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CHAPTER 16
History of Logic
Logic was born in ancient Greece and reborn a century ago. Logic keeps grow-
ing and expanding, and has contributed to the birth of the computer age. We
can better understand and appreciate logic by studying its history.
76.7 Ancient logic
The formal study of valid reasoning began with Aristotle (384-322 BC) in ancient
Greece. An unprecedented emphasis on reasoning prepared for Aristotle's logic.
Greeks used complex reasoning in geometry, to prove results like the Pythago-
rean theorem. Sophists taught rich young men to gain power by arguing effec-
tively (and often by verbal trickery). Parmenides and Heraclitus reasoned about
being and non-being, anticipating later disputes about the law of non-contradic-
tion, and Zeno reasoned about paradoxes. Socrates and Plato gave models of care-
ful philosophical reasoning; they tried to derive absurdities from proposed views
and sought beliefs that could be held consistently after careful examination.
Reasoning is an important human activity, and it didn't begin in ancient
Greece. Is this ability biologically based, built into our brains by evolution
because it aids survival? Or does it have a divine origin, since we're made in the
"image and likeness" of God? Or do both explanations have a place ? Logic raises
fascinating issues for other Cisciplines.
Aristotle began the study of logic. He was the first to formulate a correct prin-
ciple of inference, to use letters for terms, and to construct an axiomatic system.
He created syllogistic logic (Chapter 2), which studies arguments like these
(using "all A isB," "no A is B," "some A is 8," or "some A is not B"):
I i''
n1'-,
This is inctti:.
deriving them
showing that t
of syllogism'
from what. I-::
such as defir..:
Aristotle ::
states that tlE
to the same .:
true at the sa:
ments. Arist,-:
more certain :
an infinite se:
Deniers of th-;
point home. -,,,'
to stop. Arisi;
or "S is not P
Aristotle :...
Chapters 10;:
not happen (
battle tomorr,:'
necessarv u-he
Aristotle relec;
He seemed to ,
excluded midJ
he anticipateJ
t'alse ($77.7). )
box-inside/bor
them doubtfu-
After Aristc,
"arrd," and "or
defended, for e
tollens (denr ir.
Valid ,
argLtment
A11 humans are mortal. all H ls M
Al1 Greeks are humans. all G is H
.'. A11 Greeks are mortal. .'. ail G is M
This is aslidbecause of its formal structure, as given by the formulation on the
right; any argument having this same structure will be valid. If we change the
structure, we may get arr invalid argument, iike this one:
:keeps grow-
uter age. We
v.
BC) in ancient
istotle's logic.
the Pythago-
rrguing effec-
:asoned about
ron-contradic-
rodels of care-
roposed views
mination.
3in in ancient
by evolution
:e made in the
e? Logic raises
a correct prin-
rmatic system.
nts like these
B"),
;M
;H
rM
ulation on the
rve change the
HISTORY OF LOGIC 349
All Romans are mortal. all R is M
tnoatfi t All Greeks are mortal. all G is M
argument .'. All Greeks are Romans. .'. all G is R
This is inaalid because its form is wrong. Aristotle defended valid forms by
deriving them from self-evidently valid forms; he criticized invalid forms by
showing that they sometimes give true premises and a false conclusion. His logic
of syllogisms is about logic in a narrou) sense, since it deals with what follows
from what. He also pursued other topics that connect with appraising arguments,
such as definitions and fallacies; these are about logic in abroader sense.
Aristotle proposed two principles of thought. His law of non-contradiction
states that the same property cannot at the same time both belong and not belong
to the same object in the same respect. So "S is P" and "S is not P" can't both be
true at the same time, unless we take "5" or "P" differently in the two state-
ments. Aristotle saw this law as so certain that it can't be proved by anything
more certain; not all knowledge can be demonstrated, since otherwise we'd need
an infinite series of arguments that prove every premise by a further argument.
Deniers of the law of non-contradiction assume it in their practice; to drive this
point home, we might bombard them with contradictions until they plead for us
to stop. Aristotle also supported the law of exduded middle, that either "S is P"
or "S is not P" is true. Some deviant logics today dispute both laws (Chapter 17).
Aristotle also studied the logic of "necessary" and "possible" (see modal logic,
Chapters l-0 and L1). He discussed future contingents (events that may or may
not happen). Consider a possible sea battle tomorrow. If "There utill be a sea
battle tomorrow" ("S" below) is now either true or false, this seems to make
necessary whether the battle occurs:
Either it's true that S or it's false that S.
If it's true that S, then it's necessary that S.
If it's false that S, then it's necessary that not-S.
.'. Either it's necessary that S or it's necessary that not-S.
Aristotle rejected the conclusion, saying that there was no necessity either way.
He seemed to deny the first premise and thus the universal truth of the law of
excluded middle (which he elsewhere defends); if we interpret him this way, then
he anticipated many-valued logic in using a third truth rralue besides true and
false ($17.1). Another solution is possible. Many think premises 2 and 3 have a
box-inside/box-outside ambiguity ($10.1): taking them as "(A, nB)" makes
them doubtful while taking them as "E(A, B)" makes the argument invalid.
After Aristotle, Stoics and others developed a logic that focused on "if-then,"
" and," and " or ," Iike our propositional logic (Chapters 6 and, 7). Stoic logicians
defended, for example, an important inference form that came to be called modus
t ollens (denying mode) :
350 PART FOUR: FURTHER VISTAS
Valid )
argLLment
Stoics also studied modal logic. Unlike logicians today, they took "necessary"
and "possible" in a temporal sense, like "true at all times" and "true at some
times." They disputed whether there'was a good modal argument for t'atalism,
the view that all events happen of inherent necessity (see $10.3b #10). They also
disputed how to understand "lf A then B" t577.4). Philo of Megara saw it as true
if and only if it's not now the case that A is true and B is false; this fits the modern
truth table for "if-then." Diodorus Chronos saw it as true if and only rf Ais nerser
at any time true while B is false.
Aristotelian and Stoic logic were first seen as rivals, differing in three ways:
o Aristotle focused on "all," "r1o," and "some." Stoics focused on "if-then,"
" and," and " or."
o Aristotle used letter variables and expressed arguments as long conditionals,
hke "lf ail A is B, and all C is A, then a1l C is B." Stoics used number variables
and expressed arguments as sets of statements, like "If 1 then 2. But not-2.
Therefore, not-1."
o Aristotle saw logic not as part of philosophy but rather as a tool for all think-
ing. Stoics saw logic as one of philosophy's three branches (the other two being
physics and ethics). But both agreed that students should study logic early,
before going deeply into other areas.
Later thinkers combined these approaches into traditional logic. For the next two
thousand years, Aristotle's logic with Stoic additions ruled in the West.
At the same time, another tradition of logic rose up in India, China, and Tibet.
We call it Buddhist logic even though Hindus and others pursued it too. It
studied many topics important in the West, including inference, fallacies, and
Ianguage. This is a common pattern in Buddhist logic:
Here there is fire, because there is smoke.
Wherever there is sn-roke there is fire, as in a kitchen.
Here there is smoke.
.'. Here there is fire.
The last three lines are deductively valid:
Al1 cases of smoke are cases of fire.
This is a case of smoke.
.'. This is a case of fire.
This omits "as in a kitchen," which suggests inductive reasoning (Chapter 5); in
our experience of smoke and fire, smoke always seems to involve fire.
If your view is correct, then
such and such is true.
Such and such is false.
. Your view isn't correct.
If C then S
Not-S
Not-C
The Easte:
covers mai--r.'
commenta: - :
thinking .:- ;,
caution :.:.-:.
der.ianr - -- :- -,
Iai.','..=
using :.:": -,
thinhr.: :: ,
and r,r-=:-:,-
Ludrvis i'..::=
a stronE-',' :: -
76.2 l;.
