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CHAPTER1
LOGIC
1010SENTENCES
THEORYINANUTSHELL
Logicalstatement(orsentence)isanydeclarativesentence,whosetruthfulness
maybeverified.Onecandistinguishsimplestatements,whichcannotbe
reducedandthecompound(orcomplex)statements,consistingofthesimple
ones.
Simplestatementsaredenotedbylowercaseletters:p,q,r,łThelogicalvalue
(truthvalue,Booleanvalue)isdenotedbytheGreekletter
ν
(nu).Ittakesalways
oneoftwovalues:0,whenthestatementisfalse,and1,whenitistrue.Wewrite
itinthefollowingway:
ν
()
p
=
[
®
¯
1
0
,
,
if
otherwise.
p
is
true,
Asmentionedbefore,itispossibletoconstructcompoundstatementsbasedon
thesimpleones.Whatweneedaretheoperations.Fiveofthemaredescribed
below.
Negationisaone-argument(unary)operation.Wedenoteitby“¬”
1
andread
simply“not”.Theso-calledtruthtablefornegation(i.e.thelistofallthe
possiblevaluesofitdependingonthetruthvaluesoftheinitialsimple
statement)ispresentedinTable1.1.Asyoucansee,thevalueof¬pisalways
differentthatthevalueofp.
Table1010Truthtableofthenegation
p
0
1
¬p
1
0
Fourremainingoperationsareconjunction(“p∧q”,wereadit“pandq”),
disjunction(“p∨q”,wereadit“porq”),implication(“p3q”,wereadit“ifp,
thenq”)andequivalence(“p⇔q”,wereadit“pifandonlyifq”).The
conjunctionoftwostatementsistrueonlyifbothofthesestatementsaretrue.
Thedisjunctionistrue,ifatleastoneofthemistrue(notethatitmeansthat
1
Youcanalsomeet„~”.
7
