is true. These so-called paradoxes do not create contradictions.

Inclusive disjunction.
See disjunction.

Inclusive quantification theory.
See predicate logic.

Independence of an axiom.
If "S-A" denotes system S minus axiom A, then for many logicians an axiom A is independent of system S iff neither A nor ~A are theorems of S-A, that is, iff both ~A and ~~A. For some, A is independent of S iff A is not a theorem of S-A, that is, iff ~A alone. When S is consistent, then the latter definition is equivalent to the former, which is equivalent to the undecidability of wff A in S.

Individuals.
The objects or elements taken as the subjects of the predicates of first-order predicate logic. See constant; domain; variable.

Induction.
An inference in which the conclusion contains information that was not contained in the premises. See deduction; mathematical induction.

Induction hypothesis.
See mathematical induction.

Induction step.
See mathematical induction.

Inference.
A series of wffs or propositions in which some (called premises) support another (called the conclusion). Also the act of concluding the conclusion from the premises. See conclusion; deduction; derivation; induction; premise; proof.

Instantiation.
In predicate logic, to remove a quantifier from a wff and either leave the previously bound variables free or replace them with constants. See generalization; quantifier.
  • Existential instantiation. Instantiation from the existential quantifier. For example, to move from statements like (x)Px to Px or Pa; from "something is purple" to "x is purple" or "alabaster is purple". Valid only under several restrictions.
  • Universal instantiation. Instantiation from the existential quantifier. For example, to move from statements like (x)Px to Px or Pa; from "everything is purple" to "x is purple" or "alabaster is purple". Valid without restriction.

Integers.
The natural numbers supplemented by their negative counterparts. The set {...-3, -2, -1, 0, 1, 2, 3...}.

Interpolation theorem.
If (AB), and if A and B share at least one propositional symbol, then there is a wff C all of whose propositional symbols occur in A and B such that (AC) and (CB). A syntactic version of the theorem replaces "" with "".

Interpretation (of a formal language).
The assignment of objects from the domain to the constants of a formal language, truth-values to the proposition symbols, truth-functions to the connectives, other functions to the function symbols, and extensions to the predicates (when these extensions consist of subsets of the domain). These assignments are made by the human logician and are not native to the symbols of the formal language. These assignments can be captured by a function f so that (for example) for a constant, f(c) = object d from domain D; for a proposition, f(p) = true; for a truth-function, f() = material implication; for a function, f(g) = squaring the successor; or for a predicate, f(P) = the set of purple things. In propositional logic, an interpretation is just such a function; in predicate logic, it is some set (the domain) together with such a function defined for members of that domain.

Intersection of sets.
The intersection of two sets, A and B, is the set of elements that are members of both A and B. Notation: AB, or sometimes, AB. Also called the product of A and B. AB =df {x : (xA)·(xB)}

Irrational numbers.
Real numbers that are not equal to the ratio of two integers. In decimal notation, they are the fractions represented by infinite, non-repeating decimal expansions.

Isomorphism of models.
Roughly, when models are identical in form and differ (if at all) only in content. Or when their domains map onto one another in the sense that their elements can be put into one-to-one correspondence and they stand in the same relations. To define isomorphism more precisely, let us say that D and D' are the domains of the two models under comparison, that for every member d of D there is a counterpart d' of D' and vice versa, that every function f defined for D has a counterpart function f' defined for D' and vice versa, and that every predicate P defined for D has a counterpart predicate P' defined for D' and vice versa. Now the two models are isomorphic iff these three conditions are met: (1) D and D' can be put into one-to-one correspondence, (2) for all functions f and f', f(d1...dn) = dn+1 iff f'(d1'...dn') = dn+1' and (3) for all predicates P and P', Pd1...dn iff P'd1'...dn'. See categoricity of systems; Löwenheim-Skolem theorem.


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J


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K

k-validity.
A wff is k-valid iff it is true for every interpretation with a domain of exactly k members. See logical validity; omega-completeness.


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L

Language.
See formal language.

Lemma.
A theorem (or metatheorem) proved only for the sake of another theorem (or metatheorem).

Liar paradox.
"This very statement is false." If it is true, then it is false; if it is false, then it is true. Also called the Epimenides paradox.

Logical axiom.
See axioms.

Logical validity.
For a wff, to be true for every interpretation of the formal language; to have every interpretation be a model. "Every interpretation" here is understood to mean all, but only, those interpretations in which the connectives and/or quantifiers take their standard meanings. In truth-functional propositional logic, logically valid wffs are also called tautologies. In standard predicate logic, logical validity is limited to interpretions with non-empty domains. Logical validity is also called logical truth. See k-validity; model; predicate logic; tautology; true for an interpretation. Notation: A (A is a logically valid wff).

