Editing Charles Sanders Peirce (section)

== Logic, or semiotic ==
In 1918, the logician [[Clarence Irving Lewis|C.&nbsp;I. Lewis]] wrote, "The contributions of C.S. Peirce to symbolic logic are more numerous and varied than those of any other writer—at least in the nineteenth century."<ref>Lewis, Clarence Irving (1918), ''A Survey of Symbolic Logic'', see ch. 1, §7 "Peirce", pp. 79–106, see [https://archive.org/stream/surveyofsymbolic00lewiiala#page/79/mode/1up p. 79] (''Internet Archive''). Note that Lewis's bibliography lists works by Frege, tagged with asterisks as important.</ref> 

=== Relational logic ===
Beginning with his first paper on the [[Charles Sanders Peirce bibliography#LOR1870|"Logic of Relatives" (1870)]], Peirce extended the [[theory of relations]] pioneered by [[Augustus De Morgan]].{{efn|Much of the mathematics of relations now taken for granted was "borrowed" from Peirce, not always with all due credit; on that and on how the young [[Bertrand Russell]], especially his ''Principles of Mathematics'' and ''[[Principia Mathematica]]'', did not do Peirce justice, see Anellis (1995).<ref name="Anellis"/>}} Beginning in 1940, [[Alfred Tarski]] and his students rediscovered aspects of Peirce's larger vision of relational logic, developing the perspective of [[relation algebra]].

Relational logic gained applications. In mathematics, it influenced the abstract analysis of [[E. H. Moore]] and the [[Lattice (order)|lattice theory]] of [[Garrett Birkhoff]]. In computer science, the [[relational model]] for [[database]]s was developed with Peircean ideas in work of [[Edgar F. Codd]], who was a doctoral student<ref>Avery, John (2003) ''Information theory and evolution'', p. 167; also Mitchell, Melanie, "[http://web.cecs.pdx.edu/~mm/MMScientificAncestry.html My Scientific Ancestry] {{Webarchive|url=https://web.archive.org/web/20141008181914/http://web.cecs.pdx.edu/~mm/MMScientificAncestry.html|date=October 8, 2014}}".</ref> of [[Arthur W. Burks]], a Peirce scholar. In economics, relational logic was used by [[Frank P. Ramsey]], [[John von Neumann]], and [[Paul Samuelson]] to study preferences and utility and by [[Kenneth J. Arrow]] in ''[[Social Choice and Individual Values]]'', following Arrow's association with Tarski at [[City College of New York]].

=== Quantifiers ===
On Peirce and his contemporaries [[Ernst Schröder (mathematician)|Ernst Schröder]] and [[Gottlob Frege]], [[Hilary Putnam]] (1982)<ref name="Putnam" /> documented that Frege's work on the logic of quantifiers had little influence on his contemporaries, although it was published four years before the work of Peirce and his student Oscar Howard Mitchell. Putnam found that mathematicians and logicians learned about the logic of quantifiers through the independent work of Peirce and Mitchell, particularly through Peirce's "On the Algebra of Logic: A Contribution to the Philosophy of Notation"<ref name="CSP1885" /> (1885), published in the premier American mathematical journal of the day, and cited by [[Peano]] and Schröder, among others, who ignored Frege. They also adopted and modified Peirce's notations, typographical variants of those now used. Peirce apparently was ignorant of Frege's work, despite their overlapping achievements in logic, [[philosophy of language]], and the [[foundations of mathematics]].

Peirce's work on formal logic had admirers besides [[Ernst Schröder (mathematician)|Ernst Schröder]]:
* Philosophical algebraist [[William Kingdon Clifford]]<ref>Beil, Ralph G. and Ketner, Kenneth (2003), "Peirce, Clifford, and Quantum Theory", ''International Journal of Theoretical Physics'' v. 42, n. 9, pp. 1957–1972.</ref> and logician [[William Ernest Johnson]], both British;
* The Polish school of logic and foundational mathematics, including [[Alfred Tarski]];
* [[Arthur Prior]], who praised and studied Peirce's logical work in a 1964 paper<ref name="SP2" /> and in ''Formal Logic'' (saying on page 4 that Peirce "perhaps had a keener eye for essentials than any other logician before or since").

