How can and be true?Commonsensetells us that and is falsewhenevereither or is false.
We completethe last columnaccordingly.
T T T
T F F
F T F
F F F
Anotherway to statethe meaningof the truth tableis that is true only when is trueand is true.Let’s do the samefor or. Beforewe do that, we needto clarifywhich“or” we meanin mathematics.
In ordinaryspeech,or is sometimesusedto mean,“this or that, but not both.”This is calledtheexclusive
or(it excludesor keepsout both).In mathematics,or means“this,that, or both this and that.”This is called
theinclusiveor. Knowingthat or is inclusivemakesthe truth tablean easyjob.
Example 2
or is true, because is true.or is true, because is true.or is true because is true and is true.or is falsebecause is falseand is false.Example 3
Drawa truth tablefor or , whichis written.Beginby fillingin all the T/F combinationspossiblefor and. Keepingin mindthe definitionof or above(inclusive),fill in the third column. or will only befalsewhenboth and are false;it is true otherwise.
T T T
T F T
F T T
F F F
LessonSummary
Do we all haveour own versionof whatis logical?Let’s hopenot—wewouldn’tbe able to agreeon whatis
or isn’t logical!To avoidthis, thereare agreed-onrulesfor logic,just like thereare rulesfor games.The two
mostbasicrulesof logicthat we will be usingthroughoutour studiesare thelaw of detachmentand the
law of syllogism.
Pointsto Consider
Rulesof logicare universal;they applyto all fieldsof knowledge.For us, the rulesgive a powerfulmethod
for provingnew factsthat are suggestedby our explorationsof points,lines,planes,and so on. We will
structurea specificformat,the two-columnproof,for provingthesenew facts.In upcominglessonsyou will