Editing Alfred Tarski (section)

==Work in logic==
Tarski's student, [[Robert Lawson Vaught]], has ranked Tarski as one of the four greatest logicians of all time &mdash; along with [[Aristotle]], [[Gottlob Frege]], and [[Kurt Gödel]].<ref name=FFintro/><ref name="Vaught">{{cite journal|last=Vaught|first=Robert L.|date=Dec 1986|title=Alfred Tarski's Work in Model Theory|journal=[[Journal of Symbolic Logic]]|volume=51|issue=4|pages=869–882|doi=10.2307/2273900|jstor=2273900|s2cid=27153078 }}</ref><ref name="Restall">{{cite web|url=http://consequently.org/writing/logicians/|title=Great Moments in Logic |last=Restall|first=Greg|date=2002–2006|access-date=2009-01-03| archive-url= https://web.archive.org/web/20081206052240/http://consequently.org/writing/logicians/| archive-date= 6 December 2008 | url-status= live}}</ref> <!-- Of these four, he was by far the most prolific author. MORE THAN ARISTOTLE???  PROVIDE Reference! --> However, Tarski often expressed great admiration for [[Logic of relatives|Charles Sanders Peirce]], particularly for his pioneering work in the [[Finitary relation|logic of relations]].

Tarski produced axioms for ''logical consequence'' and worked on [[deductive system]]s, the algebra of logic, and the theory of definability. His semantic methods, which culminated in the model theory he and a number of his Berkeley students developed in the 1950s and 60s, radically transformed Hilbert's proof-theoretic metamathematics.
Around 1930, Tarski developed an abstract theory of logical deductions that models some properties of logical calculi. Mathematically, what he described is just a finitary closure operator on a set (the set of ''sentences'').  In [[abstract algebraic logic]], finitary closure operators are still studied under the name ''consequence operator'', which was coined by Tarski. The set ''S'' represents a set of sentences, a subset ''T'' of ''S'' a theory, and cl(''T'') is the set of all sentences that follow from the theory. This abstract approach was applied to fuzzy logic (see Gerla 2000).
{{blockquote|In [Tarski's] view, metamathematics became similar to any mathematical discipline. Not only can its concepts and results be mathematized, but they actually can be integrated into mathematics. ... Tarski destroyed the borderline between metamathematics and mathematics. He objected to restricting the role of metamathematics to the foundations of mathematics.<ref name="Sinaceur">{{cite journal|last=Sinaceur|first=Hourya|title=Alfred Tarski: Semantic Shift, Heuristic Shift in Metamathematics|journal=Synthese|volume=126|issue=1–2|pages=49–65|issn=0039-7857|doi=10.1023/A:1005268531418|year=2001|s2cid=28783841|url=https://halshs.archives-ouvertes.fr/halshs-01119489}}</ref>}}

Tarski's 1936 article "On the concept of logical consequence" argued that the conclusion of an argument will follow logically from its premises if and only if every model of the premises is a model of the conclusion.<ref>{{Cite journal |title=Tarski on Logical Consequence |journal=Notre Dame Journal of Formal Logic |year=1996 |doi=10.1305/ndjfl/1040067321 |s2cid=13217777 |last1=Gómez-Torrente |first1=Mario |volume=37 |doi-access=free }}</ref> In 1937, he published a paper presenting clearly his views on the nature and purpose of the deductive method, and the role of logic in scientific studies.<ref name="mathshistory.st-andrews.ac.uk"/> His high school and undergraduate teaching on logic and axiomatics culminated in a classic short text, published first in Polish, then in German translation, and finally in a 1941 English translation as ''Introduction to Logic and to the Methodology of Deductive Sciences''.<ref>{{Cite web |url=https://archive.org/details/in.ernet.dli.2015.471634 |title=Introduction To Logic And To The Methodology Of Deductive Sciences |website=archive.org |access-date=28 April 2023}}</ref>

Tarski's 1969 "Truth and proof" considered both [[Gödel's incompleteness theorems]] and [[Tarski's undefinability theorem]], and mulled over their consequences for the axiomatic method in mathematics.