Löwenheim-Skolem theorem.
Every first-order theory with a model has a denumerable model. The theorem implies that consistent first-order theories, including those intended to capture the real numbers or other uncountable sets, will be non-categorical; hence it implies that there is no consistent, categorical description of the reals in a first-order theory. See categoricity of systems; Skolem paradox.
  • Downward Löwenheim-Skolem theorem. If a first-order theory has a model of any infinite cardinality k, then it has a model of every infinite cardinality j, jk.
  • Upward Löwenheim-Skolem theorem. If a first-order theory has a model of any infinite cardinality, then it has a model of any arbitrary infinite cardinality, and hence, models of every infinite cardinality. Variation for systems with identity: if a first-order theory has a normal model of any infinite cardinality, then it has a normal model of any arbitrary infinite cardinality.


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M

Mathematical induction.
A powerful technique for proving that a theorem holds for all cases in a large or infinite set. The proof has two steps, the basis and induction step. Roughly, in the basis the theorem is proved to hold for the "ancestor" case, and in the induction step it is proved to hold for all "descendant" cases. For a more precise definition, see sub-items below. See heredity; induction.
  • Basis. Proof that the theorem in question holds for the minimal case.
  • Induction hypothesis. Assumption that the theorem in question holds (in weak mathematical induction) for an arbitrary case k, or that it holds (in strong mathematical induction) for all cases up to and including k.
  • Induction step. Proof that if the induction hypothesis is true, then the theorem in question holds for case k+1.
  • Strong and weak mathematical induction. Two versions of the induction hypothesis (see above).

Matrix.
In wffs of predicate logic in which all quantifiers are clustered together at the left side, the section to the right of the quantifiers. See prefix; prenex normal form.

Maximal proof-theoretic consistent set (maximal p-consistent set).
A set of wffs that cannot be enlarged without becoming p-inconsistent. See proof-theoretic consistency.

Membership.
The relation of an element to the sets to which it belongs. Notation: x S (x is a member of set S).

Metalanguage.
The language in which we talk about a formal language. The "target" of the metalanguage is called the object language. See object language.

Metalogic.
The study of formal systems, especially those intended to capture branches of logic, e.g. truth-functional propositional logic, resulting in metatheorems about those systems. Reasoning about reasoning. See metalanguage; metatheorem.

Metatheorem.
A statement about a formal system (as opposed to a wff inside it) proved either informally or by appeal to axioms and rules from another system (as opposed to proved inside the system as a theorem).

Minimization.
One of the simple function-building operations of recursive function theory. Roughly, if we are given a computable function f(x,y), then we posit another function g which computes the least value of y (a natural number) such that f(x,y) = 0. We say that g is created from f by minimization. One way that g might work is to start from 0 and test every natural number in order, stopping at the first value which makes f(x,y) = 0. See recursive function theory. Also called µ (Greek letter mu).
  • Bounded minimization. To insure that g is a total function, we create it from f by bounded minimization. We pick a bound z and try every value 0...z as the value of y; the first one to make f(x,y) = 0 is returned as the value of g; if none makes f(x,y) = 0, then g returns 0. Since g always returns a value, it is a total function; since z is always finite, g is computable.
  • Unbounded minimization. If f(x,y) never equals 0, then g is undefined; hence g is a partial, hence incomputable, function. Think of g running on a computer; when it is unbounded, it will run forever with some inputs. It might test the values 0, 1, 2... in search of a value for y which will make f(x,y) = 0. But if there is no such y, and if no bound is put on the search, then the search will never halt.

Model.
An interpretation in which expressions of interest to us (e.g. a wff, a set of wffs, a system) come out true for that interpretation. See interpretation; isomorphism of models; true for an interpretation.
  • Cardinality of a model. The cardinality of the domain of the model. See domain.
  • Model of a wff or set of wffs. An interpretation, I, that makes those wffs true for I.
  • Model of a formal system. An interpretation, I, that makes its set of theorems true for I. A model of a system is a model of its set of theorems.
  • Non-standard model. Weakly, any non-standard interpretation that is a model. Strongly, any model that is not isomorphic with the intended (standard) model. See isomorphism of models.
  • Normal model. A normal interpretation that is a model. See interpretation, normal.

