Predicate functor logic

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In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
, predicate functor logic (PFL) is one of several ways to express
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
(also known as predicate logic) by purely algebraic means, i.e., without quantified variables. PFL employs a small number of algebraic devices called predicate functors (or predicate modifiers) that operate on terms to yield terms. PFL is mostly the invention of the
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premi ...
ian and
philosopher A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...
Willard Quine.

# Motivation

The source for this section, as well as for much of this entry, is Quine (1976). Quine proposed PFL as a way of algebraizing
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
in a manner analogous to how
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
algebraizes
propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...
. He designed PFL to have exactly the expressive power of
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
with identity. Hence the
metamathematics Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the ter ...
of PFL are exactly those of first-order logic with no interpreted predicate letters: both logics are
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by ...
,
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies ...
, and undecidable. Most work Quine published on logic and mathematics in the last 30 years of his life touched on PFL in some way. Quine took "functor" from the writings of his friend
Rudolf Carnap Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism. ...
, the first to employ it in
philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. So ...
and
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
, and defined it as follows:
"The word ''functor'', grammatical in import but logical in habitat... is a sign that attaches to one or more expressions of given grammatical kind(s) to produce an expression of a given grammatical kind." (Quine 1982: 129)
Ways other than PFL to algebraize
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
include: * Cylindric algebra by
Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
and his American students. The simplified cylindric algebra proposed in Bernays (1959) led Quine to write the paper containing the first use of the phrase "predicate functor"; *The polyadic algebra of
Paul Halmos Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operato ...
. By virtue of its economical primitives and axioms, this algebra most resembles PFL; * Relation algebra algebraizes the fragment of
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
consisting of formulas having no atomic formula lying in the scope of more than three quantifiers. That fragment suffices, however, for
Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearl ...
and the
axiomatic set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
ZFC; hence relation algebra, unlike PFL, is incompletable. Most work on relation algebra since about 1920 has been by Tarski and his American students. The power of relation algebra did not become manifest until the monograph Tarski and Givant (1987), published after the three important papers bearing on PFL, namely Bacon (1985), Kuhn (1983), and Quine (1976); *
Combinatory logic Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of compu ...
builds on
combinator Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of comp ...
s, higher order functions whose domain is another combinator or function, and whose range is yet another combinator. Hence
combinatory logic Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of compu ...
goes beyond first-order logic by having the expressive power of
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, which makes combinatory logic vulnerable to paradoxes. A predicate functor, on the other hand, simply maps predicates (also called terms) into predicates. PFL is arguably the simplest of these formalisms, yet also the one about which the least has been written. Quine had a lifelong fascination with
combinatory logic Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of compu ...
, attested to by his introduction to the translation in Van Heijenoort (1967) of the paper by the Russian logician
Moses Schönfinkel Moses Ilyich Schönfinkel (russian: Моисей Исаевич Шейнфинкель, translit=Moisei Isai'evich Sheinfinkel; 29 September 1888 – 1942) was a logician and mathematician, known for the invention of combinatory logic. Life Mose ...
founding combinatory logic. When Quine began working on PFL in earnest, in 1959, combinatory logic was commonly deemed a failure for the following reasons: * Until Dana Scott began writing on the
model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...
of combinatory logic in the late 1960s, almost only
Haskell Curry Haskell Brooks Curry (; September 12, 1900 – September 1, 1982) was an American mathematician and logician. Curry is best known for his work in combinatory logic. While the initial concept of combinatory logic was based on a single paper b ...
, his students, and Robert Feys in Belgium worked on that logic; *Satisfactory axiomatic formulations of combinatory logic were slow in coming. In the 1930s, some formulations of combinatory logic were found to be inconsistent. Curry also discovered the Curry paradox, peculiar to combinatory logic; *The
lambda calculus Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation t ...
, with the same expressive power as
combinatory logic Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of compu ...
, was seen as a superior formalism.

# Kuhn's formalization

The PFL
syntax In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituency), ...
, primitives, and axioms described in this section are largely Steven Kuhn's (1983). The
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...
of the functors are Quine's (1982). The rest of this entry incorporates some terminology from Bacon (1985).