===Truth in formalized languages===
{{main|Semantic theory of truth}}
In 1933, Tarski published a very long paper in Polish, titled "Pojęcie prawdy w językach nauk dedukcyjnych",<ref>Alfred Tarski, "POJĘCIE PRAWDY W JĘZYKACH NAUK DEDUKCYJNYCH", Towarszystwo Naukowe Warszawskie, Warszawa, 1933. [http://www.archiwum.wfis.uw.edu.pl/bibfis/index.php?option=com_content&view=article&id=129:a-tarski-pojcie-prawdy-w-jzykach-nauk-dedukcyjnych&catid=56:marginalia&Itemid=106 (Text in Polish in the Digital Library WFISUW-IFISPAN-PTF)] {{Webarchive|url=https://web.archive.org/web/20160304000353/http://www.archiwum.wfis.uw.edu.pl/bibfis/index.php?option=com_content&view=article&id=129:a-tarski-pojcie-prawdy-w-jzykach-nauk-dedukcyjnych&catid=56:marginalia&Itemid=106 |date=2016-03-04 }}.</ref> "Setting out a mathematical definition of truth for formal languages." The 1935 German translation was titled "Der Wahrheitsbegriff in den formalisierten Sprachen", "The concept of truth in formalized languages", sometimes shortened to "Wahrheitsbegriff". An English translation appeared in the 1956 first edition of the volume ''[[Logic, Semantics, Metamathematics]]''. This collection of papers from 1923 to 1938 is an event in 20th-century [[analytic philosophy]], a contribution to [[Mathematical logic|symbolic logic]], [[semantics]], and the [[philosophy of language]]. For a brief discussion of its content, see [[Convention T]] (and also [[T-schema]]).

A philosophical debate examines the extent to which Tarski's theory of truth for formalized languages can be seen as a [[correspondence theory of truth]]. The debate centers on how to read Tarski's condition of material adequacy for a true definition. That condition requires that the truth theory have the following as theorems for all sentences p of the language for which truth is being defined:

: "p" is true [[if and only if]] p.

(where p is the proposition expressed by "p")

The debate amounts to whether to read sentences of this form, such as

{{blockquote|"Snow is white" is true if and only if snow is white}}

as expressing merely a [[deflationary theory of truth]] or as embodying [[truth]] as a more substantial property (see Kirkham 1992).

===Logical consequence===
In 1936, Tarski published Polish and German versions of a lecture, “On the Concept of Following Logically",<ref name="Tarski2002">{{cite journal 
|last1=Tarski 
|first1=Alfred 
|title=On the Concept of Following Logically
|date=2002 
|volume=23 
|pages=155–196 
|doi=10.1080/0144534021000036683
|journal = History and Philosophy of Logic|issue=3 
|s2cid=120956516 
}}</ref>
he had given the preceding year at the International Congress of Scientific Philosophy in Paris. <!-- (First appearance in English?) --> A new English translation of this paper, Tarski (2002), highlights the many differences between the German and Polish versions of the paper and corrects a number of mistranslations in Tarski (1983).<ref name="Tarski2002" />

This publication set out the modern [[model theory|model-theoretic]] definition of (semantic) logical consequence, or at least the basis for it. Whether Tarski's notion was entirely the modern one turns on whether he intended to admit models with varying domains (and in particular, models with domains of different [[cardinal number|cardinalities]]).{{citation needed|date=July 2023}} This question is a matter of some debate in the philosophical literature. [[John Etchemendy]] stimulated much of the discussion about Tarski's treatment of varying domains.<ref>{{cite book|last=Etchemendy|first=John|title=The Concept of Logical Consequence|publisher=Stanford CA: CSLI Publications|year=1999|isbn=978-1-57586-194-4}}</ref>

