Logic chapter 1 basics part 3

Question 1

Describe a symbolization key.

Answer 1

A symbolization key assigns natural language interpretations to a collection of atomic formulas.

Question 2

How can we use a symbolization key in logic?

Answer 2

We can use a symbolization key to assign atomic formulas (individual variables) to a collection of natural language sentences.

Question 3

Define the relationship between complex formulas and natural language sentences.

Answer 3

Under our natural language interpretation of logical connectives, we can translate between complex formulas and complex natural language sentences.

Question 4

Give an example of a natural language sentence and its corresponding atomic formula.

Answer 4

An example is: ‘It is raining outside’ corresponds to the atomic formula A.

Question 5

What is the purpose of translating between formal and natural language?

Answer 5

The purpose is to facilitate understanding and interpretation of logical statements in both formal and natural contexts.

Question 6

How does the example ‘p 2 is irrational’ relate to symbolization keys?

Answer 6

It serves as an atomic formula that can be interpreted through a symbolization key.

Question 7

Explain the significance of the example ‘Edmonton is north of Calgary’ in the context of symbolization keys.

Answer 7

It illustrates how a natural language sentence can be represented as an atomic formula using a symbolization key.

Question 8

Describe the translation of the sentence ‘Edmonton is north of Calgary and p2 is irrational’ into TFL.

Answer 8

The translation into TFL is represented by the formula C ∧ B.

Question 9

How is the sentence ‘If it is raining outside and p2 is irrational, then it is raining outside’ expressed in TFL?

Answer 9

This sentence is translated into TFL as (A ∧ B) → A.

Question 10

Define the meaning of the symbol ‘∧’ in the context of TFL.

Answer 10

The symbol ‘∧’ represents the logical conjunction, meaning ‘and’.

Question 11

What does the symbol ‘→’ signify in TFL?

Answer 11

The symbol ‘→’ signifies logical implication, meaning ‘if…then…’.

Question 12

Identify the natural language sentence that corresponds to the formula C ∧ B.

Answer 12

The natural language sentence is ‘Edmonton is north of Calgary and p2 is irrational’.

Question 13

Explain the significance of irrational numbers in the context of the provided translations.

Answer 13

Irrational numbers, such as p2, are used in the translations to illustrate logical statements involving mathematical concepts.

Question 14

List the components of the formula (A ∧ B) → A in natural language.

Answer 14

The components are ‘It is raining outside’ (A) and ‘p2 is irrational’ (B).

Question 15

Describe the translation of the sentence ‘It is not raining outside or Edmonton is north of Calgary’ into TFL.

Answer 15

The translation is represented by the formula: ¬ A ∨ C.

Question 16

Define the logical relationship expressed in the sentence ‘p 2 is irrational if and only if it is not raining outside’.

Answer 16

This relationship is translated into TFL as B ↔ ¬ A.

Question 17

How can the statement ‘It is raining outside’ be represented in TFL?

Answer 17

It can be represented as A.

Question 18

Do the sentences ‘p 2 is irrational’ and ‘Edmonton is north of Calgary’ have specific representations in TFL?

Answer 18

Yes, ‘p 2 is irrational’ is represented as B and ‘Edmonton is north of Calgary’ is represented as C.

Question 19

Explain the meaning of the symbol ‘¬’ in TFL.

Answer 19

The symbol ‘¬’ represents negation, indicating that a statement is not true.

Question 20

What does the symbol ‘∨’ signify in TFL?

Answer 20

The symbol ‘∨’ signifies logical disjunction, meaning ‘or’.

Question 21

How is the logical equivalence expressed in TFL?

Answer 21

Logical equivalence is expressed using the symbol ‘↔’, meaning ‘if and only if’.

Question 22

Describe the natural language argument presented in the content.

Answer 22

The argument states that if John cuts Bill’s lawn, then Bill owes John $20. It also states that John did cut Bill’s lawn, leading to the conclusion that Bill owes John $20.

Question 23

Define the symbolization key used in the formalization of the argument.

Answer 23

The symbolization key defines ‘A’ as ‘John cuts Bill’s lawn’ and ‘B’ as ‘Bill owes John $20’.

Question 24

How is the first premise of the argument formalized?

Answer 24

The first premise is formalized as A → B, meaning ‘If John cuts Bill’s lawn, then Bill owes John $20’.

Question 25

What does the second premise state in the formalized argument?

Answer 25

The second premise states A, which means ‘John cut Bill’s lawn’.

Question 26

Explain the conclusion of the argument in both natural and formal language.

Answer 26

In natural language, the conclusion is ‘Bill owes John $20’. In formal language, it is represented as B.

Question 27

Do the premises lead to a valid conclusion in the argument?

Answer 27

Yes, the premises lead to a valid conclusion, as the second premise affirms the antecedent of the first premise, allowing the conclusion to follow.

Question 28

How can the argument be summarized in formal logic notation?

Answer 28

The argument can be summarized as: 1. A → B, 2. A, 3. B.

Question 29

Describe the first premise of the argument involving John and Bill.

Answer 29

The first premise states that if John cuts Bill’s lawn, then Bill owes John $20.

Question 30

Define the symbolization key used in the argument.

Answer 30

The symbolization key defines A as ‘John cut Bill’s lawn’ and B as ‘Bill owes John $20’.

Question 31

How is the second premise of the argument expressed in formal language?

Answer 31

The second premise is expressed as ¬B, meaning ‘It is not the case that Bill owes John $20’.

Question 32

Do the premises lead to a conclusion in the argument?

Answer 32

Yes, the premises lead to the conclusion that it is not the case that John cut Bill’s lawn, expressed as ¬A.

Question 33

Explain the logical structure of the argument presented.

Answer 33

The argument follows a logical structure where the first premise implies a condition, the second premise negates that condition, leading to a conclusion that also negates the initial action.

Question 34

What conclusion can be drawn from the premises in the argument?

Answer 34

The conclusion drawn is that John did not cut Bill’s lawn.

Question 35

How can the argument be summarized in formal terms?

Answer 35

In formal terms, the argument can be summarized as: 1. A → B, 2. ¬B, therefore ¬A.

Question 36

Identify the role of premise 2 in the argument.

Answer 36

Premise 2 serves to negate the outcome of premise 1, indicating that Bill does not owe John $20.