Mediera- ----
logic beca:: =
The C:.:-;:
Ages. H. '.'.:
box-in'ti= : -
foreknorr-. i =
tle's 1oS-; ---:
On lnt;""-'-. ..
The -{::: "
Christia:. ::
ogy and :-.=:
taries, tex:r -
logic, co:.j-:-
ish Sparn -.'.
=:
In Ch:-.:,"
Anselm i'.:=
terior ,1,:.;.'. :
was basei ::.
Peter of S:.:
The cler'=:
Capitalized :
i:1en s
:-S
:.C
,:k "necessary"
i true at some
:'.: ior t'atalism,
=10). They also
ra saw it as true
irts the modern
:.lr- if A is nerser
:r three ways:
,r', "if-then,"
: ;rnditionals,
:.:.er variables
l, But not-2.
, :, r all think-
:,r,er two being
:',- logic ear1y,
:,rr the next two
: 'est.
:.rna, and Tibet.
:;ued it too. It
e iallacies, and
: Chapter 5); in
: iire.
HISTORY OF LOGIC 351.
The Eastern logic tradition is poorly understood in the WesU this tradition
covers many centuries, and many texts are difficult or untranslated. Some
commentators emphasize similarities between East and West; they see human
thinking as essentially the same everywhere. Others emphasize differences and
caution against imposing a Western framework on Eastern thought. And some
deviant logicians see the Eastern tradition as congenial to their views.
Many see the East as more mystical than logical; Zen Buddhism delights in
using paradoxes (like the sound of one hand clapping) to move us beyond logical
thinking toward a mystical enlightenment. But East and West both have logical
and mystical elements. Sometimes these come together in the same individual;
Ludwig Wittgenstein in the early 20th century invented truth tables but also had
a strongly mystical side.
76.2 Medieval logic
Medieval logicians carried on the basic framework of Aristotle and the Stoics, as
logic became important in higher education.
The Christian thinker Boethius (480-521) helped the transition to the Middle
Ages. He wrote on logic, including commentaries; he explained the modal
box-inside/box-outside ambiguity as he defended the compatibility of divine
foreknowledge and human freedom (S10.3b #4 and #14). He translated Aristo-
tle's logic into Latin. Many of his translations -were lost; but his Categories and
On lnterpretation became the main source for the logica aetus (old logic).
The Arab worid dominated in logic from 800-1200. Some Arab logicians were
Christian, but most were Muslim; both groups saw logic as important for theol-
ogy and medicine. They translated Aristotle into Arabic and wrote commen-
taries, textbooks, and original works. They pursued topics like syllogisms, modal
logic, conditionals, universals, predication, and existence. Baghdad and Moor-
ish Spain were centers of logic studies.
In Christian Europe, logic was reborn in the 11th and 12th centuries, with
Anselm, Peter Abelard, and Latin translations of Aristotle's Prior Analytics, Pos-
terior Analytics, Topics, and Sophistical Ret'utations; the logica rzooa (new logic)
was based on these. There was interest in universals and in how terms signify.
Peter of Spain and William of Sherwood wrote logic textbooks.
The ciever Barbara-Celarent verse was a tool for teaching syllogisms:
Barbara, Celarent, Darii, Ferioque, prioris;
Cesare, Camestres, Festino, Baroco, secundae;
tertia, Darapti, Disamis, Datisi, Felapton,
Bocardo, Ferison, habet; quarta insuper addit
Bramantip, Camenes, Dimaris, Fesapo, Fresison.
Capitalized names are valid syllogisms. Vowels are sentence forms:
352 PART FOUR: FURTHER VISTAS
A all-is-
I some-is-
E no-is-
O some - is not -
ailMisP
allSisM
allSisP
Aff -Irm universal/particular
nE-gO universal/particular
MP
SM = irgure I
So "Barbara," with AAA vowels, has three "all" statements:
fallacr' -
o If the ;:
ment t: -
. If the c--:
In the [::
tron.Ete: -. :
priori . . ;'
16.-1 :. -_
A -l --, - -
ntt:._.: i
COnil-: -:= - -
isB':-:,.--
with =--- . .
Venn :'-. = -
har-e a::= = -
logrc :._:.- -
impro..'. :., -
rer.olu:- - :
Ihe (,=-
Hegel F:-:,::
evolr-es i.-i - .
together .--. , -
applied il".:, :
logic as a:. .-,=
flicting p:. :. -
self -contr;. i, ::
and time ;--.: :
I'he p1-.--, . .
future der'. -:
reduce reai :--
their pencrl- :.:
like that oi
"c
Many thr:..
Morgan prt: -
"A08"; a lei:=
known for hr'
Aristotelian syllogisms have two premises. Middle term "M" is common to both
premises; predicate "P" occurs in the first premise, while subject "S" occurs in
the second. There are four figures (arrangements of premise letters):
1 (prioris) 2 (secundae) 3 (tertia) 4 (quarta)
MP PM MP PM
SM SM MS MS
Aristotle's four axioms are valid first-figure forms:
Barbara
allMisP
allSisM
.'. all S is P
Celarent Darii
noMisP allMisP
allSisM someSisM
noSisP .'.someSisP
Ferio
noMisP
some S is M
some S is not P
The other 15 forms can be derived as theorems. The consonants give clues on
how to do this; for example, "m" says to switch the order of the premises.
Thomas Aquinas (722+-74), the most influential medieval philosopher, had
little impact on logic's development; but he made much ,r" of logic. Since he
emphasized reasoning and wrote so much, he likely produced more philosophical
arguments than anyone else who has ever lived.
Fourteenth-century logicians include William of Ockham and |ean Buridan.
Ockham's razor says "Accept the simplest theory that adequately explains the
data." Ockham developed modal logic and tried to avoid metaphysics when
analyzing language. Buridan's ass was a fictional donkey whose action was par-
alyzedwhen he was placed exactly midway between two food bowls. Buridan also
formulated the standard rules for valid syllogisms; one version says that a syllo-
gism is aalid just if it satisfies all of these conditions:
. Ever] term distributed in the conclusion must be distributed in the premises.
(A term is distributed in a statement just if the statement makes some claim
about e-oery entity that the term refers to.)
. The middle term must be distributed in at least one premise. (The middle term
is the one common to both premises; if we violate this rule, we commit the He complarneJ
Llar
Iar
HISTORY OF LOGIC 353
fallacy of the undistributed middle.)
. If the conclusion is negative, exactly one premise must be negative. (A state-
ment is negatiae if it contains "rro" ot "not"; otherwise it's positive.)
. If the conclusion is positive, both premises must be positive.
In the Middle Ages, logic was important in philosophy and in higher educa-
tion. Even today, logic, like biology, uses many Latin terms (modus ponens, e
prioril a posteriori, de rel de dicto, and so on).
76.3 Enlightenmentlogic
Aristotelian logic dominated until the end of the 19th century. Several logicians
contributed to syllogistic logic; for example, Leonhard Euler diagrammed "all A
is B" by putting an A-circle inside a larger B-circle, Lewis Carroll entertained us
with silly syllogisms and points about logic in Alice in Wonderland, andlohn
Venn gave us diagrams for testing syllogisms (52.6). But most logicians would
have agreed with Immanuel Kant, who said that Aristotle invented and perfected
logic; nothing else of fundamental importance could be added, although we might
improve teaching techniques. Kant would have been shocked to learn about the
revolution in logic that came about a hundred years later.
The German thinkers Georg Hegel and Karl Marx provided a side current.
Hegel proposed that logic should see contradictions as explaining how thought
evolves historically; one view provokes its opposite, and then the two come
together in a higher synthesis. Marx saw contrafictions in the world as real; he
applied this to political struggles and revolution. While some saw this dialectical
logic as an alternative to traditional logig critics objected that this confuses con-
flicting properties in the world (like hot/cold or capitalist/proletariat) with logical
self-contradictions (like the same object being both white and, in the same sense
and time and respect, also non-white).