=== Philosophy of logic ===
A philosophy of logic, grounded in his categories and semiotic, can be extracted from Peirce's writings and, along with Peirce's logical work more generally, is exposited and defended in Hilary Putnam (1982);<ref name="Putnam" /> the Introduction in Nathan Houser ''et al.'' (1997);<ref>Houser, Roberts, and Van Evra, eds. (1997), ''Studies in the Logic of Charles Sanders Peirce'', Indiana U., Bloomington, IN.</ref> and [[Randall Dipert]]'s chapter in [[Cheryl Misak]] (2004).<ref>Misak, ed. (2004), ''The Cambridge Companion to Peirce'', Cambridge U., UK.</ref>{{Semiotics}}

==== Logic as philosophical ====
Peirce regarded logic ''per se'' as a division of philosophy, as a normative science based on esthetics and ethics, as more basic than metaphysics,<ref name="FRL">Peirce (1899 MS), "F.R.L." [First Rule of Logic], ''Collected Papers of Charles Sanders Peirce'', 1.135–140, [https://web.archive.org/web/20120106071421/http://www.princeton.edu/~batke/peirce/frl_99.htm Eprint]</ref> and as "the art of devising methods of research".<ref name="ars">Peirce (1882), "Introductory Lecture on the Study of Logic" delivered September 1882, ''Johns Hopkins University Circulars'', v. 2, n. 19, pp. [https://books.google.com/books?id=E0YFAAAAQAAJ&pg=PA11&dq=%22art+of+devising+methods+of+research%22 11–12] (via Google), November 1882. Reprinted (''The Essential Peirce'', 1:210–214; ''Writings of Charles S. Peirce'', 4:378–382; ''Collected Papers of Charles Sanders Peirce'', 7.59–76). The definition of logic quoted by Peirce is by [[Peter of Spain (author)|Peter of Spain]].</ref> More generally, as inference, "logic is rooted in the social principle", since inference depends on a standpoint that, in a sense, is unlimited.<ref>Peirce (1878), "The Doctrine of Chances", ''Popular Science Monthly'', v. 12, pp. 604–615 (CP 2.645–668, ''Writings of Charles S. Peirce'', 3:276–290, ''The Essential Peirce'', 1:142–154). {{quote|...&nbsp;death makes the number of our risks, the number of our inferences, finite, and so makes their mean result uncertain. The very idea of probability and of reasoning rests on the assumption that this number is indefinitely great. ... logicality inexorably requires that our interests shall ''not'' be limited. ... Logic is rooted in the social principle.}}</ref> Peirce called (with no sense of deprecation) "mathematics of logic" much of the kind of thing which, in current research and applications, is called simply "logic". He was productive in both (philosophical) logic and logic's mathematics, which were connected deeply in his work and thought.

Peirce argued that logic is formal semiotic: the formal study of signs in the broadest sense, not only signs that are artificial, linguistic, or symbolic, but also signs that are semblances or are indexical such as reactions. Peirce held that "all this universe is perfused with signs, if it is not composed exclusively of signs",<ref>Peirce, ''Collected Papers of Charles Sanders Peirce'', 5.448 footnote, from "The Basis of Pragmaticism" in 1906.</ref> along with their representational and inferential relations. He argued that, since all thought takes time, all thought is in signs<ref name="QFM">Peirce, (1868), "Questions concerning certain Faculties claimed for Man", ''Journal of Speculative Philosophy'' v. 2, n. 2, [https://books.google.com/books?id=YHkqP2JHJ_IC&pg=RA1-PA103 pp. 103–114]. On thought in signs, see p. 112. Reprinted ''Collected Papers of Charles Sanders Peirce'', 5.213–263 (on thought in signs, see 253), ''Writings of Charles S. Peirce'', 2:193–211, ''The Essential Peirce'', 2:11–27. ''Arisbe'' [http://www.cspeirce.com/menu/library/bycsp/question/qu-frame.htm Eprint] {{Webarchive|url=https://web.archive.org/web/20071014064210/http://cspeirce.com/menu/library/bycsp/question/qu-frame.htm |date=2007-10-14  }}.</ref> and sign processes ("semiosis") such as the inquiry process. He [[Classification of the sciences (Peirce)|divided]] logic into: (1) speculative grammar, or stechiology, on how signs can be meaningful and, in relation to that, what kinds of signs there are, how they combine, and how some embody or incorporate others; (2) logical critic, or logic proper, on the modes of inference; and (3) speculative or [[universal rhetoric]], or methodeutic,<ref name="rhetoric"/> the philosophical theory of inquiry, including pragmatism.

==== Presuppositions of logic ====
In his "F.R.L." [First Rule of Logic] (1899), Peirce states that the first, and "in one sense, the sole", rule of reason is that, ''to learn, one needs to desire to learn'' and desire it without resting satisfied with that which one is inclined to think.<ref name="FRL"/> So, the first rule is, ''to wonder''. Peirce proceeds to a critical theme in research practices and the shaping of theories:

<blockquote><poem>...there follows one [[corollary]] which itself deserves to be inscribed upon every wall of the city of philosophy:
::Do not block the way of inquiry.</poem></blockquote>

Peirce adds, that method and economy are best in research but no outright sin inheres in trying any theory in the sense that the investigation via its trial adoption can proceed unimpeded and undiscouraged, and that "the one unpardonable offence" is a philosophical barricade against truth's advance, an offense to which "metaphysicians in all ages have shown themselves the most addicted". Peirce in many writings holds that [[Classification of the sciences (Peirce)|logic precedes metaphysics]] (ontological, religious, and physical).