## Syntax

An ''atomic term'' is an upper case Latin letter, ''I'' and ''S'' excepted, followed by a numerical superscript called its ''degree'', or by concatenated lower case variables, collectively known as an ''argument list''. The degree of a term conveys the same information as the number of variables following a predicate letter. An atomic term of degree 0 denotes a Boolean variable or a truth value. The degree of ''I'' is invariably 2 and so is not indicated. The "combinatory" (the word is Quine's) predicate functors, all monadic and peculiar to PFL, are Inv, inv, ∃, +, and p. A term is either an atomic term, or constructed by the following recursive rule. If τ is a term, then Invτ, invτ, ∃τ, +τ, and pτ are terms. A functor with a superscript ''n'', ''n'' a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
> 1, denotes ''n'' consecutive applications (iterations) of that functor. A formula is either a term or defined by the recursive rule: if α and β are formulas, then αβ and ~(α) are likewise formulas. Hence "~" is another monadic functor, and concatenation is the sole dyadic predicate functor. Quine called these functors "alethic." The natural interpretation of "~" is
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and fals ...
; that of concatenation is any connective that, when combined with negation, forms a functionally complete set of connectives. Quine's preferred functionally complete set was
conjunction Conjunction may refer to: * Conjunction (grammar), a part of speech * Logical conjunction, a mathematical operator ** Conjunction introduction Conjunction introduction (often abbreviated simply as conjunction and also called and introduction or ...
and
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and fals ...
. Thus concatenated terms are taken as conjoined. The notation + is Bacon's (1985); all other notation is Quine's (1976; 1982). The alethic part of PFL is identical to the ''Boolean term schemata'' of Quine (1982). As is well known, the two alethic functors could be replaced by a single dyadic functor with the following
syntax In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituency), ...
and
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...
: if α and β are formulas, then (αβ) is a formula whose semantics are "not (α and/or β)" (see NAND and NOR).