The philosopher Gottfried Leibniz, the co-inventor of calculus, anticipated
future developments. He proposed the idea of a symbolic language that would
reduce reasoning to calculation. If controversies arose, the parties could take up
their pencils and say, "Let us calculate." Leibniz created a logical notation much
like that of Boole (and much earlier); but his work was published after Boole.
Many thinkers tried to invent an algebraic notation for logic. Augustus De
Morgan proposed symbolizing "all A is B" as "A))B" and "some A is B" as
"A0B"; a letter on the concave side of the parenthesis is distributed. He became
known for his De Morgan laws for propositional logic:
Not both A and B = Either not-A or not-B
Not either A or B = Both not-A and not-B
He complained that current logic couldn't handle relational arguments like "A11
; common to both
iect "5" occurs in
tters):
fta)
[,
;
Ferio
fisP
:SisM
:SisnotP
nts give clues on
e premises.
philosopher, had
of logic. Since he
rore philosophical
md Jean Buridan.
rtely explains the
retaphysics when
;e action was Par-
rwls. Buridan also
says that a syllo-
in the premises.
rkes some claim
fhe middle term
we commit the
354 PART FOUR: FURTHER VISTAS
dogs are animals; therefore all heads of dogs are heads of animals,, (S9.5b #25).
The Boolean algebra of George Boole iig15-64) was a breakthro,lgh, ,ir.,." it
used math to check the correctness of inferences. Boole used letters io, ,.,r; ,o
"M" might be the set of mortals and "H" the set of humans. putting two letters
togetherrepresents the intersectiott ol the sets; so ,,HM,,
is the set J those who
are both human and mortal. Then "All humans are mortal,, is ,,H
= HM,,, which
says that the set of humans = the set of those who are both human and mortal.
A syllogism is a series of equations:
Valid ,
argutnent
All humans are mortal. H = HM
All Greeks are humans. G = GH
. . All Greeks are mortal. ... G = GM
we can derive the conclusion by substituting equals for equals. In premise 2,
9 = GH,_replace "H" wirh /tHM/'/ (premise f ,ry, H = HMj to get G = GHM.
Then replace "GH" w_r1h "G" (premise 2 says G = GH) ,o g.i G = GM.
Boolean formulas, like those on the left below lwhiih .,r. , lrt", symbolism),
can be interpreted to be about sets or about statements:
" -4" can mean ,,the
set of non_A s,, or ,,not_A,,
" AaB" can mean "the intersection of sets A and 8,, or ,,A
and B,,
" AwB" can mean ,,the
union of sets A and 8,, or ,,
A or 8,,
so if " A" is the set of animals, then "-A" is the set of non-animals; but if ,,A,,
is
"Aristotle is a logician," then " -A" is "Aristotle isn't a logician.,, The same iaws
cover both; for example , " A^B = Ba)A" works for eith"er sers or statements.
Boolesn op-erators (like "and," " or," and "not") use the statement interpretation.
Boole, the father of.mat.hematical logic, thought rhat logic belonged with
mathematicians instead of philosoph"rr. Brrt both lroups .r-J,o have an inter-
est in logic, each getting the slice of the action thrtTits it better. While Boole was
important, a greater revolution in logic was to come.
76.4 Frege and Russell
Gottlob Frege (1848-r92s) created modern logic with his tlzg Begrit'fsschrit't
("conceptwriting"). Its BB pages introduced a symbolism that, for th"" fiirt ti-f,
let us combine in every way Aristotle,s ,,a11,,, ,;no,,,and ,,some,,
with the Stoic
" if-then," " and," and " ot.',
-So
we can symbol ize ,,If everything that,s A or B is
then C and D, then everything that's non-D is non-A.,, Thus"the gap between
Aristotle and the stoics was overcome in a higher synthesis. r.ege"rlro showed
how to analyze arguments with relations (like-"x loves y,,) and m"ultiple q.ranti-
fiers; so we can show that "There is someone thut
"r"ryone
loves,, entails
"Everyone loves someone" - but not conversely. F.ege presented logic as a
formal sv:::'
of formulas :
Frege's rrc
early 20th ce:
for "not," "ti
These can .c:
This n'as :-s -
simpler n.:::-
Frege ie '. ;-
wanted ta ::.-
purelr, lol::"-
axioms ar.; -:
condition --.-. :,
condition : -- -
some sets ::a
object) rvh:-.
not a men:i'.:
members c:::,
if x is not ; =.,
not a merr.ral
Russell ask;:
principle, R .;
SoisRam.:
way we get I :
provable in F:
life work coll:
Russeil gre,
minds worke:
needed fixins
.als" ($9.5b #25).
kthrough, since it
[etters for sets; so
)utting two letters
e set of those who
"H=H]ly'r," which
urnan and mortal.
=HM
=GH
=GM
ra1s. In premise 2,
togetG=GHM.
G=GM.
r later symbolism),
A and B"
or B"
imals; but if "A" is
m." The same laws
rets or statements.
rent interpretation.
gic belonged with
re to have an inter-
:r. While Boole was
879 Begrit'fsschrift
rt, for the first time,
me" with the Stoic
ing that's A or B is
rs the gap between
Frege also showed
nd multiple quanti-
rone loves" entails
resented logic as a
HISTORY OF TOGIC 355
t'ormal system, with purely notational rules for determining the grammaticality
of formulas and the correctness of proofs.
Frege's work was ignored until Bertrand Russell (1.872-L970) praised it in the
early 20th century. Frege's difficult symbolism alienated people. He used lines
for " not," " if-then," and " all" '.
Not-A
---rA
If A then B For all x
--pB -Y
La
These can combine to symbolize "Not all A is non-B" (our "-(x)(Ax > -Bx)"):
-=*[Nx
This was also his way to write "Some A is B" (our "(Ix)(Ax'B")"); he had no
simpler notation for "some" or "and."
Frege developed logic to help show that arithmetic is reducible to logic; he
wanted to define all basic concepts of arithmetic (like numbers and addition) in
purely logical terms and prove all basic truths of arithmetic using just logical
axioms and inference rules. Frege used a seemingly harmless axiom that every
condition on x picks out a set containing just those elements that satisfy that
condition; so the condition "x is a cat" picks out the set of cats. But consider that
some sets are members of themselves (the set of abstract objects is an abstract
object) while other sets aren't (the set of cats isn't a cat), By Frege's axiom, "x is
not a member of itself" picks out the set containing just those things that are not
members of themselves. Call this "set R." So any x is a member of R, if and only
if x is not amember of x (here "€" means "is a member of" and"e" means "is
not a member of"):
For all x, x € R if and only if x c x.
Russell asked in a 1902letter to Frege: What about set R itself ? By the above
principle, R is a member of R, if and only if R is not a member of R:
R e Rif andonlyif R e R.
So is R a member of itself? If it is, then it isn't - and if it isn't, then it is; either
way we get a contradiction. Since this contrafiction, called Russell's paradox, was
provable in Frege's system, that system was flawed. Frege was crushed, since his
life work collapsed. His attempts to fix the problem weren't successful.
Russell greatly admired Frege and his groundbreaking work in logic; the two
minds worked along similar lines. But the paradox showed that Frege's work
needed fixing. So Russell, with his former teacher Alfred North Whitehead,
355 PART FOUR: FURTHER VISTAS
worked to develop logic and set theory in a way that avoided the contradiction.
They also developed a more intuitive symbolism (much like what we use in this
book), based on the work of Giuseppe Peano. The result was their massive Prln-
cipia Mathemqtica, which was published in 1910-7973. Principiahad a huge in-
fluence and became the standard formulation of the new logic.
