Editing C. I. Lewis (section)

===Logic===
Lewis studied logic under his eventual Ph.D. thesis supervisor, [[Josiah Royce]], and is a principal architect of modern [[philosophical logic]]. In 1912, two years after the publication of the first volume of ''[[Principia Mathematica]]'', Lewis began publishing articles{{citation needed|reason=Give a few of them here.|date=November 2016}} taking exception to 
''Principia' ''s pervasive use of [[Material implication (rule of inference)|material implication]], more specifically, to [[Bertrand Russell]]'s reading of ''a''→''b'' as "''a'' implies ''b''." Lewis restated this criticism in his reviews{{citation needed|date=November 2016}} of both editions of ''Principia Mathematica''. Lewis's reputation as a promising young logician was soon assured.

[[Material implication (rule of inference)|Material implication]] (the rule of inference which claims that stating "P implies Q" is equivalent to stating "Q OR not P") allows a true consequent to follow from a false antecedent (so if P is not true still Q may be true since you only stated what a true P implies, but did not state what is implied if P is untrue).

Lewis proposed to replace the usage of material implication during discussions involving logic with the term [[Strict conditional|strict implication]], by which a ([[Contingency (philosophy)|contingently]]) false antecedent, which is false but could have been true, does not always strictly imply a (contingently) true consequent, which is true but could have been false. The same logical result is implied, but in a clearer and more explicit way. Stating '''strictly''' that P implies Q is explicitly not stating what the untrue P implies. And therefore if P is not true, Q may be true, but may be false as well.<ref name="hughes">Hughes and Cresswell (1996: chapt. 11)</ref>

As opposed to material implication, in strict implication the statement is not [[primitive notion|primitive]] - it is not defined in positive terms, but rather in the combined terms of [[negation]], [[Logical conjunction|conjunction]], and a prefixed unary [[intensional statement|intensional]] [[modal operator]], <math>\Diamond</math>. The following is its formal definition: 
::If ''X'' is a formula with a [[boolean domain|classical bivalent]] [[truth value]]
::(which must be either true or false), 
::then <math>\Diamond</math>''X'' can be read as "''X'' is possibly true".<ref name="hughes" />

Lewis then defined "''A'' strictly implies ''B''" as "<math>\neg \Diamond</math>(''A''<math>\land \neg</math>''B'')".<ref>Cooper H. Langford and C. I. Lewis, Symbolic Logic (New York, 1932), p. 124.</ref> Lewis's strict implication is now a historical curiosity, but the formal [[modal logic]] in which he grounded that notion is the ancestor of all modern work on the subject. Lewis' <math>\Diamond</math> notation is still standard, but current practice usually takes its dual, the square notation <math>\square</math>, meaning "necessity", which is stating a [[primitive notion]], while the diamond notation, <math>\Diamond</math>, is left as a defined (derived) meaning. With square notation "''A'' strictly implies ''B''" is simply written as <math>\square</math>(''A''→''B''), which states explicitly that we are only implying the truth of B when A is true, and we are not implying anything about when B can be false, nor what A implies if it is false, in which case B can be false or B can just as well be true.<ref name="hughes" />

His first published monograph about advances in logic since the time of [[Gottfried Wilhelm Leibniz|Leibniz]], ''A Survey of Symbolic Logic'' (1918), culminating a series of articles written since 1900, went out of print after selling several hundred copies. At the time of its publication, it included the only discussion in English of the logical writings of [[Charles Sanders Peirce]].<ref>[https://www.jstor.org/stable/40321109 Lewis, Peirce and the Complexity of Classical Pragmatism], Richard Robin [[JSTOR]] Scientific Periodical website. Lewis being the first to deal with Peirce is not recorded in [https://plato.stanford.edu/entries/peirce/ the Peirce entry] of the online Stanford Encyclopedia of Philosophy</ref>  This book followed Russell's 1900 monograph on Leibnitz, and in later editions he removed a section that seemed similar to it.<ref>Bertrand Russell, ''A Critical Exposition of the Philosophy of Leibniz'' (Cambridge: The University Press, 1900).</ref><ref>[https://plato.stanford.edu/entries/lewis-ci/ About Lewis] in the online [[Stanford Encyclopedia of Philosophy]]</ref>

Lewis went on to devise [[modal logic]] which he described in his next book ''Symbolic Logic'' (1932) as possible formal analyses of the [[Alethic modality|alethic modalities]], modes of logical truth such as necessity, possibility and impossibility.

Several amended versions of his first book "A Survey of Symbolic Logic" have been written over the years, designated as S1 to [[S5 (modal logic)|S5]], the last two, S4 and S5, generated much mathematical and philosophical interest, sustained to the present day and are the beginnings of what became the field of [[normal modal logic]].<ref name="hughes" />