C. I. Lewis published a system in his 1918 "Survey of Symbolic Logic".
This original system had the problem of "Modal Collapse" [Lp==p],
as Emil L. post soon pointed out. Lewis later repaired this system, in his
1920 paper
"*Strict Implication, An Emendation*". [Note that Lewis used the term
"Emendation", not the term "Amendation", observing a distinction not common
in English today.]
The details and references are at the bottom of this page.
This page covers the emended system, not the original, since the emended
system is (in my opinion -jh) the one intended by most writers citing it
at the time that referred to the "Survey System".

Note that the emended system is technically equivalent to S3 [Lewis and Langford, 1932, p500 (near bottom)] but is being kept separate for reasons of historic importance. It was studied separately (and separately referenced) for many years. Papers into the late 1930's still referred to this system as "The calculus of Strict Implication", or as "Lewis' Calculus of Strict Implication", and often as "The survey system".

Lewis attributes the basic ideas for this system to MacColl's
"*Symbolic Logic and its Applications.*"
[Lewis,
1918, p292]
This, in turn was largely a reprise of items in MacCall's articles in
"*Mind; A quarterly review of Psychology and Philosophy*".
These included:

- "
*Symbolical Reasoning*", Hugh MacColl, "*Mind; A quarterly review of Psychology and Philosophy*", Old Series, Vol 5, No. 17, January 1880, p45-60 [Yes, that is the spelling of the title in the original. -jh] - "Symbolic Reasoning (II.)", Hugh MacColl,
"
*Mind; A quarterly review of Psychology and Philosophy*", New Series, Vol. 6, No. 24, Oct., 1897 p493-510 - "Symbolic Reasoning (III.)", Hugh MacColl,
"
*Mind; A quarterly review of Psychology and Philosophy*", Mind, New Series, Vol. 9, No. 33, Jan., 1900 p75-84 - "Symbolic Reasoning (IV.)", Hugh MacColl,
"
*Mind; A quarterly review of Psychology and Philosophy*", New Series, Vol. 11, No. 43 (Jul., 1902), pp. 352-368 - "Symbolic Reasoning (V.)", Hugh MacColl,
"
*Mind; A quarterly review of Psychology and Philosophy*", New Series, Vol. 12, No. 47 (Jul., 1903), pp. 355-364 - "Symbolic Reasoning (VI.)", Hugh MacColl,
"
*Mind; A quarterly review of Psychology and Philosophy*", New Series, Vol. 14, No. 53 (Jan., 1905), pp. 74-81 - "Symbolic Reasoning (VII.)", Hugh MacColl,
"
*Mind; A quarterly review of Psychology and Philosophy*", New Series, Vol. 14, No. 55 (Jul., 1905), pp. 390-397 - "Symbolic Reasoning (VIII.)", Hugh MacColl,
"
*Mind; A quarterly review of Psychology and Philosophy*", New Series, Vol. 15, No. 60 (Oct., 1906), pp. 504-518 - "'If' and 'Imply'", Hugh MacColl,
"
*Mind; A quarterly review of Psychology and Philosophy*", New Series, Vol. 17, No. 65, Jan., 1908, p151-152 - "'If' and 'Imply'", Hugh MacColl,
"
*Mind; A quarterly review of Psychology and Philosophy*", New Series, Vol. 17, No. 67, Jul., 1908, p453-455

This is often credited as one of the first true modal logics. Lewis lists it as
"The Logic of Strict Implication" without other qualification.
(It is a continuation of research he published in
"*Mind; A quarterly review of Psychology and Philosophy*"
and "*Journal of Philosophy, Psychology, and Scientific Methods*"
[Lewis,
1918, 291]

The earlier papers included:

- "
*Implication and the Algebra of Logic*", by C. I. Lewis, in "*Mind; A quarterly review of Psychology and Philosophy*", New series, Volume 21, number 84, October 1912, p521-531 - "
*Interesting Theorems in Symbolic Logic*" by C. I. Lewis, in "*Journal of Philosophy, Psychology, and Scientific Methods*", Volume 10, number 9, (April 1913), p239-242 [Showing that there were more paradoxes of strict implication than most authors were commenting on. -jh] - "
*A New Algebra of Implications and some Consequences*", by C. I. Lewis, in "*Journal of Philosophy, Psychology, and Scientific Methods*" by C. I. Lewis, in "The Journal of Philosophy, Psychology, and Scientific Methods" Volume 10, number 16, July 1913, p428-438 [This gives an early axiom set -JH] - "
*The Matrix Algebra for Implication*" by C. I. Lewis, in "*Journal of Philosophy, Psychology, and Scientific Methods*", Volume 11, number 22, Oct. 1914, p489-600 [This gives an early axiom set -JH] - "
*The Calculus of Strict Implication*" by C.I. Lewis, in "*Mind; A quarterly review of Psychology and Philosophy*" new series, Volume 23, no. 90, April 1914, p240-247 [another axiom set. -JH] - "
*A too Brief set of Postulates for the Algebra of Logic*" by C. I. Lewis, in "*Journal of Philosophy, Psychology, and Scientific Methods*" Volume 12, no. 19, Sept 1915, p523-525 - "
*The Issues concerning Material Implication*" by C. I. Lewis, in "*Journal of Philosophy, Psychology, and Scientific Methods*" Volume 14, no. 13, Jun. 1917, p350-356