76.5 After Principia
Classical symbolic logic includes propositional and quantificational logic (Chap-
ters 6 to 9). A logic is "classical" if it accords with Frege and Russell about which
arguments are valid, regardless of differences in symbolization and proof tech-
niques. Classical symbolic logic gradually became the new orthodoxy, replacing
the older Aristotelian logic.
Much work was done to solidlfy classical symbolic logic. Different proof tech-
niques were deveioped; while Frege and Russell used an axiomatic approach, later
logicians invented inferential and truth-tree methods that were easier to use.
Different ways of symbolizing arguments were developed, including the Polish
nototion of a school of logic that was strong in Poland between the world wars.
Ludwig Wittgenstein and Emil Post independently invented truth tables, which
clarified our understanding of logical connectives (like "if-thetr," "and," and
"or") and led to a criterion of validity based on semantics - on the meaning of
the connectives and how they contribute to truth or falsity; Alfred Tarski and
others expanded the semantic approach to quantificational logic.
Much work was done in metalogic, the study of logical systems (Chapter 15).
Kurt Godel showed that Russell's axiomatization of classical logic was, given cer-
tain semantic assumptions, correct: iust the right things were provable. But he
also showed, against Frege and Russell, that arithmetic cannot be reduced to any
formal system: no consistent set of axioms and inference rules would suffice to
prove all arithmetic truths; this result, called Godel's theorem, is perhaps the
most striking and surprising result of 2Oth-century logic. Alonzo Church showed
that the problem of determining validity in quantificational logic cannot be
reduced to an mechanical algorithm (a result called Church's theorem). There
was also much activity in set theory, which after Russell's paradox became
increasingly complex and controversial.
There was also much work in philosophy of logic (Chapter 1B), which deals
with philosophical questions about logic, such as these: Are logical truths de-
pendent on human conventions (so different conventions might produce differ-
ent logical truths) or on the objective nature of reality (perhaps giving us the
framework of any possible language that would be adequate to describe reality)?
Can logic help us clarify metaphysical issues, such as what kinds of entity ulti-
mately exist? Should we assume abstract entities (like properties and proposi-
tions) when we do logic? How can we resolve logical paradoxes (such as Russell's
parado-t --, -
distort : -,
the de:-:::-
Los: i
was i:;:
lo'l:: _:. ,,.
in ::.. -:: .
tain: -',-:.- :
Neun":.: -.
electrc:..--,:
and so::'.'. ,
marn i.: ':-
T^---- -
LL)q-. _
approa-:. .
sensor- j-,
branch.= -
'
-1! L- i::
Trvo tr :=-
classic:, - :
for erar.:-:
sical lc:-: '
The :' .
"nece=ir:-.
pursue.- _:. .
Leu'is's ','.' -
arguei::,:
and qr-.a:.:-.
erties. l:.1
Kripke :, .
'
sense oi -. .
useful lr. ,:
phers r:.:-.
properii. i
Other .'.
of knortl;:,
tionship l
future ti:-..
(Chapter= -
supplerr,e:.:
De nili'..
needs to 'le
values. 1:r
fuzzy-1ogii
(0.00). or:
led the contradiction.
le what we use in this
as their rnassive Prin-
incipia had a huge in-
)sc.
iicational logic (Chap-
d Russell about which
:ation and proof tech-
'orthodoxy, replacing
. Different proof tech-
omatic approach, later
at were easier to use.
l, including the Polish
:ween the world wars.
ed truth tables, which
"if-then," "and," artd
r - on the meaning of
ity; Alfred Tarski and
.logic.
systems (Chapter 15).
al logic was, given cer-
were provable. But he
rnot be reduced to any
rules would suffice to
eorem, is perhaps the
ilonzo Church showed
ional logic cannot be
rch's theorem). There
sell's paradox became
apter 18), which deals
Are logical truths de-
might produce differ-
perhaps giving us the
te to describe reality)?
at kinds of entity ulti-
roperties and proPosi-
oxes (such as Russell's
HISTORY OF LOGIC 357
paradox and the liar paradox) ? Are logical truths empirical or a priori? Does logic
distort ordinary beliefs and ordinary language, or does it correct them? What is
the definition and scope of logic?
Logic was important in the development of computers. The key insight here
was that logical functions like "and" and " or" can be simulated electrically by
logic gates; this idea goes back to the American logician Charles Sanders Peirce
in the 1880s and was rediscovered by Claude Shannon in 1938. A computer con-
tains logic gates, plus memory and input-output devices. Logicians like |ohn von
Neumann, Alan Turing, and Arthur Burks helped design the first large-scale
electronic computers. Since logic is important for computers, in both hardware
and software, it's studied today in computer science departments. So now three
main departments study logic - philosophy, math, and computer science.
Logic today is also an important part of cognitioe science, an interdisciplinary
approach to thought that includes linguistics, psychology, biology (brain and
sensory systems), computers (especially artificial intelligence), and other
branches of philosophy (especially epistemology and philosophy of mind).
As classical symbolic logic became the orthodoxy, it started to be questioned.
Two types of non-classical logic came to be. Supplementary logics accepted that
classical logic was fine as far as it went but needed to be supplemented to deal,
for example, with "necessary" and "possible." Deviant logics thought that clas-
sical logic was wrong on some points and needed to be changed.
The most important supplementary logic is modal logic, which deals with
"necessary" and "possible" (Chapters 1,0 and tt). Ancient and medieval logicians
pursued modal logic; but 2Oth-century logicians mostly ignored it until C. L
Lewis's work in the 1930s. Modal logic then became controversial. Willard Quine
argued that it was based on a confusion; he thought logical necessity was unclear
and quantified modal logic led to an objectionable metaphysics of necessary prop-
erties. There was lively debate on modal logic for many years. In 1959, Saul
Kripke presented a possible-worlds way to explain modal logic; this made more
sense of it and gave it new respect among logicians. Possible worlds have proved
useful in other areas and are now a common tool in logic; and several philoso-
phers (including Alvin Plantinga) have defended a metaphysics of necessary
properties. Today, modal logic is a well-established extension of classical logic.
Other extensions apply to ethics ("A ought to be done" or "A is good"), theory
of knowledge ("X believes that A" or "X knows that A"), the part-whole rela-
tionship ("X is a part of Y"), temporal relationships ("It will be true at some
future time that A" and "lt was true at some past time that A"), and other areas
(Chapters 12toL4). Most logicians would agree that classical logic needs to be
supplemented in order to cover certain kinds of argument.
Deaiant logics say that classical symbolic logic is wrong on some points and
needs to be changed (Chapter 17). Some propose using more than two truth
values. Maybe we need a third truth value for "half-true." Or maybe we need a
fuzzy-logicrange of truth values, from completely true (1.00) to completely false
(0.00). Or perhaps " A" and "not-A" can both be false (intuitionist logic) or both
358 PART FOUR: FURTHER VISTAS
be true (paraconsistent logic). Or perhaps the classical approach to "if-then" is
flawed; some views even reject modus ponens ("If A then B, A .'. B") and modus
tollens ("lf A then B, not-B .'. not-A"). These and other devrant logics have been
proposed. Today there is much questioning of basic logical principles.
This brief history of logic has focused on deductive logic and related areas.
There has also been much interest in informal logic (Chapters 3 and 4), inductive
logic (Chapter 5), and history of logic (this chapter).
So logic has a complex history - from Aristotle and the Stoics in ancient
Greece, through the Middle Ages and the Enlightenment, to the turmoil of the
19th century and logic's transformation with Frege and Russell, and into recent
classical and non-classical logics and the birth of the computer age.1
1
For *or. on the history of logic, I suggest P. H. Nidditch's The Development ot' Mathematical
Logic (London: Routledge & Kegan Paul,1962) and, for primary sources, Irving Copi and James
Gould's Readings on Logic (New York: Macmillan, 1954). Also useful are William and Martha
Kneale's The Deaelopment of Logic (Oxford: Clarendon, 1962) and Joseph Bocheisl<t's A History
of Formal Logic, trans.Ivo Thomas (Notre Dame, Ind.: University of Notre Dame, 1951).
