translating english to propositional logic

Translating English to Propositional Logic

Phil 57 section 3San Jose State University

Fall 2010

Does order always matter?

John went to school and Mary went to school.

Mary went to school and John went to school.

P = John went to school.Q = Mary went to school.(PQ) means the same as (QP), logically speaking.

Does order always matter?

John went to school or Mary went to school.

Mary went to school or John went to school.

P = John went to school.Q = Mary went to school.(PQ) means the same as (QP), logically speaking.

Conjunction and disjunction are commutative.

(PQ) means the same as (QP)

P Q (PQ) (QP)

T T T T

T F F F

F T F F

F F F F

Conjunction and disjunction are commutative.

(PQ) means the same as (QP)

P Q (PQ) (QP)

T T T T

T F T T

F T T T

F F F F

Conjunction and disjunction are also associative.

((PQ)R) means the same as (P(QR)).

((PQ)R) means the same as (P(QR)).

With mixed operators, order does matter.

P = I like peanut butter.Q = I like jelly.

~(PQ) means “it is not the case that I like peanut butter and jelly”.

(~P~Q) means “I don’t like peanut butter and I don’t like jelly.”

With mixed operators, order does matter.

P = Tom will work late.Q = Dick will work late.R = Harry will call in sick.~((PQ)R) means “it is not the case that Tom

and Dick will work late or that Harry will call in sick.”

Translating material conditionals.

If [antecedent], then [consequent].

P= antecedentQ= consequent

PQ

Material conditionals and necessary conditions.

Getting an A on the final exam is a necessary condition for getting an A in the class.

Necessary because if I don’t meet this condition, I don’t bring about the outcome.

P= I get an A on the final exam.Q= I get an A in the class.

QP

Material conditionals and necessary conditions.

Getting an A on the final exam is a necessary condition for getting an A in the class.

P= I get an A on the final exam.Q= I get an A in the class.

P Q QPT T T

T F T

F T F

F F T

Material conditionals and sufficient conditions.

Getting a B on all the exams is a sufficient condition for getting a B in the class.

Sufficient because it’s enough to bring the outcome, but it’s not the only way to bring it.

P= I get a B on all the exams.Q= I get a B in the class.

PQ

Material conditionals and sufficient conditions.

Getting a B on all the exams is a sufficient condition for getting a B in the class.

P= I get a B on all the exams.Q= I get a B in the class.

P Q PQT T T

T F F

F T T

F F T

Translating material conditionals.

If … , then ...

It taxes go up, then inflation will rise.T= Taxes go upR= Inflation will rise

TR

Translating material conditionals.

… only if...

Iran will supply arms to Syria only if Syria helps Hezbollah.

R= Iran will supply arms to SyriaS= Syria helps Hezbollah

RS

Translating material conditionals.

Only if ... will …

Only if Jenna passes the exam will Jenna get her license.

P= Jenna passes the examQ= Jenna will get her license

QP

Translating material conditionals.

… if ...

I will pass the muffins if you ask me nicely.M= I will pass the muffinsN= You ask me nicely.

NM

Translating material conditionals.

Construction Translation

If P, then Q (PQ)

P, if Q (QP)

P only if Q (PQ)

Only if P, Q (QP)

Translating biconditionals.

… if and only if …

Jill needs a parachute if and only if she is planning to jump from the plane.

P= Jill needs a parachute.Q= Jill is planning to jump from the plane.PQ

Translating biconditionals.

… just in case …Bill will take the geology course just in case it fulfils the science requirement.

T= Bill will take the geology course R= The geology course fulfils the science

requirement.TR

A biconditional is the conjunction of two material conditionals.

Jill needs a parachute if and only if she is planning to jump from the plane.

If Jill needs a parachute, then she is planning to jump from the plane,

ANDIf Jill is planning to jump from the plane, then

she needs a parachute.

Biconditionals are commutative.

PQ is the same as QP

“Bill will take the geology course just in case it fulfils the science requirement” is equivalent to

“The geology course fulfils the science requirement just in case Bill will take it.”

Conjunction of two conditionals (and conjunction is commutative)

Biconditionals are associative.

(P(QR)) is the same as ((PQ)R)

Conjunction of two conditionals (and conjunction is associative)

But note that material conditionals are neither commutative nor associative.

(PQ) ≠ (QP) (P(QR)) ≠ ((PQ)R)

Order matters translating conditionals.

P = Ben will answer the phones.Q = Liz will work out the budget.

(PQ): If Ben will answer the phones, then Liz will work out the budget.

(QP): If Liz will work out the budget, then Ben

will answer the phones.

Order matters translating conditionals.

P = Ben will answer the phones.Q = Liz will work out the budget.

P Q (PQ) (QP)T T T T

T F F T

F T T F

F F T T

Translating English to PL.

“Propositional Logic translation guide” on course website

(http://www.stemwedel.org/logic-and-critical-reasoning/PL-TranslationGuide.pdf)

Practice (like on HW #7) will help!