Model-theoretic consistency (m-consistency).
The state of having a model. See model; proof-theoretic consistency.
  • Model-theoretic consistent wff. A wff that has a model.
  • Model-theoretic consistent set of wffs (m-consistent set). A set of wffs for which there is a model, I, in which each member of the set is true for I.
  • Model-theoretic inconsistency (m-inconsistency). The state of not having a model; being true in no interpretations.

Model theory.
The study of the interpretations of formal languages of formal systems and associated questions of the truth and isomorphism of interpretations. See categoricity; interpretation; isomorphism of models; Löwenheim-Skolem theory; model; proof theory.

Modus ponens. A rule of inference by which we infer B when given A and A B.

Modus tollens. A rule of inference by which we infer ~A when given A B and ~B.

Molecule.
In propositional logic, a compound proposition as opposed to a simple proposition or atom. See atom; compound proposition.

Monotonicity.
The property of a system by which adding new wffs to a set of wffs cannot invalidate previously valid derivations from that set. If A is any wff, and Γ and Δ any sets of wffs, then a system is monotonic iff ΓA implies Δ,ΓA. In non-monotonic logics, derivations valid from Γ can be invalid from ΓΔ. Less formally, in non-monotonic logics, a conclusion that follows from a set of premises might no longer follow when new propositions are added to the set of premises and none subtracted.


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N

N-adic function or predicate.
A function of predicate that takes n arguments. Also called n-ary functions and predicates.

Natural numbers.
The set {0, 1, 2, 3...}. It excludes negative numbers and fractions. The cardinality of this set is Ą0 by definition.

Negation.
A truth function that is true when its argument is false, and false when its argument is true (in 2-valued logics). Also the operator or connective denoting this function; also the proposition built from this operator. Notation: ~p, also ¬p, also, p with a bar on top. (By convention an unadorned propositional symbol, p, is the affirmation of proposition p.)

Negation completeness.
A system is negation-complete when for every closed wff A of its language, either A or ~A is a theorem. That is, all closed wffs are decidable. See decidable wff.
  • Negation incompleteness. A system is negation-incomplete when for at least one closed wff A neither A nor ~A is a theorem, or when there is at least one undecidable closed wff.

N-formula.
See wff, open.

Normal.
See interpretation, normal; model, normal.

Noun numeral.
See numeral.

N-tuple.
A sequence of n terms. See sequence.

Null set.
The set with zero members. Notation: Ų, sometimes 0 or empty set.

Numbers.
See integers; irrational numbers; natural numbers; numeral; rational numbers; real numbers.

Numeral.
A symbol or wff whose intended interpretation is a number. Notation: ("n" with a bar over it), also "n" (numeral for number n).
  • Adjectival numerals. Numerals that adjectivally modify nouns, as in "three bags full" and "two turtle doves". Predicate logic can express adjectival numerals unambiguously, but not noun numerals.
  • Noun numerals. Numerals that function as nouns in the propositions in which they occur, or objects in the domain of the interpretation, as "three is the successor of two".

N-valued logics.
See truth-value.

N-wff.
See wff, open.


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O

Object language.
The formal language of a system. The language used to talk about the object language is called the metalanguage. Outside logic, an object language need not be formal; it is any referent language of any metalanguage. See formal language; metalanguage.

Omega-completeness.
A system is omega-complete iff there is no wff W with one free variable such that (1) Wn is a theorem for every natural number n, and (2) (x)Wx is not a theorem. See closure of a system.
  • Omega-incompleteness. There is a wff W with one free variable such that (1) Wn is a theorem for every natural number n, and (2) (x)Wx is not a theorem. Wx is true for every n by instantiation but not by generalization. See k-validity.

Omega-consistency.
A system is omega-consistent iff there is no wff W with one free variable such that (1) Wn is a theorem for every natural number n, and (2) ~(x)Wx is also a theorem.
  • Omega-inconsistency. There is a wff W with one free variable such that (1) Wn is a theorem for every natural number n, and (2) ~(x)Wx is also a theorem.

One-to-one correspondence.
The pairing-off of the members of one set with the members of another set such that each member of the first has exactly one counterpart in the second and each member of the second has exactly one counterpart in the first. The method of pairing off need not be effective. Notation: AB (set A can be put into one-to-one correspondence with set B)

Operator.
See connective.


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P

Paradox.
Originally, any surprising, puzzling, or counter-intuitive claim, especially a counter-intuitive truth. In modern logic, a concept or proposition that is not only self-contradictory, but for which the obvious alternatives are either self-contradictory or very costly. See Grelling's paradox; Liar paradox; material implication, paradoxes of; Russell paradox; Skolem paradox.