Deviant l:+:
5LaL(1,ic-.-: i-:
onh :r;:: . '.
t4'O tru::. '. : '.
trUelPa:::-,:
IF-THE.- .=
Der-t,":-. - , ,
SOme ai:;-=. .
ard 1oe-; =- :
would h.:::.:
are r.aiii ": i '
17.7 l::
Most 1oE-:.-
proposit - :.=.
and false ..
tences th.;: ..:
is whrte ".:
SentenCe: :,:
stipulate :: -.:
Many-vi'-
might api,, -
or plausi'b,=
decided. - :-
This table s:

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hist-logic.pdf History of logic chapter 16 Aristotle the father of logic

  • 1. CHAPTER 16 History of Logic Logic was born in ancient Greece and reborn a century ago. Logic keeps grow- ing and expanding, and has contributed to the birth of the computer age. We can better understand and appreciate logic by studying its history. 76.7 Ancient logic The formal study of valid reasoning began with Aristotle (384-322 BC) in ancient Greece. An unprecedented emphasis on reasoning prepared for Aristotle's logic. Greeks used complex reasoning in geometry, to prove results like the Pythago- rean theorem. Sophists taught rich young men to gain power by arguing effec- tively (and often by verbal trickery). Parmenides and Heraclitus reasoned about being and non-being, anticipating later disputes about the law of non-contradic- tion, and Zeno reasoned about paradoxes. Socrates and Plato gave models of care- ful philosophical reasoning; they tried to derive absurdities from proposed views and sought beliefs that could be held consistently after careful examination. Reasoning is an important human activity, and it didn't begin in ancient Greece. Is this ability biologically based, built into our brains by evolution because it aids survival? Or does it have a divine origin, since we're made in the "image and likeness" of God? Or do both explanations have a place ? Logic raises fascinating issues for other Cisciplines. Aristotle began the study of logic. He was the first to formulate a correct prin- ciple of inference, to use letters for terms, and to construct an axiomatic system. He created syllogistic logic (Chapter 2), which studies arguments like these (using "all A isB," "no A is B," "some A is 8," or "some A is not B"): I i'' n1'-, This is inctti:. deriving them showing that t of syllogism' from what. I-:: such as defir..: Aristotle :: states that tlE to the same .: true at the sa: ments. Arist,-: more certain : an infinite se: Deniers of th-; point home. -,,,' to stop. Arisi; or "S is not P Aristotle :... Chapters 10;: not happen ( battle tomorr,:' necessarv u-he Aristotle relec; He seemed to , excluded midJ he anticipateJ t'alse ($77.7). ) box-inside/bor them doubtfu- After Aristc, "arrd," and "or defended, for e tollens (denr ir. Valid , argLtment A11 humans are mortal. all H ls M Al1 Greeks are humans. all G is H .'. A11 Greeks are mortal. .'. ail G is M This is aslidbecause of its formal structure, as given by the formulation on the right; any argument having this same structure will be valid. If we change the structure, we may get arr invalid argument, iike this one:
  • 2. :keeps grow- uter age. We v. BC) in ancient istotle's logic. the Pythago- rrguing effec- :asoned about ron-contradic- rodels of care- roposed views mination. 3in in ancient by evolution :e made in the e? Logic raises a correct prin- rmatic system. nts like these B"), ;M ;H rM ulation on the rve change the HISTORY OF LOGIC 349 All Romans are mortal. all R is M tnoatfi t All Greeks are mortal. all G is M argument .'. All Greeks are Romans. .'. all G is R This is inaalid because its form is wrong. Aristotle defended valid forms by deriving them from self-evidently valid forms; he criticized invalid forms by showing that they sometimes give true premises and a false conclusion. His logic of syllogisms is about logic in a narrou) sense, since it deals with what follows from what. He also pursued other topics that connect with appraising arguments, such as definitions and fallacies; these are about logic in abroader sense. Aristotle proposed two principles of thought. His law of non-contradiction states that the same property cannot at the same time both belong and not belong to the same object in the same respect. So "S is P" and "S is not P" can't both be true at the same time, unless we take "5" or "P" differently in the two state- ments. Aristotle saw this law as so certain that it can't be proved by anything more certain; not all knowledge can be demonstrated, since otherwise we'd need an infinite series of arguments that prove every premise by a further argument. Deniers of the law of non-contradiction assume it in their practice; to drive this point home, we might bombard them with contradictions until they plead for us to stop. Aristotle also supported the law of exduded middle, that either "S is P" or "S is not P" is true. Some deviant logics today dispute both laws (Chapter 17). Aristotle also studied the logic of "necessary" and "possible" (see modal logic, Chapters l-0 and L1). He discussed future contingents (events that may or may not happen). Consider a possible sea battle tomorrow. If "There utill be a sea battle tomorrow" ("S" below) is now either true or false, this seems to make necessary whether the battle occurs: Either it's true that S or it's false that S. If it's true that S, then it's necessary that S. If it's false that S, then it's necessary that not-S. .'. Either it's necessary that S or it's necessary that not-S. Aristotle rejected the conclusion, saying that there was no necessity either way. He seemed to deny the first premise and thus the universal truth of the law of excluded middle (which he elsewhere defends); if we interpret him this way, then he anticipated many-valued logic in using a third truth rralue besides true and false ($17.1). Another solution is possible. Many think premises 2 and 3 have a box-inside/box-outside ambiguity ($10.1): taking them as "(A, nB)" makes them doubtful while taking them as "E(A, B)" makes the argument invalid. After Aristotle, Stoics and others developed a logic that focused on "if-then," " and," and " or ," Iike our propositional logic (Chapters 6 and, 7). Stoic logicians defended, for example, an important inference form that came to be called modus t ollens (denying mode) :
  • 3. 350 PART FOUR: FURTHER VISTAS Valid ) argLLment Stoics also studied modal logic. Unlike logicians today, they took "necessary" and "possible" in a temporal sense, like "true at all times" and "true at some times." They disputed whether there'was a good modal argument for t'atalism, the view that all events happen of inherent necessity (see $10.3b #10). They also disputed how to understand "lf A then B" t577.4). Philo of Megara saw it as true if and only if it's not now the case that A is true and B is false; this fits the modern truth table for "if-then." Diodorus Chronos saw it as true if and only rf Ais nerser at any time true while B is false. Aristotelian and Stoic logic were first seen as rivals, differing in three ways: o Aristotle focused on "all," "r1o," and "some." Stoics focused on "if-then," " and," and " or." o Aristotle used letter variables and expressed arguments as long conditionals, hke "lf ail A is B, and all C is A, then a1l C is B." Stoics used number variables and expressed arguments as sets of statements, like "If 1 then 2. But not-2. Therefore, not-1." o Aristotle saw logic not as part of philosophy but rather as a tool for all think- ing. Stoics saw logic as one of philosophy's three branches (the other two being physics and ethics). But both agreed that students should study logic early, before going deeply into other areas. Later thinkers combined these approaches into traditional logic. For the next two thousand years, Aristotle's logic with Stoic additions ruled in the West. At the same time, another tradition of logic rose up in India, China, and Tibet. We call it Buddhist logic even though Hindus and others pursued it too. It studied many topics important in the West, including inference, fallacies, and Ianguage. This is a common pattern in Buddhist logic: Here there is fire, because there is smoke. Wherever there is sn-roke there is fire, as in a kitchen. Here there is smoke. .'. Here there is fire. The last three lines are deductively valid: Al1 cases of smoke are cases of fire. This is a case of smoke. .'. This is a case of fire. This omits "as in a kitchen," which suggests inductive reasoning (Chapter 5); in our experience of smoke and fire, smoke always seems to involve fire. If your view is correct, then such and such is true. Such and such is false. . Your view isn't correct. If C then S Not-S Not-C The Easte: covers mai--r.' commenta: - : thinking .:- ;, caution :.:.-:. der.ianr - -- :- -, Iai.','..= using :.:": -, thinhr.: :: , and r,r-=:-:,- Ludrvis i'..::= a stronE-',' :: - 76.2 l;. Mediera- ---- logic beca:: = The C:.:-;: Ages. H. '.'.: box-in'ti= : - foreknorr-. i = tle's 1oS-; ---: On lnt;""-'-. .. The -{::: " Christia:. :: ogy and :-.=: taries, tex:r - logic, co:.j-:- ish Sparn -.'. =: In Ch:-.:," Anselm i'.:= terior ,1,:.;.'. : was basei ::. Peter of S:.: The cler'=: Capitalized :