A significant rebuttal to the article
"*A too Brief set of Postulates for the Algebra of Logic*"
is Norbert Weiner's "*Mr. Lewis and Implication*" in
"The Journal of Philosophy, Psychology, and Scientific Methods"
Volume 13, 1916, p656-663

Emil L. Post pointed out that this this system exhibits "modal collapse" (Lp==p), and Lewis
pubilished "*Strict Implication, An Emendation*" in "The Journal of Philosophy, and Scientific Methods"
(Volume 17, number 11, May 1920, p300-302) to address the issue.
[Details of the fix are at end of table.]

Many other logicians of the time (such as H. B. Smith) were responding to perceived problems with "Russell and Whitehead's Material Implication". And many argued that material implication didn't capture what they thought of as implication. Lewis was one of them. Lewis's original system is based on an "impossibility" operator, and not (as his later work) on a possibility operator. Kurt Gödel's paper [Gödel, 1933 ] was responsible for this switch in the field.

On why "impossibility" instead of what we use today, Max Cresswell offers the theory that "...[Lewis] thought of Russell's logic as incomplete, and so a second kind of negation would reveal the ambiguity he discerned."

Peter K. Schotch offers the observation that "When Lewis came to Harvard he was put in Pierce's old office which contained all of Pierce's papers including his logical works (the so-called existential graphs). In a not particularly well-known work, Pierce attempted to extend this system to include modality. That system was called the "gamma graphs" and the primitive modality was impossibility." Dr. Schotch asks: "Just a coincidence?"

And I personally observe that Aristotle's system of modality ALSO had an impossibility operator (which he eventually shows equivalent to the negation of the possibility operator. -JH)

*** NOTE *** This is his emended system, not his original system. See the note at the bottom of the table.

Lewis's notation has a number of compromises, including reusing some "Principia Mathematica" notations with different meanings. The stated "excuse" (to use Lewis' terminology) is "typographical convenience". [Lewis, 1918, 292, fn1]

[I really can't manage all the symbols Lewis used, because they aren't available on all browsers. So I've compromised, given that his symbols were (by his own admission) a compromise for typographical convenience, I can claim I'm following tradition. -JH]

- Operators [Lewis, 1918, p292]
- Negation: -p, "p is false" [Note that this is a minus sign not a tilde sign]
- Impossibility: ~p, "p is impossible" or "It is impossible for p to be true" [Note that this is a tilde sign, not a minus sign]
- Logical Product: p & q, [Lewis uses p x q] "p and q both" or "p is true and q is true" [Lewis frequently just uses concatenation for this, but has the explicit symbol.]
- Equivalence: p = q

- Definitions [Lewis, 1918, 293]
- 1.01 Consistency: p*q =def ~~(p&q) [Lewis uses centered open circle]
- 1.02 Strict Implication: p => q =def ~(p&-q) [Lewis uses the fishhook implication symbol, I'm using "=>".] (*** Note that in [Lewis, 1936, p78] Lewis states that this definition is redundant -JH)
- 1.03 Material Implication: p > q =def -(p&-q) [Lewis uses backwards implication sign]
- 1.04 Strict Logical Sum: p ^ q =def ~(-p&-q)
- 1.05 Material Logical Sum: p + q =def -(-p&-q)
- 1.06 Strict Equivalence: (p=q) =def (p=>q)&(q=>p)
- 1.07 Material Equivalence: (p==q) =def (p>q)&(q>p) [Lewis uses triple bar for equivalence]

- Rules
- Axioms [Called "Postulates" by Lewis]
[Lewis,
1918, p294]
- 1.1 (p & q) => (q & p)
- 1.2 (q & p) => p
- 1.3 p => (p & p)
- 1.4 (p & (q & r)) => (q & (p & r))
- 1.5 p => -(-p))
- 1.6 ((p=>q) & (q => r)) => (p => r)
- 1.7 ~p => -p
- 1.8 (p => q) => (~q => ~p)

Actually, the original 1.8 says

"(p => q) = (~q => ~p)"

but Lewis (in responce to Emil L. Post's objections) published
"*Strict Implication, An Emendation*" in "The Journal of Philosophy, and Scientific Methods"
(Volume 17, number 11, May 1920, p300-302) which
changed it to the current

"(p=>q)=>(~q=>~p)"

© Copyright 2006, by John Halleck, All Rights Reserved.

This page is http://www.cc.utah.edu/~nahaj/logic/structures/systems/s.html

This page was last modified on April 22nd, 2010