Partial function.
A function whose value is undefined for some arguments. See total function.

Power set.
The set of all the subsets of a set. Notation: *S is the power set of set S; sometimes P(S) or 2S.

Predicate.
Intuitively, whatever is said of the subject of a sentence. A function from individuals (or a sequence of individuals) to truth-values. See attribute; matrix; n-adic predicate; predicate logic; prefix; propositional function; relation. Notation: in "Px", P is the predicate.
  • Argument of a predicate. Any of the individuals of which the predicate is asserted. Notation: in "Pxyz", x, y, and z are the arguments of predicate P. In first-order predicate logic, only terms can be arguments; see term.
  • Extension of a predicate. The set of all objects of which the predicate is true. Notation: the extension of predicate P is {x : Px}. See Russell's paradox.

Predicate logic.
The branch of logic dealing with propositions in which subject and predicate are separately signified, reasoning whose validity depends on this level of articulation, and systems containing such propositions and reasoning. Also called quantification theory. See predicate; first-order theory.
  • First-order predicate logic. Predicate logic in which predicates take only individuals as arguments and quantifiers only bind individual variables.
  • Higher-order predicate logic. Predicate logic in which predicates take other predicates as arguments and quantifiers bind predicate variables. For example, second-order predicates take first-order predicates as arguments. Order n predicates take order n-1 predicates as arguments (n > 1). See Grelling's paradox.
  • Inclusive predicate logic. Predicate logic that does not exclude interpretations with empty domains. Standard predicate logic excludes empty domains and defines logical validity accordingly, i.e. true for all interpretations with non-empty domains. Also called inclusive quantification theory. See existential import; logical validity.
  • Monadic predicate logic. Predicate logic in which predicates take only one argument; the logic of attributes.
  • Polyadic predicate logic. Predicate logic in which predicates take more than one argument; the logic of n-adic predicates (n > 1); the logic of relations.
  • Predicate logic with identity. A system of predicate logic with (x)(x=x) as an axiom, and the following axiom schema, [(x=y) (AA')]c, when A' differs from A only in that y may replace any free occurrence of x in A so long as y is free wherever it replaces x (y need not replace every occurrence of x in A), and when Bc is an arbitrary closure of B. See first-order theory with identity; identity.
  • Pure predicate calculus. A system of predicate logic whose language contains no function symbols or individual constants. As opposed to a number-theoretic predicate calculus which contains these things.

Prefix.
In predicate logic wffs in which all quantifiers are clustered at the left, the section of quantifiers. See matrix; prenex normal form.

Premise.
Also spelled "premiss". A wff from which other wffs are derived or inferred. In an argument, the propositions cited in support of the conclusion are the argument's premises. See argument; conclusion; derivation; inference.

Prenex normal form.
A wff of predicate logic is in prenex normal form iff (1) all its quantifiers are clustered at the left, (2) no quantifier is negated, (3) the scope of each quantifier extends to the end of the wff, (4) no two quantifiers quantify the same variable, (5) every quantified variable occurs in the matrix of the wff. See matrix; prefix; Skolem normal form.

Primitive recursion.
One of the simple function-building operations of recursive function theory. If we are given the functions f(x) and g(x), then we can create a new function h(x) from f and g by primitive recursion thus: when x = 0, then h(x) = f(x); but when x > 0, then h(x) = g(h(x-1)). (For rigor, the minus sign in the last expression should be replaced by another function, but I leave it this way for informal clarity.) Not to be mistaken for "primitive recursive functions". See recursive function theory.

Product of sets.
See intersection.

Proof.
A finite, non-empty sequence of wffs in which the last member is the wff proved and each of the others is either an axiom or the result of applying a rule of inference to wffs preceding it in the sequence. In short, a derivation in which all premises are theorems. See constructive proof; derivation; existence proof.

Proof-theoretic consistency (p-consistency).
The state of not implying a contradiction. See maximal p-consistent set; model-theoretic consistency.
  • Proof-theoretic consistent set of wffs (p-consistent set). A set of wffs is p-consistent there is no wff A such that both A and ~A can be derived from the set.
  • Proof-theoretic inconsistency (p-inconsistency). The state of implying a contradiction.

Proof theory.
The study of the deductive apparatuses of formal systems and associated questions of what is provable in a system (hence, consistency, completeness, and decidability, even though these concepts have a semantic motivation). Broadly, any study of formal systems that makes no reference to the interpretation of the language. See completeness; consistency; decidable system; deductive apparatus; model theory; proof.