  • 4. i:1en s :-S :.C ,:k "necessary" i true at some :'.: ior t'atalism, =10). They also ra saw it as true irts the modern :.lr- if A is nerser :r three ways: ,r', "if-then," : ;rnditionals, :.:.er variables l, But not-2. , :, r all think- :,r,er two being :',- logic ear1y, :,rr the next two : 'est. :.rna, and Tibet. :;ued it too. It e iallacies, and : Chapter 5); in : iire. HISTORY OF LOGIC 351. The Eastern logic tradition is poorly understood in the WesU this tradition covers many centuries, and many texts are difficult or untranslated. Some commentators emphasize similarities between East and West; they see human thinking as essentially the same everywhere. Others emphasize differences and caution against imposing a Western framework on Eastern thought. And some deviant logicians see the Eastern tradition as congenial to their views. Many see the East as more mystical than logical; Zen Buddhism delights in using paradoxes (like the sound of one hand clapping) to move us beyond logical thinking toward a mystical enlightenment. But East and West both have logical and mystical elements. Sometimes these come together in the same individual; Ludwig Wittgenstein in the early 20th century invented truth tables but also had a strongly mystical side. 76.2 Medieval logic Medieval logicians carried on the basic framework of Aristotle and the Stoics, as logic became important in higher education. The Christian thinker Boethius (480-521) helped the transition to the Middle Ages. He wrote on logic, including commentaries; he explained the modal box-inside/box-outside ambiguity as he defended the compatibility of divine foreknowledge and human freedom (S10.3b #4 and #14). He translated Aristo- tle's logic into Latin. Many of his translations -were lost; but his Categories and On lnterpretation became the main source for the logica aetus (old logic). The Arab worid dominated in logic from 800-1200. Some Arab logicians were Christian, but most were Muslim; both groups saw logic as important for theol- ogy and medicine. They translated Aristotle into Arabic and wrote commen- taries, textbooks, and original works. They pursued topics like syllogisms, modal logic, conditionals, universals, predication, and existence. Baghdad and Moor- ish Spain were centers of logic studies. In Christian Europe, logic was reborn in the 11th and 12th centuries, with Anselm, Peter Abelard, and Latin translations of Aristotle's Prior Analytics, Pos- terior Analytics, Topics, and Sophistical Ret'utations; the logica rzooa (new logic) was based on these. There was interest in universals and in how terms signify. Peter of Spain and William of Sherwood wrote logic textbooks. The ciever Barbara-Celarent verse was a tool for teaching syllogisms: Barbara, Celarent, Darii, Ferioque, prioris; Cesare, Camestres, Festino, Baroco, secundae; tertia, Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison, habet; quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison. Capitalized names are valid syllogisms. Vowels are sentence forms:
  • 5. 352 PART FOUR: FURTHER VISTAS A all-is- I some-is- E no-is- O some - is not - ailMisP allSisM allSisP Aff -Irm universal/particular nE-gO universal/particular MP SM = irgure I So "Barbara," with AAA vowels, has three "all" statements: fallacr' - o If the ;: ment t: - . If the c--: In the [:: tron.Ete: -. : priori . . ;' 16.-1 :. -_ A -l --, - - ntt:._.: i COnil-: -:= - - isB':-:,.-- with =--- . . Venn :'-. = - har-e a::= = - logrc :._:.- - impro..'. :., - rer.olu:- - : Ihe (,=- Hegel F:-:,:: evolr-es i.-i - . together .--. , - applied il".:, : logic as a:. .-,= flicting p:. :. - self -contr;. i, :: and time ;--.: : I'he p1-.--, . . future der'. -: reduce reai :-- their pencrl- :.: like that oi "c Many thr:.. Morgan prt: - "A08"; a lei:= known for hr' Aristotelian syllogisms have two premises. Middle term "M" is common to both premises; predicate "P" occurs in the first premise, while subject "S" occurs in the second. There are four figures (arrangements of premise letters): 1 (prioris) 2 (secundae) 3 (tertia) 4 (quarta) MP PM MP PM SM SM MS MS Aristotle's four axioms are valid first-figure forms: Barbara allMisP allSisM .'. all S is P Celarent Darii noMisP allMisP allSisM someSisM noSisP .'.someSisP Ferio noMisP some S is M some S is not P The other 15 forms can be derived as theorems. The consonants give clues on how to do this; for example, "m" says to switch the order of the premises. Thomas Aquinas (722+-74), the most influential medieval philosopher, had little impact on logic's development; but he made much ,r" of logic. Since he emphasized reasoning and wrote so much, he likely produced more philosophical arguments than anyone else who has ever lived. Fourteenth-century logicians include William of Ockham and |ean Buridan. Ockham's razor says "Accept the simplest theory that adequately explains the data." Ockham developed modal logic and tried to avoid metaphysics when analyzing language. Buridan's ass was a fictional donkey whose action was par- alyzedwhen he was placed exactly midway between two food bowls. Buridan also formulated the standard rules for valid syllogisms; one version says that a syllo- gism is aalid just if it satisfies all of these conditions: . Ever] term distributed in the conclusion must be distributed in the premises. (A term is distributed in a statement just if the statement makes some claim about e-oery entity that the term refers to.) . The middle term must be distributed in at least one premise. (The middle term is the one common to both premises; if we violate this rule, we commit the He complarneJ
  • 6. Llar Iar HISTORY OF LOGIC 353 fallacy of the undistributed middle.) . If the conclusion is negative, exactly one premise must be negative. (A state- ment is negatiae if it contains "rro" ot "not"; otherwise it's positive.) . If the conclusion is positive, both premises must be positive. In the Middle Ages, logic was important in philosophy and in higher educa- tion. Even today, logic, like biology, uses many Latin terms (modus ponens, e prioril a posteriori, de rel de dicto, and so on). 76.3 Enlightenmentlogic Aristotelian logic dominated until the end of the 19th century. Several logicians contributed to syllogistic logic; for example, Leonhard Euler diagrammed "all A is B" by putting an A-circle inside a larger B-circle, Lewis Carroll entertained us with silly syllogisms and points about logic in Alice in Wonderland, andlohn Venn gave us diagrams for testing syllogisms (52.6). But most logicians would have agreed with Immanuel Kant, who said that Aristotle invented and perfected logic; nothing else of fundamental importance could be added, although we might improve teaching techniques. Kant would have been shocked to learn about the revolution in logic that came about a hundred years later. The German thinkers Georg Hegel and Karl Marx provided a side current. Hegel proposed that logic should see contradictions as explaining how thought evolves historically; one view provokes its opposite, and then the two come together in a higher synthesis. Marx saw contrafictions in the world as real; he applied this to political struggles and revolution. While some saw this dialectical logic as an alternative to traditional logig critics objected that this confuses con- flicting properties in the world (like hot/cold or capitalist/proletariat) with logical self-contradictions (like the same object being both white and, in the same sense and time and respect, also non-white). The philosopher Gottfried Leibniz, the co-inventor of calculus, anticipated future developments. He proposed the idea of a symbolic language that would reduce reasoning to calculation. If controversies arose, the parties could take up their pencils and say, "Let us calculate." Leibniz created a logical notation much like that of Boole (and much earlier); but his work was published after Boole. Many thinkers tried to invent an algebraic notation for logic. Augustus De Morgan proposed symbolizing "all A is B" as "A))B" and "some A is B" as "A0B"; a letter on the concave side of the parenthesis is distributed. He became known for his De Morgan laws for propositional logic: Not both A and B = Either not-A or not-B Not either A or B = Both not-A and not-B He complained that current logic couldn't handle relational arguments like "A11 ; common to both iect "5" occurs in tters): fta) [, ; Ferio fisP :SisM :SisnotP nts give clues on e premises. philosopher, had of logic. Since he rore philosophical md Jean Buridan. rtely explains the retaphysics when ;e action was Par- rwls. Buridan also says that a syllo- in the premises. rkes some claim fhe middle term we commit the
