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What rule of inference is used in each of these argu- ments?
a) Alice is a mathematics major. Therefore, Alice is ei- ther a mathematics major or a computer science major.
b) Jerry is a mathematics major and a computer science major. Therefore, Jerry is a mathematics major.
c) If it is rainy, then the pool will be closed. It is rainy. Therefore, the pool is closed.
d) If it snows today, the university will close. The uni- versity is not closed today. Therefore, it did not snow today.
e) If I go swimming, then I will stay in the sun too long. If I stay in the sun too long, then I will sunburn. There- fore, if I go swimming, then I will sunburn.

Respuesta :

profantoniofonte

Answer:

a) Addition b) Simplification c) Modus Ponens d) Modus Tollens e) Hypothetical Syllogism

Step-by-step explanation:

Firstly before applying the Rules of Inference, we have to translate those arguments into a symbolic form.

a) Alice is a mathematics major. Therefore, Alice is either a mathematics major or a computer science major.

Alice is a mathematics major = B

A computer Science major = R

1. P

2. ∴P ∨ Q

3. P→(P ∨ Q)

What we have here is a corresponding Tautology called Addition

Addition

b) Jerry is a mathematics major and a computer science major. Therefore, Jerry is a mathematics major.

Making the whole process of translating into symbolic language.

1. Jerry is a mathematics major and a computer science major: P ∧ Q

2. P

3 (P∧Q)→(P)

Since those premises in 1 and 2  are composed by and implication and a conjuction, this corresponding tautology is a

Simplification.

c) If it is rainy, then the pool will be closed. It is rainy. Therefore, the pool is closed.

Decomposing the premises and translating into Symbolic:

.  it is rainy= P.  the pool  closed= Q  

1st. Premise: P→Q

2nd. P

3rd.  (P→Q. ∧P)→Q

Notice  these Corresponding Tautologies. How they reinforce the argument, making it valid.

Modus Ponens

d)  If it snows today, the university will close. The university is not closed today. Therefore, it did not snow today.

It snows today =P

The university will close=Q.

The university is not closed=¬Q

it did not snow today=¬P

1. (P→Q)

2.∧¬Q

3. ¬P

4.((P→Q)∧ ¬Q)→¬P

Modus Tollens

e)  If I go swimming, then I will stay in the sun too long. If I stay in the sun too long, then I will sunburn. Therefore, if I go swimming, then I will sunburn.

I go swimming= P,  I will stay in the sun too long= Q

I stay in the sun too long = R  I will sunburn=S

1: P→Q

2:R→S

3. P→S

(P→Q)∧(R→S)→(P→S)

Hypothetical Syllogism

Martebi

The rule of inference that is used in Alice as a mathematics major is Addition

  • In terms of Jerry is a mathematics major and a computer science major, The rule is Simplification.

  • In terms of If it is rainy, then the pool will be closed. It is rainy, The rule is Modus ponens.

  • In terms of If it snows today, the university will close. The university is not closed today, the rule is Modus tollens.

  • In terms of If I go swimming, then I will stay in the sun too long, the rule is Hypothetical syllogism.

What are the rules of inference?

The rule of inference is a term that is also called the transformation rule. They are known to be a logical methods that is made up of different function that does takes the first step, solves their syntax, and then they bring a conclusion.

The names of the Rules of Inference are

  • Name
  • Addition
  • Simplification
  • Conjunction
  • Resolution, etc.

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