A little Propositional Logic

Propositional Logic• Basics of propositional logic are a way to

symbolize the parts of arguments so that we can analyze whole arguments for validity.

Statement: which can be regarded as either True or False.

Simple Statement: consists of ONE statement.Ex. “Tina is tall.”

Compound Statement: consists of TWO or MORE statements.

Ex. “Tina is tall and Sarah is tall.”

Putting statements in symbolic form

• For Simple StatementsEx. “Tina is tall” = p

• For Compound StatementsEx. “Tina is tall and Sarah is tall.”: “Tina is tall.” = p“Sarah is tall” = q: p & q

Example• Put the following statements in symbolic form.1. “Ken hit a home.”

= p

2. “Hank went fishing and Drik went hunting”: Hank went fishing = p Drik went hunting = q: p & q

3. “France is in Europe and China in Asia.”: France is in Europe = p China is in Asia = q: p & q

4. “France and Germany are in Europe”: France and Germany are in Europe. = p

Making Truth Tables

• For single variable ‘p’

• For two variables ‘p’ and ‘q’

Making Truth Tables

• For three variables ‘p’ ‘q’ and ‘r’

Important Rule• All the variables are represented in small-caps.

example: p, q, r

• Premises are represented with an asterisk mark on top of the variable. example: p*

• Conclusion is represented with capital letter ‘C’ at the end of the variable.example: qC

Important Rule• In a valid argument it is impossible for all

premises to be true and the conclusion to be false.

Important Rule• Look for Invalidity first and then go for

Validity.i.e First identify that combination in which all premises are true and the conclusion is false. (T-T-F) or (T-T-T-F)

• If this combination is not present then the argument automatically becomes Valid.(T-T-T) or (T-T-T-T)

Conjunction

….usage of word “AND”

Making Truth Tables for conjunction

Example: “Tina is tall and Sarah is tall.”

: Tina is tall = p Sarah is tall = q: p & q

Example• Assume a, b, c are True and x, y, z are False.• Determine whether following are True or False.

1. a & b = T & T = T2. b & z = T & F = F3. x & y= F & F = F4. y & a= F & T = F

Conjunction and Validity

Example:Tina is tall. Sarah is tall.So, Tina is tall, and Sarah is tall.

p* : Premiseq* : Premisep & qC : Conclusion

Conjunction and ValidityAnswer:

The possibility of all premises being true and conclusion being false is absent.

Alternatively; the possibility of all premises as well as the conclusion being true is present in the first case; as shown in the truth table.

Hence, The Argument is Valid in nature.

Example:The sky is blue.The grass is green.Therefore, the sky is blue, and the grass is green.

Answer:p*q*p & qC

The possibility of all premises being true and conclusion being false is absent.

Alternatively; the possibility of all premises as well as the conclusion being true is present in the first case; as shown in the truth table.

Hence, The Argument is Valid in nature.

Example:The grass is green.So, grass is green, and the sky is blue.

Answer:p*p & qC The possibility of all premises being true and

conclusion being false is present in the second case.

Hence, The Argument is Invalid in nature.

Example:Franklin is short and stout.So, Franklin is short.

Answer:p & q*pC

The possibility of all premises being true and conclusion being false is absent.

Alternatively; the possibility of all premises as well as the conclusion being true is present in the first case; as shown in the truth table.

Hence, The Argument is Valid in nature.

Example:The train was on time. The bus was late. So, the train was on time and the bus was late.

Answer:p*q*p & qC The possibility of all premises being true and conclusion

being false is absent. Alternatively; the possibility of all premises as well as

the conclusion being true is present in the first case; as shown in the truth table.

Hence, The Argument is Valid in nature.

Example:The train was on time. So, the train was on time and the bus was late.

Answer:p*p & qC

The possibility of all premises being true and conclusion being false is present in the second case.

Hence, The Argument is Invalid in nature.

Example:The train was on time. So, the bus was late.

Answer:p*qC

The possibility of all premises being true and conclusion being false is present in the second case.

Hence, The Argument is Invalid in nature.

Negation

Negation• Use of word ‘NOT’• Symbolized as: ~ : called as negation

ExampleTina is NOT tall. = ~p

Negation

ExampleFrank is not angry and Hank is not tired.~p & ~q

• Assume that a, b, c are true and x, y and z are false. Then answer the following is true or false.

1. ~a & b = F&T = F

2. ~x & y = T&F = F

3. ~z & c = T&T = T

• Assume that a, b, c are true and x, y and z are false. Then answer the following is true or false.

4. ~(a & b) = ~(T&T) = ~T = F

5. ~(c & ~z) = ~(T&T) = ~T = F

6. ~x & ~z = T&T = T

Example“Tina is not tall and Sarah is tall. So, Tina is

not tall.”

Answer:~p & q*~pC

The possibility of all premises being true and conclusion being false is absent.

Alternatively; the possibility of all premises as well as the conclusion being true is present in the fourth case; as shown in the truth table.