  • 7. 354 PART FOUR: FURTHER VISTAS dogs are animals; therefore all heads of dogs are heads of animals,, (S9.5b #25). The Boolean algebra of George Boole iig15-64) was a breakthro,lgh, ,ir.,." it used math to check the correctness of inferences. Boole used letters io, ,.,r; ,o "M" might be the set of mortals and "H" the set of humans. putting two letters togetherrepresents the intersectiott ol the sets; so ,,HM,, is the set J those who are both human and mortal. Then "All humans are mortal,, is ,,H = HM,,, which says that the set of humans = the set of those who are both human and mortal. A syllogism is a series of equations: Valid , argutnent All humans are mortal. H = HM All Greeks are humans. G = GH . . All Greeks are mortal. ... G = GM we can derive the conclusion by substituting equals for equals. In premise 2, 9 = GH,_replace "H" wirh /tHM/'/ (premise f ,ry, H = HMj to get G = GHM. Then replace "GH" w_r1h "G" (premise 2 says G = GH) ,o g.i G = GM. Boolean formulas, like those on the left below lwhiih .,r. , lrt", symbolism), can be interpreted to be about sets or about statements: " -4" can mean ,,the set of non_A s,, or ,,not_A,, " AaB" can mean "the intersection of sets A and 8,, or ,,A and B,, " AwB" can mean ,,the union of sets A and 8,, or ,, A or 8,, so if " A" is the set of animals, then "-A" is the set of non-animals; but if ,,A,, is "Aristotle is a logician," then " -A" is "Aristotle isn't a logician.,, The same iaws cover both; for example , " A^B = Ba)A" works for eith"er sers or statements. Boolesn op-erators (like "and," " or," and "not") use the statement interpretation. Boole, the father of.mat.hematical logic, thought rhat logic belonged with mathematicians instead of philosoph"rr. Brrt both lroups .r-J,o have an inter- est in logic, each getting the slice of the action thrtTits it better. While Boole was important, a greater revolution in logic was to come. 76.4 Frege and Russell Gottlob Frege (1848-r92s) created modern logic with his tlzg Begrit'fsschrit't ("conceptwriting"). Its BB pages introduced a symbolism that, for th"" fiirt ti-f, let us combine in every way Aristotle,s ,,a11,,, ,;no,,,and ,,some,, with the Stoic " if-then," " and," and " ot.', -So we can symbol ize ,,If everything that,s A or B is then C and D, then everything that's non-D is non-A.,, Thus"the gap between Aristotle and the stoics was overcome in a higher synthesis. r.ege"rlro showed how to analyze arguments with relations (like-"x loves y,,) and m"ultiple q.ranti- fiers; so we can show that "There is someone thut "r"ryone loves,, entails "Everyone loves someone" - but not conversely. F.ege presented logic as a formal sv:::' of formulas : Frege's rrc early 20th ce: for "not," "ti These can .c: This n'as :-s - simpler n.:::- Frege ie '. ;- wanted ta ::.- purelr, lol::"- axioms ar.; -: condition --.-. :, condition : -- - some sets ::a object) rvh:-. not a men:i'.: members c:::, if x is not ; =., not a merr.ral Russell ask;: principle, R .; SoisRam.: way we get I : provable in F: life work coll: Russeil gre, minds worke: needed fixins
  • 8. .als" ($9.5b #25). kthrough, since it [etters for sets; so )utting two letters e set of those who "H=H]ly'r," which urnan and mortal. =HM =GH =GM ra1s. In premise 2, togetG=GHM. G=GM. r later symbolism), A and B" or B" imals; but if "A" is m." The same laws rets or statements. rent interpretation. gic belonged with re to have an inter- :r. While Boole was 879 Begrit'fsschrift rt, for the first time, me" with the Stoic ing that's A or B is rs the gap between Frege also showed nd multiple quanti- rone loves" entails resented logic as a HISTORY OF TOGIC 355 t'ormal system, with purely notational rules for determining the grammaticality of formulas and the correctness of proofs. Frege's work was ignored until Bertrand Russell (1.872-L970) praised it in the early 20th century. Frege's difficult symbolism alienated people. He used lines for " not," " if-then," and " all" '. Not-A ---rA If A then B For all x --pB -Y La These can combine to symbolize "Not all A is non-B" (our "-(x)(Ax > -Bx)"): -=*[Nx This was also his way to write "Some A is B" (our "(Ix)(Ax'B")"); he had no simpler notation for "some" or "and." Frege developed logic to help show that arithmetic is reducible to logic; he wanted to define all basic concepts of arithmetic (like numbers and addition) in purely logical terms and prove all basic truths of arithmetic using just logical axioms and inference rules. Frege used a seemingly harmless axiom that every condition on x picks out a set containing just those elements that satisfy that condition; so the condition "x is a cat" picks out the set of cats. But consider that some sets are members of themselves (the set of abstract objects is an abstract object) while other sets aren't (the set of cats isn't a cat), By Frege's axiom, "x is not a member of itself" picks out the set containing just those things that are not members of themselves. Call this "set R." So any x is a member of R, if and only if x is not amember of x (here "€" means "is a member of" and"e" means "is not a member of"): For all x, x € R if and only if x c x. Russell asked in a 1902letter to Frege: What about set R itself ? By the above principle, R is a member of R, if and only if R is not a member of R: R e Rif andonlyif R e R. So is R a member of itself? If it is, then it isn't - and if it isn't, then it is; either way we get a contradiction. Since this contrafiction, called Russell's paradox, was provable in Frege's system, that system was flawed. Frege was crushed, since his life work collapsed. His attempts to fix the problem weren't successful. Russell greatly admired Frege and his groundbreaking work in logic; the two minds worked along similar lines. But the paradox showed that Frege's work needed fixing. So Russell, with his former teacher Alfred North Whitehead,