Proper axiom.
See axioms.

Proper subset.
See subset.

Proposition.
(1) In truth-functional propositional logic, any statement. (2) In predicate logic, a closed wff, as opposed to a propositional function or open wff. (3) In logic generally (for some), the meaning of a sentence that is invariant through all the paraphrases and translations of the sentence. See compound proposition; contingency; contradiction; simple proposition; tautology.

Propositional function.
In predicate logic, a function from individuals to truth-values. A wff of predicate logic with at least one free variable. An open wff. A propositional function becomes a proposition when it is closed; it is closed either by generalization or instantiation, that is, either by binding free variables or replacing them with constants. See closure of a wff; free variable; function; generalization; instantiation; wff, open.


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Q

Quantification theory.
See predicate logic.

Quantifier.
In predicate logic, a symbol telling us of how many objects (in the domain) the predicate is asserted. The quantifier applies to, or binds, variables which stand as the arguments of predicates. In first-order logic these variables must range over individuals; in higher-order logics they may range over predicates. See bound variable; existential import; free variable; generalization; instantiation; predicate logic.
  • Existential quantifier. The quantifier asserting, "there are some" or "there is at least one". Notation: $ For example, the natural translation of ∃xPx is, "There is at least one thing with property P."
  • Universal quantifier. The quantifier asserting, "for all" or "for all things". Notation: (x); also  ". For example, the natural translation of ("x)Px is, "All things have property P." The "all" in the universal quantifier refers to all the objects in the domain of the interpretation, not to all objects whatsoever. See universe of discourse.
  • Vacuous quantifier. A quantifier that binds no variables, e.g. "("y)" in ("x)("y)(Ax ® Bx).


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R

Range.
The set of objects that may serve as the values (outputs) of a function.

Rational numbers.
All numbers that are equal to the ratio of two integers.

Real numbers.
The rational plus the irrational numbers. The real number line is also called the real number (or numerical) continuum.

Recursive function.
Any function obtained from a small number of intuitively computable functions by a finite number of applications of function-building operations. See computable function; recursive function theory.
  • General recursive functions. The set of primitive recursive functions plus those that can be built with terminating unbounded minimization. See minimization.
  • Partial recursive functions. The set of general recursive functions plus those that can be built with non-terminating unbounded minimization. See minimization.
  • Primitive recursive functions. The set of recursive functions that can be built using only composition, primitive recursion, and bounded (hence terminating) minimization. See composition; primitive recursion; minimization.

Recursive function theory.
If we start with a small number of intuitively computable functions, and a small number of operations that create new computable functions from old ones, then we can generate a large set of functions called recursive functions. If we pick the initial functions and building operations so as to capture what we take to be all the intuitively computable functions, then we generate a set with the same extension as the set of Turing-computable functions. (This led Church to conjecture that all intuitively computable functions or effective methods are recursive functions.) Recursive function theory studies these functions, their method of generation, ways to prove that some functions are not recursive in this sense, and related matters. See Church's thesis; composition; effective method; minimization; primitive recursion; recursive function.

Recursive set.
A set for which there is a recursive function to determine whether any given object is a member. See decidable set; recursive function.

Relation.
A way in which two or more objects are connected, associated, or related, or (at a different level) a polyadic predicate symbolizing such a relation. See attribute; predicate logic.

Relative complement of a set.
See complement.

Relative consistency proof.
The proof that some system S is consistent by appeal to theorems and methods of reasoning from some other system S'. The result is that we know that S is consistent only if system S' is consistent. See Hilbert's program.

Representation of a function.
A function f of n arguments from natural numbers to natural numbers is represented in a system iff there is a wff A with n+1 free variables such that A is a theorem when its variables are are instantiated to the natural numbers k1...kn+1 when f(k1...kn) = kn+1, and not a theorem otherwise.
  • Strong representation of a function. A function f is strongly represented in a system iff it is represented in the system by wff A, and A is a theorem iff ~A is not a theorem.

Representation of a set.
A set N is represented in a system iff there is some propositional function with exactly one free variable, Px, such that Px is a theorem whenever x is instantiated to a member of the set, and a non-theorem otherwise. Or if N is a set of natural numbers, n is a natural number, and is a numeral for n, then N is represented iff P/x nN.

