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Chapter 1: Problem 29

Express each of these statements using logical operators, predicates, and quantifiers. a) Some propositions are tautologies. b) The negation of a contradiction is a tautology. c) The disjunction of two contingencies can be a tautology. d) The conjunction of two tautologies is a tautology.

Short Answer

Expert verified
a) \( \exists x (P(x) \land T(x)) \)b) \( \forall x (T(eg C(x))) \)c) \( \exists y_1 \exists y_2 (C(y_1) \land C(y_2) \land T(y_1 \lor y_2)) \)d) \( \forall z_1 \forall z_2 (T(z_1) \land T(z_2) \rightarrow T(z_1 \land z_2)) \)

Step by step solution

01

- Identify the key elements for each statement

Determine the main concepts in each statement: propositions, tautologies, contradictions, disjunction, and conjunction.
02

- Define predicates and variables

Let P(x) denote 'x is a proposition' and T(x) denote 'x is a tautology'.
03

- Translate statement (a)

Statement: 'Some propositions are tautologies'. Using existential quantifier (\forall), it can be written as: \( \exists x (P(x) \land T(x)) \)
04

- Translate statement (b)

Statement: 'The negation of a contradiction is a tautology'. Using logical and predicate notation, let C(x) denote 'x is a contradiction'. Then it can be written as: \( \forall x (T(eg C(x))) \)
05

- Translate statement (c)

Statement: 'The disjunction of two contingencies can be a tautology'. Let C(y) denote 'y is a contingency'. Then it can be written as: \( \exists y_1 \exists y_2 (C(y_1) \land C(y_2) \land T(y_1 \lor y_2)) \)
06

- Translate statement (d)

Statement: 'The conjunction of two tautologies is a tautology'. It can be written as: \( \forall z_1 \forall z_2 (T(z_1) \land T(z_2) \rightarrow T(z_1 \land z_2)) \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Predicates
In logic, a predicate is a statement or expression that can be true or false depending on the values of its variables. For example, in the exercise, we use the predicate P(x) to denote 'x is a proposition' and T(x) to denote 'x is a tautology'. Predicates are fundamental because they help us express logical statements systematically.

Instead of dealing with vague expressions, we can translate sentences into precise logical forms using predicates. This clarifies our understanding and helps in logical calculations.
Quantifiers
Quantifiers are symbols used in logic to specify the quantity of specimens in the domain of discourse that satisfy an open formula or predicate. There are two main types of quantifiers:
  • Universal quantifier (鈭�): Indicates that a predicate is true for all elements in a specific mathematical domain. For example, 鈭�x P(x) means 'P(x) is true for every x'.
  • Existential quantifier (鈭�): Indicates that there is at least one element in the domain that makes the predicate true. For instance, 鈭儀 P(x) translates to 'there exists at least one x such that P(x) is true'.
In the exercise, we use these quantifiers to precisely capture the meaning of statements such as 'Some propositions are tautologies'.
Tautologies
A tautology is a proposition or logical formula that is true in every possible interpretation. For instance, the statement 'p 鈭� 卢p' (either p is true or not p is true) is always true regardless of the truth value of p. Tautologies are significant in logic and mathematics because they represent statements that are universally valid.

In the given exercise, one of the statements is 'The conjunction of two tautologies is a tautology'. This can be formalized as 鈭�z鈧� 鈭�z鈧� (T(z鈧�) 鈭� T(z鈧�) 鈫� T(z鈧� 鈭� z鈧�)), meaning if both z鈧� and z鈧� are tautologies, their conjunction is also a tautology.
Contradictions
A contradiction is a logical statement that is always false, regardless of the truth values of its components. An example of a contradiction is the statement 'p 鈭� 卢p' (p and not p), which can never be true. Contradictions are essential in logic because they indicate impossible scenarios.

In the context of the exercise, the statement 'The negation of a contradiction is a tautology' is covered. It is expressed using logical notation as 鈭�x (卢C(x) 鈫� T(卢C(x))), where C(x) represents 'x is a contradiction'. This tells us that negating a contradiction will always result in a tautology.
Contingencies
A contingency is a logical statement that can be either true or false depending on the truth values of its variables. Unlike tautologies and contradictions, contingencies do not have a fixed truth value in all interpretations. An example of a contingency is the statement 'p 鈭� q', which can be true or false based on the truth values of p and q.

In the exercise, we deal with the statement 'The disjunction of two contingencies can be a tautology'. This translates to 鈭儁鈧� 鈭儁鈧� (C(y鈧�) 鈭� C(y鈧�) 鈭� T(y鈧� 鈭� y鈧�)), meaning there exist some contingencies y鈧� and y鈧� such that their disjunction is a tautology.

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Most popular questions from this chapter

Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase "It is not the case that.") a) No one has lost more than one thousand dollars playing the lottery. b) There is a student in this class who has chatted with exactly one other student. c) No student in this class has sent e-mail to exactly two other students in this class. d) Some student has solved every exercise in this book. e) No student has solved at least one exercise in every section of this book.

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