Tarski ends by pointing out that his definition of logical consequence depends upon a division of terms into the logical and the extra-logical and he expresses some skepticism that any such objective division will be forthcoming. "What are Logical Notions?" can thus be viewed as continuing "On the Concept of Logical Consequence".{{citation needed|date=July 2023}}

===Logical notions===
[[File:Alfred Tarski.jpeg|thumb|Alfred Tarski at Berkeley]]
Tarski's "What are Logical Notions?" (Tarski 1986) is the published version of a talk that he gave originally in 1966 in London and later in 1973 in [[Buffalo, New York|Buffalo]]; it was edited without his direct involvement by [[John Corcoran (logician)|John Corcoran]]. It became the most cited paper in the journal ''History and Philosophy of Logic''.<ref>{{Cite web | url=http://www.tandfonline.com/action/showMostCitedArticles?journalCode=thpl20 | title=History and Philosophy of Logic}}</ref>

In the talk, Tarski proposed demarcation of logical operations (which he calls "notions") from non-logical. The suggested criteria were derived from the [[Erlangen program]] of the 19th-century German mathematician [[Felix Klein]]. Mautner (in 1946), and possibly{{clarify|date=July 2023}} an article by the Portuguese mathematician [[José Sebastião e Silva]], anticipated Tarski in applying the Erlangen Program to logic.{{citation needed|date=July 2023}}

The Erlangen program classified the various types of geometry ([[Euclidean geometry]], [[affine geometry]], [[topology]], etc.) by the type of one-one transformation of space onto itself that left the objects of that geometrical theory invariant. (A one-to-one transformation is a functional map of the space onto itself so that every point of the space is associated with or mapped to one other point of the space. So, "rotate 30 degrees" and  "magnify by a factor of 2" are intuitive descriptions of simple uniform one-one transformations.) Continuous transformations give rise to the objects of topology, similarity transformations to those of Euclidean geometry, and so on.{{citation needed|date=July 2023}}

As the range of permissible transformations becomes broader, the range of objects one is able to distinguish as preserved by the application of the transformations becomes narrower. Similarity transformations are fairly narrow (they preserve the relative distance between points) and thus allow us to distinguish relatively many things (e.g., equilateral triangles from non-equilateral triangles). Continuous transformations (which can intuitively be thought of as transformations which allow non-uniform stretching, compression, bending, and twisting, but no ripping or glueing) allow us to distinguish a [[polygon]] from an [[annulus (mathematics)|annulus]] (ring with a hole in the centre), but do not allow us to distinguish two polygons from each other.{{citation needed|date=July 2023}}

Tarski's proposal{{which|date=July 2023}} was to demarcate the logical notions by considering all possible one-to-one transformations ([[automorphism]]s) of a domain onto itself. By domain is meant the [[universe of discourse]] of a model for the semantic theory of logic. If one identifies the [[truth value]] True with the domain set and the truth-value False with the empty set, then the following operations are counted as logical under the proposal:

