Logic chapter 1 basics part 3 Flashcards
Describe a symbolization key.
A symbolization key assigns natural language interpretations to a collection of atomic formulas.
How can we use a symbolization key in logic?
We can use a symbolization key to assign atomic formulas (individual variables) to a collection of natural language sentences.
Define the relationship between complex formulas and natural language sentences.
Under our natural language interpretation of logical connectives, we can translate between complex formulas and complex natural language sentences.
Give an example of a natural language sentence and its corresponding atomic formula.
An example is: ‘It is raining outside’ corresponds to the atomic formula A.
What is the purpose of translating between formal and natural language?
The purpose is to facilitate understanding and interpretation of logical statements in both formal and natural contexts.
How does the example ‘p 2 is irrational’ relate to symbolization keys?
It serves as an atomic formula that can be interpreted through a symbolization key.
Explain the significance of the example ‘Edmonton is north of Calgary’ in the context of symbolization keys.
It illustrates how a natural language sentence can be represented as an atomic formula using a symbolization key.
Describe the translation of the sentence ‘Edmonton is north of Calgary and p2 is irrational’ into TFL.
The translation into TFL is represented by the formula C ∧ B.
How is the sentence ‘If it is raining outside and p2 is irrational, then it is raining outside’ expressed in TFL?
This sentence is translated into TFL as (A ∧ B) → A.
Define the meaning of the symbol ‘∧’ in the context of TFL.
The symbol ‘∧’ represents the logical conjunction, meaning ‘and’.
What does the symbol ‘→’ signify in TFL?
The symbol ‘→’ signifies logical implication, meaning ‘if…then…’.
Identify the natural language sentence that corresponds to the formula C ∧ B.
The natural language sentence is ‘Edmonton is north of Calgary and p2 is irrational’.
Explain the significance of irrational numbers in the context of the provided translations.
Irrational numbers, such as p2, are used in the translations to illustrate logical statements involving mathematical concepts.
List the components of the formula (A ∧ B) → A in natural language.
The components are ‘It is raining outside’ (A) and ‘p2 is irrational’ (B).
Describe the translation of the sentence ‘It is not raining outside or Edmonton is north of Calgary’ into TFL.
The translation is represented by the formula: ¬ A ∨ C.
Define the logical relationship expressed in the sentence ‘p 2 is irrational if and only if it is not raining outside’.
This relationship is translated into TFL as B ↔ ¬ A.
How can the statement ‘It is raining outside’ be represented in TFL?
It can be represented as A.
Do the sentences ‘p 2 is irrational’ and ‘Edmonton is north of Calgary’ have specific representations in TFL?
Yes, ‘p 2 is irrational’ is represented as B and ‘Edmonton is north of Calgary’ is represented as C.
Explain the meaning of the symbol ‘¬’ in TFL.
The symbol ‘¬’ represents negation, indicating that a statement is not true.
What does the symbol ‘∨’ signify in TFL?
The symbol ‘∨’ signifies logical disjunction, meaning ‘or’.
How is the logical equivalence expressed in TFL?
Logical equivalence is expressed using the symbol ‘↔’, meaning ‘if and only if’.
Describe the natural language argument presented in the content.
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.
Define the symbolization key used in the formalization of the argument.
The symbolization key defines ‘A’ as ‘John cuts Bill’s lawn’ and ‘B’ as ‘Bill owes John $20’.
How is the first premise of the argument formalized?
The first premise is formalized as A → B, meaning ‘If John cuts Bill’s lawn, then Bill owes John $20’.
What does the second premise state in the formalized argument?
The second premise states A, which means ‘John cut Bill’s lawn’.
Explain the conclusion of the argument in both natural and formal language.
In natural language, the conclusion is ‘Bill owes John $20’. In formal language, it is represented as B.
Do the premises lead to a valid conclusion in the argument?
Yes, the premises lead to a valid conclusion, as the second premise affirms the antecedent of the first premise, allowing the conclusion to follow.
How can the argument be summarized in formal logic notation?
The argument can be summarized as: 1. A → B, 2. A, 3. B.
Describe the first premise of the argument involving John and Bill.
The first premise states that if John cuts Bill’s lawn, then Bill owes John $20.
Define the symbolization key used in the argument.
The symbolization key defines A as ‘John cut Bill’s lawn’ and B as ‘Bill owes John $20’.
How is the second premise of the argument expressed in formal language?
The second premise is expressed as ¬B, meaning ‘It is not the case that Bill owes John $20’.
Do the premises lead to a conclusion in the argument?
Yes, the premises lead to the conclusion that it is not the case that John cut Bill’s lawn, expressed as ¬A.
Explain the logical structure of the argument presented.
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.
What conclusion can be drawn from the premises in the argument?
The conclusion drawn is that John did not cut Bill’s lawn.
How can the argument be summarized in formal terms?
In formal terms, the argument can be summarized as: 1. A → B, 2. ¬B, therefore ¬A.
Identify the role of premise 2 in the argument.
Premise 2 serves to negate the outcome of premise 1, indicating that Bill does not owe John $20.