Peirce goes on to list four common barriers to inquiry: (1) Assertion of absolute certainty; (2) maintaining that something is absolutely unknowable; (3) maintaining that something is absolutely inexplicable because absolutely basic or ultimate; (4) holding that perfect exactitude is possible, especially such as to quite preclude unusual and anomalous phenomena. To refuse absolute theoretical certainty is the heart of ''fallibilism'', which Peirce unfolds into refusals to set up any of the listed barriers. Peirce elsewhere argues (1897) that logic's presupposition of fallibilism leads at length to the view that chance and continuity are very real ([[tychism]] and [[synechism]]).<ref name="FCE"/>

The First Rule of Logic pertains to the mind's presuppositions in undertaking reason and logic; presuppositions, for instance, that truth and the real do not depend on yours or my opinion of them but do depend on representational relation and consist in the destined end in investigation taken far enough ([[#defs|see below]]). He describes such ideas as, collectively, hopes which, in particular cases, one is unable seriously to doubt.<ref>Peirce (1902), The Carnegie Institute Application, Memoir 10, MS L75.361–362, ''Arisbe'' [http://www.cspeirce.com/menu/library/bycsp/l75/ver1/l75v1-04.htm#m10 Eprint] {{Webarchive|url=https://web.archive.org/web/20110524021037/http://www.cspeirce.com/menu/library/bycsp/l75/ver1/l75v1-04.htm#m10 |date=2011-05-24  }}.</ref>

==== Four incapacities ====
<div style="padding:5px;font-size:94%;width:auto; border: 1px solid #a2a9b1; float:right;">The ''Journal of Speculative Philosophy'' series (1868–1869), including
* Questions concerning certain Faculties claimed for Man (1868)
* Some Consequences of Four Incapacities (1868)
* Grounds of Validity of the Laws of Logic:<br>Further Consequences of Four Incapacities (1869)</div> In three articles in 1868–1869,<ref name="QFM"/><ref name="SCFI">Peirce (1868), "Some Consequences of Four Incapacities", ''Journal of Speculative Philosophy'' v. 2, n. 3, [https://books.google.com/books?id=YHkqP2JHJ_IC&pg=RA1-PA140 pp. 140–157]. Reprinted ''Collected Papers of Charles Sanders Peirce'', 5.264–317, ''Writings of Charles S. Peirce'', 2:211–242, ''The Essential Peirce'', 1:28–55. ''Arisbe'' [http://www.cspeirce.com/menu/library/bycsp/conseq/cn-frame.htm Eprint].</ref><ref name="GVLL">Peirce, "Grounds of Validity of the Laws of Logic: Further Consequences of Four Incapacities", ''Journal of Speculative Philosophy'' v. II, n. 4, [https://books.google.com/books?id=YHkqP2JHJ_IC&pg=RA1-PA193 pp. 193–208]. Reprinted ''Collected Papers of Charles Sanders Peirce'', 5.318–357, ''Writings of Charles S. Peirce'', 2:242–272 (''Peirce Edition Project'', [http://www.iupui.edu/~peirce/writings/v2/w2/w2_23/v2_23.htm Eprint] {{Webarchive|url=https://web.archive.org/web/20100528064903/http://www.iupui.edu/~peirce/writings/v2/w2/w2_23/v2_23.htm |date=2010-05-28  }}), ''The Essential Peirce'', 1:56–82.</ref> Peirce rejected mere verbal or [[hyperbolic doubt]] and first or ultimate principles, and argued that we have (as he numbered them<ref name="SCFI"/>):
# No power of Introspection. All knowledge of the internal world comes by hypothetical reasoning from known external facts.
# No power of Intuition (cognition without logical determination by previous cognitions). No cognitive stage is absolutely first in a process. All mental action has the form of inference.
# No power of thinking without signs. A cognition must be interpreted in a subsequent cognition in order to be a cognition at all.
# No conception of the absolutely incognizable.
(The above sense of the term "intuition" is almost Kant's, said Peirce. It differs from the current looser sense that encompasses instinctive or anyway half-conscious inference.)

Peirce argued that those incapacities imply the reality of the general and of the continuous, the validity of the modes of reasoning,<ref name="GVLL"/> and the falsity of philosophical [[René Descartes|Cartesianism]] ([[#Against Cartesianism|see below]]).