## Axioms and semantics

Quine set out neither axiomatization nor proof procedure for PFL. The following axiomatization of PFL, one of two proposed in Kuhn (1983), is concise and easy to describe, but makes extensive use of
free variable In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...
s and so does not do full justice to the spirit of PFL. Kuhn gives another axiomatization dispensing with free variables, but that is harder to describe and that makes extensive use of defined functors. Kuhn proved both of his PFL axiomatizations
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by ...
and
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies ...
. This section is built around the primitive predicate functors and a few defined ones. The alethic functors can be axiomatized by any set of axioms for sentential logic whose primitives are negation and one of ∧ or ∨. Equivalently, all tautologies of sentential logic can be taken as axioms. Quine's (1982) semantics for each predicate functor are stated below in terms of
abstraction Abstraction in its main sense is a conceptual process wherein general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods. "An abs ...
(set builder notation), followed by either the relevant axiom from Kuhn (1983), or a definition from Quine (1976). The notation $\$ denotes the set of ''n''-tuples satisfying the atomic formula $Fx_1\cdots x_n.$ *''Identity'', , is defined as: :$IFx_1x_2\cdots x_n \leftrightarrow \left(Fx_1x_1\cdots x_n \leftrightarrow Fx_2x_2\cdots x_n\right)\text$ Identity is reflexive (), symmetric (), transitive (), and obeys the substitution property: :$\left(Fx_1\cdots x_n \land Ix_1y\right) \rightarrow Fyx_2\cdots x_n.$ *''Padding'', +, adds a variable to the left of any argument list. :$\ +F^n \ \overset\ \.$ :$+Fx_1\cdots x_n \leftrightarrow Fx_2\cdots x_n.$ *''Cropping'', ∃, erases the leftmost variable in any argument list. :$\exist F^n \ \overset\ \.$ :$Fx_1\cdots x_n \rightarrow \exist Fx_2\cdots x_n.$ ''Cropping'' enables two useful defined functors: * ''Reflection'', S: :$SF^n \ \overset\ \.$ :$SF^n \leftrightarrow \exist IF^n.$ S generalizes the notion of reflexivity to all terms of any finite degree greater than 2. N.B: S should not be confused with the primitive combinator S of combinatory logic. *''
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
'', $\times$; :$F^m \times G^n \leftrightarrow F^m \exist^m G^n.$ Here only, Quine adopted an infix notation, because this infix notation for Cartesian product is very well established in mathematics. Cartesian product allows restating conjunction as follows: :$F^mx_1\cdots x_mG^nx_1\cdots x_n \leftrightarrow \left(F^m \times G^n\right)x_1\cdots x_mx_1\cdots x_n.$ Reorder the concatenated argument list so as to shift a pair of duplicate variables to the far left, then invoke S to eliminate the duplication. Repeating this as many times as required results in an argument list of length max(''m'',''n''). The next three functors enable reordering argument lists at will. *''Major inversion'', Inv, rotates the variables in an argument list to the right, so that the last variable becomes the first. :$\operatorname F^n \ \overset\ \.$ :$\operatorname Fx_1\cdots x_n \leftrightarrow Fx_nx_1\cdots x_.$ *''Minor inversion'', inv, swaps the first two variables in an argument list. :$\operatorname F^n \ \overset\ \.$ :$\operatorname Fx_1\cdots x_n \leftrightarrow Fx_2x_1\cdots x_n.$ *''Permutation'', p, rotates the second through last variables in an argument list to the left, so that the second variable becomes the last. :$\ pF^n \ \overset\ \.$ :$pFx_1\cdots x_n \leftrightarrow \operatorname \operatorname Fx_1x_3\cdots x_nx_2.$ Given an argument list consisting of ''n'' variables, p implicitly treats the last ''n''−1 variables like a bicycle chain, with each variable constituting a link in the chain. One application of p advances the chain by one link. ''k'' consecutive applications of p to ''F''''n'' moves the ''k''+1 variable to the second argument position in ''F''. When ''n''=2, Inv and inv merely interchange ''x''1 and ''x''2. When ''n''=1, they have no effect. Hence p has no effect when ''n'' < 3. Kuhn (1983) takes ''Major inversion'' and ''Minor inversion'' as primitive. The notation p in Kuhn corresponds to inv; he has no analog to ''Permutation'' and hence has no axioms for it. If, following Quine (1976), p is taken as primitive, Inv and inv can be defined as nontrivial combinations of +, ∃, and iterated p. The following table summarizes how the functors affect the degrees of their arguments.

## Rules

All instances of a predicate letter may be replaced by another predicate letter of the same degree, without affecting validity. The
rules Rule or ruling may refer to: Education * Royal University of Law and Economics (RULE), a university in Cambodia Human activity * The exercise of political or personal control by someone with authority or power * Business rule, a rule pert ...
are: *
Modus ponens In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...
; * Let ''α'' and ''β'' be PFL formulas in which $x_1$ does not appear. Then if $\left(\alpha \land Fx_1...x_n\right) \rightarrow \beta$ is a PFL theorem, then $\left(\alpha \land \exist Fx_2...x_n\right) \rightarrow \beta$ is likewise a PFL theorem.

## Some useful results

Instead of axiomatizing PFL, Quine (1976) proposed the following conjectures as candidate axioms. : $\exist I$ ''n''−1 consecutive iterations of p restores the ''status quo ante'': : $F^n \leftrightarrow p^F^n$ + and ∃ annihilate each other: : $\beginF^n \rightarrow +\exist F^n\\ F^n \leftrightarrow \exist +F^n\end$ Negation distributes over +, ∃, and p: : $\begin+\lnot F^n \leftrightarrow \lnot +F^n\\ \lnot\exist F^n \rightarrow \exist \lnot F^n\\ p\lnot F^n \leftrightarrow \lnot pF^n\end$ + and p distributes over conjunction: : $\begin+\left(F^nG^m\right) \leftrightarrow \left(+F^n+G^m\right)\\ p\left(F^nG^m\right) \leftrightarrow \left(pF^npG^m\right)\end$ Identity has the interesting implication: : $IF^n \rightarrow p^ \exist p+F^n$ Quine also conjectured the rule: If is a PFL theorem, then so are , and $\lnot \exist \lnot \alpha$.