# ''[[Truth-function]]s'': All truth-functions are admitted by the proposal. This includes, but is not limited to, all ''n''-ary truth-functions for finite ''n''. (It also admits of truth-functions with any infinite number of places.)
# ''Individuals'': No individuals, provided the domain has at least two members.
# ''Predicates'':
#* the one-place total and null predicates, the former having all members of the domain in its extension and the latter having no members of the domain in its extension
#* two-place total and null predicates, the former having the set of all ordered pairs of domain members as its extension and the latter with the empty set as extension
#* the two-place identity predicate, with the set of all order-pairs <''a'',''a''> in its extension, where ''a'' is a member of the domain
#* the two-place diversity predicate, with the set of all order pairs <''a'',''b''> where ''a'' and ''b'' are distinct members of the domain
#* ''n''-ary predicates in general: all predicates definable from the identity predicate together with [[Logical conjunction|conjunction]], [[disjunction]] and [[negation]] (up to any ordinality, finite or infinite)
# ''[[Quantifier (logic)|Quantifiers]]'': Tarski explicitly discusses only monadic quantifiers and points out that all such numerical quantifiers are admitted under his proposal. These include the standard universal and existential quantifiers as well as numerical quantifiers such as "Exactly four", "Finitely many", "Uncountably many", and "Between four and 9 million", for example. While Tarski does not enter into the issue, it is also clear that polyadic quantifiers are admitted under the proposal. These are quantifiers like, given two predicates ''Fx'' and ''Gy'', "More(''x, y'')", which says "More things have ''F'' than have ''G''."
# ''Set-Theoretic relations'': Relations such as [[inclusion (set theory)|inclusion]], [[Intersection (set theory)|intersection]] and [[Union (set theory)|union]] applied to [[subset]]s of the domain are logical in the present sense.
# ''Set membership'': Tarski ended his lecture with a discussion of whether the set membership relation counted as logical in his sense. (Given the reduction of (most of) mathematics to set theory, this was, in effect, the question of whether most or all of mathematics is a part of logic.) He pointed out that set membership is logical if set theory is developed along the lines of [[type theory]], but is extralogical if set theory is set out axiomatically, as in the canonical [[Zermelo–Fraenkel set theory]].
# ''Logical notions of higher order'': While Tarski confined his discussion to operations of first-order logic, there is nothing about his proposal that necessarily restricts it to first-order logic. (Tarski likely restricted his attention to first-order notions as the talk was given to a non-technical audience.) So, higher-order quantifiers and predicates are admitted as well.{{citation needed|date=July 2023}}

In some ways the present proposal is the obverse of that of Lindenbaum and Tarski (1936), who proved that all the logical operations of  [[Bertrand Russell]]'s and [[Alfred North Whitehead|Whitehead]]'s ''[[Principia Mathematica]]'' are invariant under one-to-one transformations of the domain onto itself. The present proposal is also employed in Tarski and [[Steven Givant|Givant]] (1987).<ref>{{Cite journal |url=https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/alfred-tarski-and-steven-givant-a-formalization-of-set-theory-without-variables-american-mathematical-society-colloquium-publications-vol-41-american-mathematical-society-providence1987-xxi-318-pp/9B436E22D3B3785570ABE9C85F870308 |title=Alfred Tarski and Steven Givant. A formalization of set theory without variables. American Mathematical Society colloquium publications, vol. 41. American Mathematical Society, Providence1987, xxi + 318 pp. |journal=The Journal of Symbolic Logic |date=12 March 2014 |volume=55 |issue=1 |pages=350–352 |doi=10.2307/2274990 |jstor=2274990 |access-date=28 April 2023 |last1=Németi |first1=István }}</ref>

[[Solomon Feferman]] and [[Vann McGee]] further discussed Tarski's proposal{{which|date=July 2023}} in work published after his death. Feferman (1999) raises problems for the proposal and suggests a cure: replacing Tarski's preservation by automorphisms with preservation by arbitrary [[homomorphism]]s. In essence, this suggestion circumvents the difficulty Tarski's proposal has in dealing with a sameness of logical operation across distinct domains of a given cardinality and across domains of distinct cardinalities. Feferman's proposal results in a radical restriction of logical terms as compared to Tarski's original proposal. In particular, it ends up counting as logical only those operators of standard first-order logic without identity.{{citation needed|date=July 2023}}

[[Vann McGee]] (1996) provides a precise account of what operations are logical in the sense of Tarski's proposal in terms of expressibility in a language that extends first-order logic by allowing arbitrarily long conjunctions and disjunctions, and quantification over arbitrarily many variables. "Arbitrarily" includes a countable infinity.<ref>{{Cite journal |url=https://www.jstor.org/stable/1523019 |title=Revision |journal=Philosophical Issues |jstor=1523019 |access-date=28 April 2023 |last1=McGee |first1=Vann |year=1997 |volume=8 |pages=387–406 |doi=10.2307/1523019 }}</ref>