Peirce rejected the conception (usually ascribed to Kant) of the unknowable thing-in-itself<ref name="SCFI"/> and later said that to "dismiss make-believes" is a prerequisite for pragmatism.<ref>Peirce (1905), "What Pragmatism Is", ''The Monist'', v. XV, n. 2, pp. 161–181, [https://archive.org/details/monist18instgoog/page/n201 see 167]. Reprinted ''Collected Papers of Charles Sanders Peirce'', 5.411–437, see 416. ''Arisbe'' [http://www.cspeirce.com/menu/library/bycsp/whatis/whatpragis.htm Eprint].</ref>

==== Logic as formal semiotic ====
Peirce sought, through his wide-ranging studies through the decades, formal philosophical ways to articulate thought's processes, and also to explain the workings of science. These inextricably entangled questions of a dynamics of inquiry rooted in nature and nurture led him to develop his semiotic with very broadened conceptions of signs and inference, and, as its culmination, a theory of inquiry for the task of saying 'how science works' and devising research methods. This would be logic by the medieval definition taught for centuries: art of arts, science of sciences, having the way to the principles of all methods.<ref name="ars"/> Influences radiate from points on parallel lines of inquiry in [[Aristotle]]'s work, in such ''loci'' as: the basic terminology of [[psychology]] in ''[[On the Soul]]''; the founding description of [[sign relation]]s in ''[[On Interpretation]]''; and the differentiation of [[inference]] into three modes that are commonly translated into English as ''[[Abductive reasoning|abduction]]'', ''[[Deductive reasoning|deduction]]'', and ''[[Inductive reasoning|induction]]'', in the ''[[Prior Analytics]]'', as well as inference by [[analogy]] (called ''paradeigma'' by Aristotle), which Peirce regarded as involving the other three modes.

Peirce began writing on semiotic in the 1860s, around the time when he devised his system of three categories. He called it both ''[[semiotic]]'' and ''semeiotic''. Both are current in singular and plural. He based it on the conception of a triadic [[sign relation]], and defined ''[[semiosis]]'' as "action, or influence, which is, or involves, a cooperation of ''three'' subjects, such as a sign, its object, and its interpretant, this tri-relative influence not being in any way resolvable into actions between pairs".<ref>Peirce 1907, ''Collected Papers of Charles Sanders Peirce'', 5.484. Reprinted, ''The Essential Peirce'', 2:411 in "Pragmatism" (398–433).</ref> As to signs in thought, Peirce emphasized the reverse: "To say, therefore, that thought cannot happen in an instant, but requires a time, is but another way of saying that every thought must be interpreted in another, or that all thought is in signs."<ref name="QFM"/>

Peirce held that all thought is in signs, issuing in and from interpretation, where ''sign'' is the word for the broadest variety of conceivable semblances, diagrams, metaphors, symptoms, signals, designations, symbols, texts, even mental concepts and ideas, all as determinations of a mind or ''quasi-mind'', that which at least functions like a mind, as in the work of crystals or bees<ref>See "[http://www.helsinki.fi/science/commens/terms/quasimind.html Quasi-mind]" in ''Commens Digital Companion to C.S. Peirce''.</ref>—the focus is on sign action in general rather than on psychology, linguistics, or social studies (fields which he also pursued).

Inquiry is a kind of inference process, a manner of thinking and semiosis. Global divisions of ways for phenomena to stand as signs, and the subsumption of inquiry and thinking within inference as a sign process, enable the study of inquiry on semiotics' three levels:

# Conditions for meaningfulness. Study of significatory elements and combinations, their grammar.
# Validity, conditions for true representation. Critique of arguments in their various separate modes.
# Conditions for determining interpretations. Methodology of inquiry in its mutually interacting modes.

Peirce uses examples often from common experience, but defines and discusses such things as assertion and interpretation in terms of philosophical logic. In a formal vein, Peirce said:

{{quote|''On the Definition of Logic''. Logic is ''formal semiotic''. A sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign, determined or created by it, into the same sort of correspondence (or a lower implied sort) with something, ''C'', its ''object'', as that in which itself stands to ''C''. This definition no more involves any reference to human thought than does the definition of a line as the place within which a particle lies during a lapse of time. It is from this definition that I deduce the principles of logic by mathematical reasoning, and by mathematical reasoning that, I aver, will support criticism of [[Weierstrass]]ian severity, and that is perfectly evident. The word "formal" in the definition is also defined.<ref>Peirce, "Carnegie Application", ''[[#NEM|The New Elements of Mathematics]]'' v. 4, p. 54.</ref>}}