# Bacon's work

Bacon (1985) takes the conditional,
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and fals ...
, ''Identity'', ''Padding'', and ''Major'' and ''Minor inversion'' as primitive, and ''Cropping'' as defined. Employing terminology and notation differing somewhat from the above, Bacon (1985) sets out two formulations of PFL: * A natural deduction formulation in the style of Frederick Fitch. Bacon proves this formulation
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by ...
and
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies ...
in full detail. *An axiomatic formulation, which Bacon asserts, but does not prove, equivalent to the preceding one. Some of these axioms are simply Quine conjectures restated in Bacon's notation. Bacon also: *Discusses the relation of PFL to the
term logic In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, ...
of Sommers (1982), and argues that recasting PFL using a syntax proposed in Lockwood's appendix to Sommers, should make PFL easier to "read, use, and teach"; *Touches on the group theoretic structure of Inv and inv; *Mentions that sentential logic, monadic predicate logic, the
modal logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
S5, and the Boolean logic of (un)permuted relations, are all fragments of PFL.

# From first-order logic to PFL

The following
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
is adapted from Quine (1976: 300-2). Given a closed formula of
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
, first do the following: * Attach a numerical subscript to every predicate letter, stating its degree; * Translate all
universal quantifier In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In othe ...
s into
existential quantifier In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, w ...
s and negation; * Restate all
atomic formula In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformu ...
s of the form ''x''=''y'' as ''Ixy''. Now apply the following algorithm to the preceding result: The reverse translation, from PFL to first-order logic, is discussed in Quine (1976: 302-4). The canonical foundation of mathematics is
axiomatic set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, with a background logic consisting of
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
with identity, with a universe of discourse consisting entirely of sets. There is a single predicate letter of degree 2, interpreted as set membership. The PFL translation of the canonical
axiomatic set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
ZFC is not difficult, as no ZFC axiom requires more than 6 quantified variables.Metamath axioms.
/ref>

* Algebraic logic

{{reflist

# References

*Bacon, John, 1985, "The completeness of a predicate-functor logic," ''Journal of Symbolic Logic 50'': 903–26. *
Paul Bernays Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a Swiss mathematician who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator of D ...
, 1959, "Uber eine naturliche Erweiterung des Relationenkalkuls" in Heyting, A., ed., ''Constructivity in Mathematics''. North Holland: 1–14. * Kuhn, Steven T., 1983,
An Axiomatization of Predicate Functor Logic
" ''Notre Dame Journal of Formal Logic 24'': 233–41. * Willard Quine, 1976, "Algebraic Logic and Predicate Functors" in ''Ways of Paradox and Other Essays'', enlarged ed. Harvard Univ. Press: 283–307. *Willard Quine, 1982. ''Methods of Logic'', 4th ed. Harvard Univ. Press. Chpt. 45. *Sommers, Fred, 1982. ''The Logic of Natural Language''. Oxford Univ. Press. *
Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
and Givant, Steven, 1987. ''A Formalization of Set Theory Without Variables''.
AMS AMS or Ams may refer to: Organizations Companies * Alenia Marconi Systems * American Management Systems * AMS (Advanced Music Systems) * ams AG, semiconductor manufacturer * AMS Pictures * Auxiliary Medical Services Educational institution ...
. *
Jean Van Heijenoort Jean Louis Maxime van Heijenoort (; July 23, 1912 – March 29, 1986) was a historian of mathematical logic. He was also a personal secretary to Leon Trotsky from 1932 to 1939, and an American Trotskyist until 1947. Life Van Heijenoort was born ...
, 1967. ''From Frege to Gödel: A Source Book on Mathematical Logic''. Harvard Univ. Press.