Hence, The Argument is Valid in nature.

ExampleFrank does not drive a truck. So, Frank does not drive a truck, and Vinny does not drive a minivan.

Answer:~p*~p & ~qC

The possibility of all premises being true and conclusion being false is present in the fourth case.

Hence, The Argument is Invalid in nature.

ExampleIt is not the case that Tina is tall and Sarah is tall. So, Tina is not tall, and Sarah is not tall.

Answer:~(p & q)* ~p & ~qC

The possibility of all premises being true and conclusion being false is present in the fourth case.

Hence, The Argument is Invalid in nature.

ExampleIt is not the case that John failed Calculus and Chemistry. John didn’t fail Calculus. So, he didn’t fail Chemistry.

Answer:~(p & q)* ~p*~qC

The possibility of all premises being true and conclusion being false is present in the fourth case.

Hence, The Argument is Invalid in nature.

ExampleIt is not the case that John failed Calculus and Chemistry. So, John didn’t fail Calculus and didn’t fail Chemistry.

Answer:~(p & q)* ~ p & ~qC

The possibility of all premises being true and conclusion being false is present in the fourth case.

Hence, The Argument is Invalid in nature.

ExampleIt is not the case that John failed Calculus and Chemistry. John failed Calculus. So he didn’t fail Chemistry.

Answer:~(p & q)*

p* ~qC

The possibility of all premises being true and conclusion being false is absent.

Alternatively; the possibility of all premises as well as the conclusion being true is present in the second case; as shown in the truth table.

Hence, The Argument is Valid in nature.

ExampleJack didn’t commit the crime, and the Mac didn’t commit the crime. So, it is not the case that Jack and the Mac committed the crime.

Answer:~p & ~q* ~(p & q)C

The possibility of all premises being true and conclusion being false is absent.

Alternatively; the possibility of all premises as well as the conclusion being true is present in the fourth case; as shown in the truth table.

Hence, The Argument is Valid in nature.

Deeper Analysis of Negation & ConjunctionExampleTina is tall. Sarah is not tall, but Missy is tall.So, Tina is tall, and Missy is tall.

p*~q & r*p & rC

The possibility of all premises being true and conclusion being false is absent.

Alternatively; the possibility of all premises as well as the conclusion being true is present in the second case; as shown in the truth table.

Hence, The Argument is Valid in nature.

ExampleThe train was on time, but the bus was not on time. The plane was on time. Therefore, it’s not the case that the bus and the plane were on time.

Answer:p & ~q*r* ~ (q & r)C

The possibility of all premises being true and conclusion being false is absent.

Alternatively; the possibility of all premises as well as the conclusion being true is present in the third case; as shown in the truth table.

Hence, The Argument is Valid in nature.

ExampleBill Clinton didn’t serve two terms in office, but Jimmy Carter did. George H. W Bush served two terms in office. So, it is not the case that both Carter and Bush did not serve two terms in office.

Answer:~ p & q*r* ~ (~ (q & r)) C

The possibility of all premises being true and conclusion being false is present in the third case.

Hence, The Argument is Invalid in nature.

Disjunction

Disjunction• Use of word ‘OR’• Symbolized as: ‘v’ : called as wedge

ExampleFrank is angry or Hank is tired.p v q

ExampleFrank is angry or Hank is tired. So, Frank is angry.

Answer:p v q*pC

The possibility of all premises being true and conclusion being false is present in the fourth case.

Hence, The Argument is Invalid in nature.

Example (Disjunction + Negation + Conjunction)It’s not the case that Frank is angry or Hank is tired. So, Frank is not angry and Hank is not tired.

Answer:~(p v q)*~p & ~qC

The possibility of all premises being true and conclusion being false is absent.

Alternatively; the possibility of all premises as well as the conclusion being true is present in the fourth case; as shown in the truth table.

Hence, The Argument is Valid in nature.

Example Frank is angry or Hank is tired. It’s not the case that Hank is tired and Larry is lonely. So, Hank is tired.

Answer:p v q*~(q & r)*qC

The possibility of all premises being true and conclusion being false is present in the third case.

Hence, The Argument is Invalid in nature.

Conditional Statement

…presence of If – then relationship.

Conditional Statement• If – then statement• Antecedent & Consequent• “If it rained, then the ground is wet.”• “it rained”= antecedent = p• “ground is wet”= consequent = q

ExampleIf it rained, then the ground is wet. It rained. So, the ground is wet.

Answer: p → q*pq The possibility of all premises being true and

conclusion being false is absent. Alternatively; the possibility of all premises as well as

the conclusion being true is present in the first case; as shown in the truth table.

Hence, The Argument is Valid in nature.

ExampleIf it rained, then the ground is wet. The ground is not wet. So, it did not rain.

Answer: p → q*~q*~pC

The possibility of all premises being true and conclusion being false is absent.

Alternatively; the possibility of all premises as well as the conclusion being true is present in the fourth case; as shown in the truth table.

Hence, The Argument is Valid in nature.