A Modern Logician Said:A Modern Logician Said:

By the aid of symbolism, we can make By the aid of symbolism, we can make transitions in reasoning almost transitions in reasoning almost mechanically by eye, which otherwise mechanically by eye, which otherwise would call into play the higher faculties of would call into play the higher faculties of the brain.the brain.

Symbolic Logic: Symbolic Logic: The Language of Modern LogicThe Language of Modern Logic

Technique for analysis of deductive argumentsTechnique for analysis of deductive argumentsEnglish (or any) language: can make any English (or any) language: can make any argument appear vague, ambiguous; especially with argument appear vague, ambiguous; especially with use of things like metaphors, idioms, emotional use of things like metaphors, idioms, emotional appeals, etc.appeals, etc.Avoid these difficulties to move into logical heart of Avoid these difficulties to move into logical heart of argument: use symbolic languageargument: use symbolic language

Now can formulate an argument with precisionNow can formulate an argument with precisionSymbols facilitate our thinking about an argumentSymbols facilitate our thinking about an argument

These are called “logical connectives”These are called “logical connectives”

Logical ConnectivesLogical Connectives

The relations between elements that every The relations between elements that every deductive argument must employdeductive argument must employ

Helps us focus on internal structure of Helps us focus on internal structure of propositions and argumentspropositions and arguments We can We can translatetranslate arguments from sentences and arguments from sentences and

propositions into symbolic logic formpropositions into symbolic logic form ““Simple statement”: does not contain any other Simple statement”: does not contain any other

statement as a componentstatement as a component ““Charlie is neat”Charlie is neat”

““Compound statement”: does contain another Compound statement”: does contain another statement as a componentstatement as a component ““Charlie is neat and Charlie is sweet” Charlie is neat and Charlie is sweet”

ConjunctionConjunction

Conjunction of two statements: “…and…”Conjunction of two statements: “…and…”Each statement is called a conjunctEach statement is called a conjunct

““Charlie is neat” (conjunct 1) Charlie is neat” (conjunct 1) ““Charlie is sweet” (conjunct 2)Charlie is sweet” (conjunct 2)

The symbol for conjunction is a dot •The symbol for conjunction is a dot • (Can also be “&”)(Can also be “&”)p p • q• q

P and q (2 conjuncts)P and q (2 conjuncts)

Truth ValuesTruth Values

Truth valueTruth value: every statement is either T or : every statement is either T or F; the truth value of a true statement is F; the truth value of a true statement is true;true; the truth value of a false statement is the truth value of a false statement is falsefalse

Truth Values of ConjunctionTruth Values of Conjunction

Truth value of conjunction of 2 statements Truth value of conjunction of 2 statements is determined entirely by the truth values is determined entirely by the truth values of its two conjunctsof its two conjunctsA conjunction statement isA conjunction statement is truth-functional truth-functional

compound statementcompound statementTherefore our symbol Therefore our symbol “•” (or “&”) is a truth-“•” (or “&”) is a truth-

functional connectivefunctional connective

Truth Table of Truth Table of Conjunction •Conjunction •

pp qq p p •• q q

TT TT TT

TT FF FF

FF TT FF

FF FF FF

Given any two statements, p and qGiven any two statements, p and q

A conjunction is true if and only if both conjuncts are trueA conjunction is true if and only if both conjuncts are true

Abbreviation of StatementsAbbreviation of Statements ““Charlie’s neat and Charlie’s sweet.”Charlie’s neat and Charlie’s sweet.”

N N • S• S DictionaryDictionary: N=“Charlie’s neat” S=“Charlie’s sweet”: N=“Charlie’s neat” S=“Charlie’s sweet”

Can choose any letter to symbolize each conjunct, but it is Can choose any letter to symbolize each conjunct, but it is best to choose one relating to the content of that conjunct to best to choose one relating to the content of that conjunct to make your job easiermake your job easier

““Byron was a great poet and a great adventurer.”Byron was a great poet and a great adventurer.” PP • A• A

““Lewis was a famous explorer and Clark was a Lewis was a famous explorer and Clark was a famous explorer.”famous explorer.” LL • C• C

““Jones entered the country at New York and Jones entered the country at New York and went straight to Chicago.”went straight to Chicago.” ““and” here does not signify a conjunctionand” here does not signify a conjunction Can’t say “Jones went straight to Chicago and Can’t say “Jones went straight to Chicago and

entered the country at New York.”entered the country at New York.” Therefore cannot use the Therefore cannot use the • here• here

Some other words that can signify conjunction:Some other words that can signify conjunction: ButBut YetYet AlsoAlso StillStill HoweverHowever MoreoverMoreover NeverthelessNevertheless (comma)(comma) (semicolon)(semicolon)

NegationNegation

Negation: contradictory or denial of a statementNegation: contradictory or denial of a statement ““not”not”

i.e. “It is not the case that…”i.e. “It is not the case that…”

The symbol for negation is tilde ~The symbol for negation is tilde ~ If M=“All humans are mortal,” thenIf M=“All humans are mortal,” then

~M=“It is not the case that all humans are mortal.” ~M=“It is not the case that all humans are mortal.” ~M=“Some humans are not mortal.”~M=“Some humans are not mortal.” ~M=“Not all humans are mortal.”~M=“Not all humans are mortal.” ~M=“It is false that all humans are mortal.”~M=“It is false that all humans are mortal.”

All these can be symbolized with ~MAll these can be symbolized with ~M

Truth Table for NegationTruth Table for Negation

pp ~p~p

TT FF

FF TT

Where p is any statement, its negation is ~pWhere p is any statement, its negation is ~p

DisjunctionDisjunction

Disjunction of two statements: “…or…” Disjunction of two statements: “…or…” Symbol is “ v ” (wedge) (i.e. A v B = A or B)Symbol is “ v ” (wedge) (i.e. A v B = A or B)

Weak (inclusive) sense: can be either case, and Weak (inclusive) sense: can be either case, and possibly bothpossibly both

Ex. “Salad or dessert” (well, you Ex. “Salad or dessert” (well, you cancan have both) have both) We will treat all disjunctions in this sense (unless a problem We will treat all disjunctions in this sense (unless a problem

explicitly says otherwise)explicitly says otherwise)

Strong (exclusive) sense: one and only oneStrong (exclusive) sense: one and only one Ex. “A or B” (you can have A Ex. “A or B” (you can have A oror B, B, at least oneat least one but not bothbut not both))

The two component statements so combined are The two component statements so combined are called “disjuncts”called “disjuncts”

Disjunction Truth TableDisjunction Truth Table

pp qq p v qp v q

TT TT TT

TT FF TT

FF TT TT

FF FF FF

A (weak) disjunction is false only in the case that both its disjuncts are falseA (weak) disjunction is false only in the case that both its disjuncts are false

DisjunctionDisjunction

TranslateTranslate::“You will do poorly on the exam unless “You will do poorly on the exam unless you study.”you study.”P=“You will do poorly on the exam.”P=“You will do poorly on the exam.”S=“You study.”S=“You study.”

P v SP v S

““Unless” = vUnless” = v

PunctuationPunctuation

As in mathematics, it is important to correctly As in mathematics, it is important to correctly punctuate logical parts of an argumentpunctuate logical parts of an argument Ex. (2x3)+6 = 12 whereas 2x(3+6)= 18Ex. (2x3)+6 = 12 whereas 2x(3+6)= 18 Ex. p Ex. p • q v r (this is ambiguous)• q v r (this is ambiguous)

To avoid ambiguity and make meaning clearTo avoid ambiguity and make meaning clear Make sure to order sets of parentheses when Make sure to order sets of parentheses when

necessary:necessary: Example: { A Example: { A •• [(B v C) [(B v C) • (C v D)• (C v D)] } ] } • ~E• ~E

{ [ ( ) ] }{ [ ( ) ] }

PunctuationPunctuation

““Either Fillmore or Harding was the Either Fillmore or Harding was the greatest American president.”greatest American president.”F v HF v H

To say “Neither Fillmore nor Harding was To say “Neither Fillmore nor Harding was the greatest American president.” (the the greatest American president.” (the negation of the first statement)negation of the first statement)~(F v H) OR (~F) ~(F v H) OR (~F) • (~H)• (~H)

PunctuationPunctuation

““Jamal and Derek will both not be elected.”Jamal and Derek will both not be elected.”~J ~J • ~D• ~D

In any formula the negation symbol will be In any formula the negation symbol will be understood to apply to the smallest statement that understood to apply to the smallest statement that the punctuation permitsthe punctuation permits

i.e. above is NOT taken to mean “~[J • (~D)]”i.e. above is NOT taken to mean “~[J • (~D)]”

““Jamal and Derek both will not be elected.”Jamal and Derek both will not be elected.”~(J •D)~(J •D)

ExampleExample

Rome is the capital of Italy or Rome is the Rome is the capital of Italy or Rome is the capital of Spain.capital of Spain. I=“Rome is the capital of Italy”I=“Rome is the capital of Italy” S=“Rome is the capital of Spain”S=“Rome is the capital of Spain”

I v SI v S Now that we have the logical formula, we can use the Now that we have the logical formula, we can use the

truth tables to figure out the truth value of this truth tables to figure out the truth value of this statementstatement

When doing truth values, do the innermost When doing truth values, do the innermost conjunctions/disjunctions/negations first, working your way conjunctions/disjunctions/negations first, working your way outwardsoutwards

I v S1. We know that Rome is the capital of Italy and that Rome is not the

capital of Spain.

1. So we know that “I” is True, and that “S” is False. We put these values directly under their corresponding letter

I v SI v S TT FF

• We know that for a disjunction, if at least one of the disjuncts is T, this is enough to make the whole disjunction T

• We put this truth value (that of the whole disjunction) under the v (wedge)

I v SI v S

TT FF

TT

3 Laws of Thought3 Laws of Thought

The principle of identityThe principle of identityThe principle of non-contradictionThe principle of non-contradictionThe principle of excluded middlemThe principle of excluded middlem

The Principle of IdentitiyThe Principle of Identitiy

If any statement is true, then it is true.If any statement is true, then it is true.

The Principle of Non- The Principle of Non- contradictioncontradiction

No statement can be both true and false.No statement can be both true and false.

The Principle of Excluded The Principle of Excluded MiddleMiddle

Every statement is either true or false.Every statement is either true or false.

Laws of ThoughtLaws of Thought

3 Laws of thoughts are the principles 3 Laws of thoughts are the principles governing the construction of truth table.governing the construction of truth table.

Used in completing truth tables.Used in completing truth tables.