Rules of inference.
Explicit rules for producing a theorem when given one or more other theorems. Functions from sequences of theorems to theorems. In a formal system they should be formal (that is, syntactical or typographical) in nature, and work without reference to the meanings of the strings they manipulate. Also called rules of transformation, rules of production. See for example modus ponens and modus tollens.

Russell's paradox.
Let S be the set of all sets that are not members of themselves. Is S a member of itself? If it is, then it is not; and if not, then it is. This contradiction infects set theory when it is permissible to speak of "all sets" or set complements without qualification, or when a set is defined loosely as any collection of any elements, or when every predicate (intension) determines a set (extension). See complement.


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S

Satisfaction.
(1) In truth-functional propositional logic, a wff A is satisfied iff at least one row of its truth-table makes it true, i.e. iff A is either a contingency or a tautology, i.e. iff A has a model. (2) In predicate logic, a wff A is satisfied iff there is a sequence of objects from the domain of some interpretation such that A is true for that sequence. What it means for a wff to be true for a sequence is defined differently for several distinct kinds of predicate logic wff, e.g. quantified wffs, atomic wffs, wffs compounded with the various connectives of the language, and naked propositional symbols. Among atomic wffs, satisfaction is defined differently depending on whether the arguments of the predicate are constants, variables, or functions, and in the last case, whether the arguments of the functions are constants, variables, or functions. Hence the concept of satisfaction cannot be made more precise without giving each of perhaps a dozen precise tests for different kinds of wff. See truth for an interpretation.
  • Satisfiability. A wff is satisfiable iff there is some interpretation in which it is satisfied.
  • Simultaneous satisfiability. A set of wffs is simultaneously satisfiable iff there is an interpretation in which each member of the set is satisfied. In predicate logic, each member may be satisfied by a different sequence from that interpretation, but all must be satisfied in the same interpretation.

Schema (plural: schemata). See axiom schema; tautology schema; theorem schema.

Semantic completeness.
The condition of a formal system in which (1) the formal language has the power to express as wffs all the propositions intended by the maker to be meaningful, and (2) the deductive apparatus has the power to prove as theorems all the propositions intended by the maker to be true. The second condition can be put more succinctly: all logically valid wffs of the language are theorems of the system. The first of these is also called expressive completeness; the second is sometimes called deductive completeness.
  • Semantic incompleteness. Failure of semantic completeness, but especially when not all logically valid wffs are theorems.

Semantic consequence.
(1) In truth-functional propositional logic, B is the semantic consequence of A iff there is no interpretation, I, in which A is true for I and B is false for I, or in short, if all models of A are models of B. (2) In predicate logic, B is the semantic consequence of A iff for every interpretation, every sequence that satisfies A also satisfies B. Or, there is no sequence in any interpretation that satisfies A but not B. Notation: A B. See satisfaction.

Semantic tautology.
A wff of truth-functional propositional logic whose truth table column contains nothing but T's when these are interpreted as the truth-value Truth. See syntactic tautology.

Semantic validity.
An inference is semantically valid iff it cannot be the case that all the premises are true and the conclusion false at the same time. See syntactic validity.

Sentence.
See closure; wff; proposition.

Sequence.
An ordered series or string of elements (called terms). Also called an n-tuple when n is the number of terms in the sequence. Notation: angled braces, <...>. Notation: s(d/k), the sequence s when the kth member is replaced by object d from the domain. Notation: t*s, the member of the domain of an interpretation I assigned by I to term t for sequence s.

Set.
Intuitively, a collection of elements (called members). The intuitive notion of a set leads to paradoxes, and there is considerable mathematical and philosophical disagreement on how best to refine the intuitive notion. In a set, the order of members is irrelevant, and repetition of members is not meaningful. Notation: curly braces, {...}. See complement; countable set; decidable set; denumerable set; difference; disjoint sets; enumerable set; equivalent sets; intersection; membership; null set; power set; proper subset; representation of a set; Russell's paradox; set theory; subset; superset; symmetric difference; uncountable set; union; universal set.