  • 9. 355 PART FOUR: FURTHER VISTAS worked to develop logic and set theory in a way that avoided the contradiction. They also developed a more intuitive symbolism (much like what we use in this book), based on the work of Giuseppe Peano. The result was their massive Prln- cipia Mathemqtica, which was published in 1910-7973. Principiahad a huge in- fluence and became the standard formulation of the new logic. 76.5 After Principia Classical symbolic logic includes propositional and quantificational logic (Chap- ters 6 to 9). A logic is "classical" if it accords with Frege and Russell about which arguments are valid, regardless of differences in symbolization and proof tech- niques. Classical symbolic logic gradually became the new orthodoxy, replacing the older Aristotelian logic. Much work was done to solidlfy classical symbolic logic. Different proof tech- niques were deveioped; while Frege and Russell used an axiomatic approach, later logicians invented inferential and truth-tree methods that were easier to use. Different ways of symbolizing arguments were developed, including the Polish nototion of a school of logic that was strong in Poland between the world wars. Ludwig Wittgenstein and Emil Post independently invented truth tables, which clarified our understanding of logical connectives (like "if-thetr," "and," and "or") and led to a criterion of validity based on semantics - on the meaning of the connectives and how they contribute to truth or falsity; Alfred Tarski and others expanded the semantic approach to quantificational logic. Much work was done in metalogic, the study of logical systems (Chapter 15). Kurt Godel showed that Russell's axiomatization of classical logic was, given cer- tain semantic assumptions, correct: iust the right things were provable. But he also showed, against Frege and Russell, that arithmetic cannot be reduced to any formal system: no consistent set of axioms and inference rules would suffice to prove all arithmetic truths; this result, called Godel's theorem, is perhaps the most striking and surprising result of 2Oth-century logic. Alonzo Church showed that the problem of determining validity in quantificational logic cannot be reduced to an mechanical algorithm (a result called Church's theorem). There was also much activity in set theory, which after Russell's paradox became increasingly complex and controversial. There was also much work in philosophy of logic (Chapter 1B), which deals with philosophical questions about logic, such as these: Are logical truths de- pendent on human conventions (so different conventions might produce differ- ent logical truths) or on the objective nature of reality (perhaps giving us the framework of any possible language that would be adequate to describe reality)? Can logic help us clarify metaphysical issues, such as what kinds of entity ulti- mately exist? Should we assume abstract entities (like properties and proposi- tions) when we do logic? How can we resolve logical paradoxes (such as Russell's parado-t --, - distort : -, the de:-:::- Los: i was i:;: lo'l:: _:. ,,. in ::.. -:: . tain: -',-:.- : Neun":.: -. electrc:..--,: and so::'.'. , marn i.: ':- T^---- - LL)q-. _ approa-:. . sensor- j-, branch.= - ' -1! L- i:: Trvo tr :=- classic:, - : for erar.:-: sical lc:-: ' The :' . "nece=ir:-. pursue.- _:. . Leu'is's ','.' - arguei::,: and qr-.a:.:-. erties. l:.1 Kripke :, . ' sense oi -. . useful lr. ,: phers r:.:-. properii. i Other .'. of knortl;:, tionship l future ti:-.. (Chapter= - supplerr,e:.: De nili'.. needs to 'le values. 1:r fuzzy-1ogii (0.00). or:
  • 10. led the contradiction. le what we use in this as their rnassive Prin- incipia had a huge in- )sc. iicational logic (Chap- d Russell about which :ation and proof tech- 'orthodoxy, replacing . Different proof tech- omatic approach, later at were easier to use. l, including the Polish :ween the world wars. ed truth tables, which "if-then," "and," artd r - on the meaning of ity; Alfred Tarski and .logic. systems (Chapter 15). al logic was, given cer- were provable. But he rnot be reduced to any rules would suffice to eorem, is perhaps the ilonzo Church showed ional logic cannot be rch's theorem). There sell's paradox became apter 18), which deals Are logical truths de- might produce differ- perhaps giving us the te to describe reality)? at kinds of entity ulti- roperties and proPosi- oxes (such as Russell's HISTORY OF LOGIC 357 paradox and the liar paradox) ? Are logical truths empirical or a priori? Does logic distort ordinary beliefs and ordinary language, or does it correct them? What is the definition and scope of logic? Logic was important in the development of computers. The key insight here was that logical functions like "and" and " or" can be simulated electrically by logic gates; this idea goes back to the American logician Charles Sanders Peirce in the 1880s and was rediscovered by Claude Shannon in 1938. A computer con- tains logic gates, plus memory and input-output devices. Logicians like |ohn von Neumann, Alan Turing, and Arthur Burks helped design the first large-scale electronic computers. Since logic is important for computers, in both hardware and software, it's studied today in computer science departments. So now three main departments study logic - philosophy, math, and computer science. Logic today is also an important part of cognitioe science, an interdisciplinary approach to thought that includes linguistics, psychology, biology (brain and sensory systems), computers (especially artificial intelligence), and other branches of philosophy (especially epistemology and philosophy of mind). As classical symbolic logic became the orthodoxy, it started to be questioned. Two types of non-classical logic came to be. Supplementary logics accepted that classical logic was fine as far as it went but needed to be supplemented to deal, for example, with "necessary" and "possible." Deviant logics thought that clas- sical logic was wrong on some points and needed to be changed. The most important supplementary logic is modal logic, which deals with "necessary" and "possible" (Chapters 1,0 and tt). Ancient and medieval logicians pursued modal logic; but 2Oth-century logicians mostly ignored it until C. L Lewis's work in the 1930s. Modal logic then became controversial. Willard Quine argued that it was based on a confusion; he thought logical necessity was unclear and quantified modal logic led to an objectionable metaphysics of necessary prop- erties. There was lively debate on modal logic for many years. In 1959, Saul Kripke presented a possible-worlds way to explain modal logic; this made more sense of it and gave it new respect among logicians. Possible worlds have proved useful in other areas and are now a common tool in logic; and several philoso- phers (including Alvin Plantinga) have defended a metaphysics of necessary properties. Today, modal logic is a well-established extension of classical logic. Other extensions apply to ethics ("A ought to be done" or "A is good"), theory of knowledge ("X believes that A" or "X knows that A"), the part-whole rela- tionship ("X is a part of Y"), temporal relationships ("It will be true at some future time that A" and "lt was true at some past time that A"), and other areas (Chapters 12toL4). Most logicians would agree that classical logic needs to be supplemented in order to cover certain kinds of argument. Deaiant logics say that classical symbolic logic is wrong on some points and needs to be changed (Chapter 17). Some propose using more than two truth values. Maybe we need a third truth value for "half-true." Or maybe we need a fuzzy-logicrange of truth values, from completely true (1.00) to completely false (0.00). Or perhaps " A" and "not-A" can both be false (intuitionist logic) or both
  • 11. 358 PART FOUR: FURTHER VISTAS be true (paraconsistent logic). Or perhaps the classical approach to "if-then" is flawed; some views even reject modus ponens ("If A then B, A .'. B") and modus tollens ("lf A then B, not-B .'. not-A"). These and other devrant logics have been proposed. Today there is much questioning of basic logical principles. This brief history of logic has focused on deductive logic and related areas. There has also been much interest in informal logic (Chapters 3 and 4), inductive logic (Chapter 5), and history of logic (this chapter). So logic has a complex history - from Aristotle and the Stoics in ancient Greece, through the Middle Ages and the Enlightenment, to the turmoil of the 19th century and logic's transformation with Frege and Russell, and into recent classical and non-classical logics and the birth of the computer age.1 1 For *or. on the history of logic, I suggest P. H. Nidditch's The Development ot' Mathematical Logic (London: Routledge & Kegan Paul,1962) and, for primary sources, Irving Copi and James Gould's Readings on Logic (New York: Macmillan, 1954). Also useful are William and Martha Kneale's The Deaelopment of Logic (Oxford: Clarendon, 1962) and Joseph Bocheisl<t's A History of Formal Logic, trans.Ivo Thomas (Notre Dame, Ind.: University of Notre Dame, 1951). Deviant l:+: 5LaL(1,ic-.-: i-: onh :r;:: . '. t4'O tru::. '. : '. trUelPa:::-,: IF-THE.- .= Der-t,":-. - , , SOme ai:;-=. . ard 1oe-; =- : would h.:::.: are r.aiii ": i ' 17.7 l:: Most 1oE-:.- proposit - :.=. and false .. tences th.;: ..: is whrte ".: SentenCe: :,: stipulate :: -.: Many-vi'- might api,, - or plausi'b,= decided. - :- This table s: