Chapter 13

Question 1

Define a tautology in the context of TFL.

Answer 1

A tautology is a sentence that is true on every valuation.

Question 2

How can truth tables be used to determine if a sentence is a tautology?

Answer 2

Truth tables can be used to check if a sentence is true on every line of its complete truth table; if it is, then the sentence is a tautology.

Question 3

Describe the relationship between necessary truth and tautology.

Answer 3

A tautology serves as a surrogate for necessary truth, as it is true under all valuations, similar to how necessary truth is always true.

Question 4

What is the significance of the example ‘(H ∧ I) → H’ in understanding tautologies?

Answer 4

The example ‘(H ∧ I) → H’ is a tautology, illustrating that it is true on every valuation.

Question 5

Explain the concept of necessary falsity in relation to TFL.

Answer 5

Necessary falsity refers to statements that are false under all valuations, which has a counterpart in TFL but is not explicitly defined in the provided content.

Question 6

How does the chapter relate to previous discussions on valuation and truth values?

Answer 6

The chapter builds on the previous discussion by introducing tautologies and contradictions, showing how truth tables can be used to evaluate these concepts.

Question 7

What are the limitations mentioned regarding necessary truths in TFL?

Answer 7

The text notes that there are some necessary truths that cannot be adequately symbolized in TFL.

Question 8

Describe a tautology in TFL.

Answer 8

A tautology in TFL is a sentence that is true under every valuation, meaning it cannot be false regardless of the truth values assigned to its components.

Question 9

How can we determine if a sentence is a contradiction in TFL?

Answer 9

A sentence is a contradiction in TFL if it is false on every line of its complete truth table, indicating it is false under every valuation.

Question 10

Define equivalence in the context of TFL.

Answer 10

A and B are equivalent in TFL if their truth values agree for every valuation, meaning there is no valuation in which they have opposite truth values.

Question 11

What is the significance of truth tables in TFL?

Answer 11

Truth tables are used to determine whether sentences are tautologies, contradictions, or equivalent by systematically evaluating their truth values across all possible valuations.

Question 12

How can we test for the equivalence of two sentences in TFL?

Answer 12

To test for equivalence, we construct a truth table for both sentences and check if their truth values match for every possible valuation.

Question 13

Explain the example of the contradiction given in the content.

Answer 13

The example of a contradiction is the sentence ‘[(C ↔ C) → C] ∧ ¬(C → C)’, which is false on every valuation, thus qualifying as a contradiction.

Question 14

What does it mean if two sentences have opposite truth values in TFL?

Answer 14

If two sentences have opposite truth values in TFL, they are not equivalent, indicating that there exists at least one valuation where one sentence is true and the other is false.

Question 15

How is the sentence ‘¬(P ∨ Q)’ related to ‘¬P ∧ ¬Q’ in TFL?

Answer 15

The relationship between ‘¬(P ∨ Q)’ and ‘¬P ∧ ¬Q’ can be tested for equivalence using a truth table, which shows whether they yield the same truth values across all valuations.

Question 16

What is the role of necessary truth in TFL?

Answer 16

Necessary truth in TFL is expressed by sentences that can be symbolized as tautologies, indicating that they are true in all possible scenarios.

Question 17

Describe the process of constructing a truth table.

Answer 17

Constructing a truth table involves listing all possible truth value combinations for the variables involved and then calculating the truth values of the compound sentences based on these combinations.

Question 18

Describe the relationship between truth values in logical operators for two sentences.

Answer 18

The truth values for the main logical operators can be compared in a table; if they match on every row, the two sentences are considered equivalent.

Question 19

Define jointly satisfiable sentences in TFL.

Answer 19

Sentences A1, A2, …, An are jointly satisfiable in TFL if there exists some valuation that makes all of them true.

Question 20

How can one test for joint satisfiability of sentences?

Answer 20

Joint satisfiability can be tested using truth tables.

Question 21

Explain the concept of entailment in TFL.

Answer 21

Sentences A1, A2, …, An entail sentence C in TFL if no valuation makes all of A1, A2, …, An true while making C false.

Question 22

What is the method to check entailment using truth tables?

Answer 22

To check entailment, one must determine if there is any valuation that makes both the premises true and the conclusion false.

Question 23

Describe the significance of the final row in a truth table comparison.

Answer 23

In a truth table comparison, if the final row shows both sentences as true, it indicates that the sentences are equivalent.

Question 24

How are jointly unsatisfiable sentences defined?

Answer 24

Sentences are jointly unsatisfiable if no valuation can make all of them true at the same time.

Question 25

What logical operators are mentioned in the context of the first two sentences?

Answer 25

Negation is mentioned for the first sentence, and conjunction for the second.

Question 26

Explain the process of using a truth table to check entailment for specific sentences.

Answer 26

To check whether ‘¬L → (J ∨ L)’ and ‘¬L’ entail ‘J’, one must verify if there is a valuation that makes the first two true and ‘J’ false.

Question 27

Describe the relationship between ‘¬ L → (J ∨ L)’ and ‘¬ L’ in the context of entailment.

Answer 27

The only row where both ‘¬ L → (J ∨ L)’ and ‘¬ L’ are true is the second row, which also shows that ‘J’ is true. Therefore, ‘¬ L → (J ∨ L)’ and ‘¬ L’ entail ‘J’.

Question 28

Define entailment in the context of TFL (Truth-Functional Logic).