Set theory.
The mathematical study of sets. See set.
  • Axiomatic set theory. The study of formal systems whose theorems, on the intended interpretation, are the truths of set theory.
  • Cantorian set theory. Set theory in which either the generalized continuum hypothesis or the axiom of choice is an axiom. See axiom of choice; continuum hypothesis.
  • Constructible set theory. Set theory limited to sets whose existence is assured by the axioms of restricted set theory (see below). In 1938 Gödel proved that the axiom of choice, continuum hypothesis, and generalized continuum hypothesis are theorems (even if not axioms) of constructible set theory.
  • Non-Cantorian set theory. Set theory in which either the negation of the generalized continuum hypothesis (GCH) or the negation of the axiom of choice (AC) is an axiom. Since GCH AC, if ~AC is an axiom, then ~GCH will be a theorem.
  • Restricted set theory. Standard set theory minus the axiom of choice. Gödel proved in 1938 that if restricted set theory is consistent, then it remains consistent when the axiom of choice is added (and also when the continuum hypothesis is added). See axiom of choice.
  • Standard set theory. The formal system first formulated by Ernst Zermelo and Abraham Frankel. Also called Zermelo-Frankel set theory or ZF.

Simple consistency.
A system is simply consistent iff there is no wff A such that both A and ~A are theorems.
  • Simple inconsistency. A system is simply inconsistent if there is some wff A such that both A and ~A are theorems.

Simple propositions.
A proposition whose internal structure does not interest us; hence a proposition whose internal structure we do not make visible in our notation. Notation: p, q, r, etc. Also called atoms.

Skolem normal form.
A wff of predicate logic is in Skolem normal form iff (1) it is in prenex normal form, (2) it contains no function symbols, and (3) all existential quantifiers are to the left of all universal quantifiers. See prenex normal form.

Skolem paradox.
The paradox that results from the Löwenheim-Skolem theorem (LST). Does LST mean that the real numbers have the same cardinality as the natural numbers? Does it mean that the difference between the real numbers and the natural numbers that explains the greater cardinality of the reals cannot be described unambiguously in a first-order theory or proved to exist? Does it mean that no set is "absolutely" uncountable but only "relatively" to a given set of axioms and a given interpretation? See Löwenheim-Skolem theorem; model theory.

Soundness.
An argument or inference is sound iff its reasoning is valid and all its premises are true. It is unsound otherwise, i.e. if either its reasoning is invalid, or at least one premise is false, or both. See validity.

Stroke function.
The dyadic truth function "not both". One of only two dyadic truth functions capable of expressing all truth functions by itself. Notation: p|q. Also called the Sheffer stroke, and alternative denial. See dagger function.

Subset.
A set all of whose members belong to a second set (a superset of the subset). Notation: A B (A is a subset of B). See superset.
  • Proper subset. A subset lacking at least one member of its superset. Set A is a proper subset of set B iff all members of A are members of B, but at least one member of B is not a member of A. Notation: A B.

Substitution.
To replace one symbol with another or with a wff. In axiom schemata, to replace metalanguage variables with object language wffs. In instantiation, to replace a variable with a constant. In generalization, to replace a constant with a variable. Notation (for one of these): At/v (the result of substituting term t for the free occurrences of variable v in wff A).

Sum of sets.
See union.

Superset.
A set some of whose members form a reference set. If A is a subset of B, then B is a superset of A. See subset.

Symmetric difference of sets.
The symmetric difference of two sets, A and B, is the set of objects that are members of either A or B but not both. No standard notation. The symmetric difference of sets A and B is the set {x : (xA) (xB)}. The symmetric difference of sets is the set-theoretic equivalent of exclusive disjunction; for the equivalent of inclusive disjunction, see union of sets.

Syntactic completeness.
A system is syntactically complete iff there is no unprovable schema B that could be added to the system as an axiom schema without creating simple inconsistency.
  • Syntactic incompleteness. The failure of syntactic completeness; there is at least one unprovable schema that could be added as an axiom schema without creating simple inconsistency.

Syntactic consequence.
A is the syntactic consequence of a set Γ of wffs iff A can be derived from Γ (and the axioms). Notation: Γ A.

Syntactic tautology.
A wff of truth-functional logic whose truth table column contains nothing but T's when these T's are uninterpreted tokens rather than, say, truth-values. The rules for generating the truth table column tell us to use one of these uninterpreted T's in exactly those cases where semantic considerations would have led us to use the truth-value Truth. See semantic tautology.

Syntactic validity.
An inference is syntactically valid iff the conclusion can be derived from the premises by means of stipulated rules of inference. See semantic validity.

System.
See closure of a system; formal system.


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T

Tautology.
A logically valid wff of truth-functional propositional logic. A compound proposition that is true in every row of its truth table or in every interpretation. See contingency; contradiction; logical validity; semantic tautology; syntactic tautology.
  • Tautology schema (plural: schemata). A formula containing variables of the metalanguage which becomes a tautology when the variables are instantiated to wffs of the formal language.

Term.
Grammatically, the type of expression that can serve as the argument of a predicate or function. The subject of predication; the input of a function. As such (in first-order predicate logic) either an individual constant, individual variable, or a function (with its own arguments) defined for a domain and range of individuals. See constant; function; variable.
  • Closed term. A term without variables.
  • Open term. A term with at least one variable.

Theorem.
A wff that is proved or provable. Axioms are special cases of theorems. Notation: A (A is a theorem); or SA (A is a theorem in system S). See antitheorem; proof.
  • Theorem schema (plural: schemata). A formula containing variables of the metalanguage which becomes a theorem when the variables are instantiated to wffs of the formal language.

Total function.
A function whose value is defined for all possible arguments (from that domain). See partial function.

Transfinite cardinal.
Any infinite cardinal number, that is, any cardinality greater than or equal to Ą0.

Transformation rules.
See rules of inference.

Truth for an interpretation.
(1) For a wff of propositional logic, to be true under the assignments of a given interpretation. (2) For a wff of predicate logic, to be true for all sequences of some intepretation. Also called true for I. How a wff can be true for an interpretation must be defined separately for each connective in the language. See connective; logical validity; satisfaction.

Truth function.
A total function from truth-values (or sequences of truth-values) to truth-values. See conjunction; dagger function; disjunction; equivalence; function; implication; negation; stroke function.

Truth-functional compound proposition.
A compound proposition whose truth-value can be determined solely on the basis of the truth-values of its components and the definitions of its connectives.

Truth-functional connective.
A connective that makes only truth-functional compounds. See connective.

Truth-functional propositional logic.
The branch of logic that deals with the truth-functional connectives and the relations they permit among propositions. The logic of the relations between or among propositions, as opposed to predicate logic which covers the structure within propositions.

Truth-value.
The state of being true or the state of being false.
  • 2-valued logics. Logics in which there are only two truth-values, namely, truth and falsehood.
  • Many-valued logics. Logics that recognize more than two truth-values. In 3-valued logics, for example, the third truth-value is often "unknown" or "unprovable" or "neither true nor false". Also called n-valued logics.

Uncountable set.
A set whose cardinality is greater than Ą0. See countable set.

Undecidable set.
See decidable set; decidable system; decidable wff.


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U

Undecidable set.
See decidable set.

Undecidable system.
See decidable system.

Undecidable wff.
See decidable wff.

Union of sets.
The union of two sets, A and B, is the set of objects that are members of either A or B or both. Notation: AB. Also called the sum of two sets. AB =df {x : (xA)(xB)}. The union of sets is the set-theoretic equivalent of inclusive disjunction; for the equivalent of exclusive disjunction, see symmetric difference of sets.

Universal quantifier.
See quantifier.

Universal set.
The set of all things. Cantor's theorem (that the power set of a given set has a greater cardinality than the given set) implies that there is no largest set or all-inclusive set, at least if every set has a power set. Hence as "the set of all things", the universal set is not recognized in standard set theory. Sometimes the set of all things under consideration in the context; the universe of discourse. The complement of the null set. Notation: 1 (numeral one), or V. See complement; power set; universe of discourse.

Universe of discourse.
The set of all things under consideration in the context; the set of things covered by universal quantification. See complement; universal set.


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V

Validity.
(1) For wffs or propositions, see logical validity. (2) For arguments and inferences, see semantic validity; soundness; syntactic validity.

Variable.
A symbol whose referent varies or is unknown. A place-holder, as opposed to an abbreviation or name (a constant). See bound variables; constant; free variables.
  • Individual variable. A variable ranging over individual objects from the domain of a system. Only individual variables and constants can serve as the arguments of functions and first order predicates. See domain.
  • Metalanguage variable. A variable in the metalanguage of some system S which ranges over wffs of S.
  • Predicate variable. A variable ranging over attributes and relations in higher order logic.


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W

Wff.
Acronym of "well-formed formula", pronounced whiff. A string of symbols from the alphabet of the formal language that conforms to the grammar of the formal language. See decidable wff, formal language.
  • Closed wff. In predicate logic, a wff with no free occurrences of any variable; either it has constants in place of variables, or its variables are bound, or both. Also called a sentence. See bound variables; free variables; closure of a wff.
  • Open wff. In predicate logic, a wff with at least one free occurrence a variable. See free variables; propositional function. Some logicians use the terms, 1-wff, 2-wff,...n-wff for open wffs with 1 free variable, 2 free variables, ...n free variables. (Others call these 1-formula, 2-formula,...n-formula.)


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