Answer 28

Entailment in TFL means that if a set of premises A1, A2, …, An is true, then the conclusion C must also be true. If A1, A2, …, An entail C, the argument A1, A2, …, An ∴ C is valid.

Question 29

How can one test for the validity of arguments in English using TFL?

Answer 29

To test for the validity of English arguments, first symbolize them in TFL, then check for entailment using truth tables.

Question 30

Explain why an argument A1, A2, …, An ∴ C cannot be invalid if A1, A2, …, An entail C.

Answer 30

If A1, A2, …, An entail C, it is impossible to find a case where all premises are true and C is false, which would be required for the argument to be invalid.

Question 31

What is the significance of truth tables in determining entailment in TFL?

Answer 31

Truth tables provide a systematic way to evaluate the truth values of premises and conclusions, allowing one to determine if the premises entail the conclusion.

Question 32

How does the truth value of sentence letters affect the evaluation of arguments in TFL?

Answer 32

The truth values of sentence letters determine the truth values of the entire sentences based on the connectives used, which is essential for evaluating the validity of arguments.

Question 33

Describe the process of generating a valuation for sentence letters in TFL.

Answer 33

A valuation is generated by assigning truth values to sentence letters based on a specific case, which reflects the truth values of the corresponding sentences in that case.

Question 34

What conclusion can be drawn if an argument in TFL is found to be valid?

Answer 34

If an argument in TFL is valid, it means that whenever the premises are true, the conclusion must also be true, confirming the logical relationship between them.

Question 35

Describe the symbol used to represent entailment in the context of TFL sentences.

Answer 35

The symbol used to represent entailment is ‘⊨’, known as the double turnstile, which abbreviates the statement that a set of TFL sentences together entails another sentence.

Question 36

How is the double turnstile symbol used in metalanguage sentences?

Answer 36

In metalanguage sentences, the double turnstile symbol ‘⊨’ is used to indicate that a set of TFL sentences entails another sentence, such as in the example: A, A → B ⊨ B.

Question 37

Define the limiting case of the double turnstile symbol.

Answer 37

The limiting case of the double turnstile symbol is written as ⊨ C, which indicates that there is no valuation that makes all the sentences on the left side true while making C false, meaning C is true in every valuation.

Question 38

What does it mean if a sentence is represented as A ⊨ ?

Answer 38

If a sentence is represented as A ⊨, it means that no valuation makes A true, indicating that A is a contradiction.

Question 39

Explain the significance of the double turnstile in the context of entailment.

Answer 39

The double turnstile is significant as it provides a concise way to express the relationship of entailment between TFL sentences and allows for the discussion of tautologies and contradictions in a formalized manner.

Question 40

How can we express the idea that a sentence C is a tautology using the double turnstile?

Answer 40

We can express that a sentence C is a tautology by writing ⊨ C, indicating that every valuation makes C true.

Question 41

What is the relationship between object language and metalanguage in the context of the double turnstile?

Answer 41

The double turnstile is a symbol of the metalanguage, which is used to discuss the properties of the object language (TFL sentences) rather than being a part of the object language itself.

Question 42

Describe a scenario where we might want to deny a tautological entailment.

Answer 42

We might want to deny a tautological entailment when we want to assert that there is a situation or valuation where the entailment does not hold, indicating a more complex relationship between the sentences involved.

Question 43

Describe the meaning of A1, A2, …, An ⊭ C.

Answer 43

It indicates that there exists some valuation that makes all of A1, A2, …, An true while making C false.

Question 44

How does A1, A2, …, An ⊨ C differ from A1, A2, …, An ⊭ C?

Answer 44

A1, A2, …, An ⊨ C means that every valuation that makes A1, A2, …, An true also makes C true, while A1, A2, …, An ⊭ C indicates that at least one valuation makes A1, A2, …, An true and C false.

Question 45

Define the relationship between A ⊨ C and A → C.

Answer 45

A ⊨ C holds true if no valuation makes A true and C false, while A → C is a tautology if no valuation makes A true and C false.

Question 46

How is the symbol ‘→’ characterized in relation to TFL?

Answer 46

The symbol ‘→’ is a sentential connective of TFL, meaning it combines two TFL sentences into a longer TFL sentence.

Question 47

Explain the significance of the symbol ‘⊨’.

Answer 47

The symbol ‘⊨’ represents a metalinguistic sentence that refers to the surrounding TFL sentences, distinguishing it from the sentential connective ‘→’.

Question 48

What does it mean when A → C is described as a tautology?

Answer 48

A → C is a tautology if it is true in all cases except when A is true and C is false.

Question 49

How do the terms ‘entails’ and ‘implies’ differ in logical context?

Answer 49

‘Entails’ refers to the relationship denoted by ‘⊨’, while ‘implies’ can refer to both ‘⊨’ and ‘→’, which can lead to confusion.

Question 50

What is the implication of using ‘implies’ in logical discussions?

Answer 50

Using ‘implies’ can create confusion, so it is recommended to use it only to mean ‘entails’.

Question 51

Describe the conditions under which A → C is a tautology.

Answer 51

A → C is a tautology if no valuation makes A true and C false, meaning it is true in all other cases.

Question 52

How can one express the relationship between A ⊨ C and A → C?

Answer 52

A → C is a tautology if and only if A ⊨ C, indicating a strong connection between entailment and the truth of the conditional.