2.3 Logical Implication: Rules of Inference

• From the notion of a valid argument, we begin a formal study of what we shall mean by an argument and when such an argument is valid.

• Consider the implication

(p1p2p3 … pn)→q.

The statements p1, p2, p3, … , pn are called the premises( 前提） of the argument, and the statement q is the conclusion for the argument.

推論證明

Example 2.19: page 69.

Example 2.20: page 70. • (we know that (p1p2)→q is a valid argument, and we may say

that the truth of the conclusion q is deduced or inferred from the truth of the premises p1, p2.)

Definition 2.4:

• If p, q are arbitrary statements such that p→q is a tautology, then we say that p logically implies q and we write pq to denote this situation. Note:

• 1) if p q, we have p q and q p. • 2) if p q and q p, then we have p q. • 3) p ≠> q is used to indicate that p→q is not a taut

ology – so the given implication (namely, p→q) is not a logical implication. (See the paragraphs from the bottom of page 70 to the top of page 71.)

Example 2.21: page 71.

• To prove the correctness of a statement, a great deal of the effort must put into constructing the truth tables. And since we want to avoid even larger tables, we are persuaded to develop a list of techniques called rules of inference that help us. (See the top paragraph of page 72.)

Example 2.22: page 72. (the Rule of Detachment)

Example 2.23: page 73. (the Law of the Syllogism)

Example 2.24: page 74.

Example 2.25: page 75. (Modus Tollens: method of denying)

Example 2.26: page 76~77. (the Rule of Conjunction)

Example 2.27: page 77

Example 2.28: page 77~78. (the Rule of Contradiction)

Table 2.19: page 79. (Rules of Inference)

Example 2.29: page 80

Example 2.30

Example 2.31: page 81~82.

Example 2.32:page 82 反證法

(p(q r)) ((p^q) r)

Note:

• [(p1p2p3 … pn) → (q→r)] [(p1p2p3 … pnq)→r]. (page 83)

Example 2.33: page 84.

2.4 The Use of Quantifiers

Open statement• Sentences that involve a variable, such as x, need not be

statements. • For example, the sentence "The number x + 2 is an even

integer" is not necessarily true or false unless we know what value is substituted for x.

• If we restrict our choices to integers, then when x is replaced by ‑5, ‑1, or 3, for instance, the resulting statement is false.

• In fact, it is false whenever x is replaced by an odd integer. • When an even integer is substituted for x, however, the

resulting statement is true.• We refer to the sentence "The number x + 2 is an even

integer" as an open statement, which we formally define as follows.

Definition 2.5:

• A declarative sentence is an open statement if – it contains one or more variables, and– it is not a statement, but– it becomes a statement when the variables in it are

replaced by certain allowable choices.

Example:

• “The number x + 2 is an even integer” is an open statement and is denoted by p(x). The allowable choices for x is called the universe (set) for p(x). If x = 3, p(3) is a false statement.

Example:

• “q(x,y): The numbers y + 2, x – y, and x + 2y are even integers.”, then q(4,2) is true.

• From the above examples, we can say for some x, p(x) (TRUE), for some x, y, q(x,y) (TRUE), or for all x, p(x) (FALSE).

Quantifiers

• Two types of quantifiers, which are called the existential and the universal quantifiers, can quantify the open statements p(x) and q(x,y).

• the existential quantifier (means “for some x”, “for at least one x”, or “there exists an x such that”): “for some x, p(x)” is denoted as “ x, p(x)”.

• the universal quantifier (means “for all x”, “for any x”, “for each x”, or “for every x”): “for all x, all y” is denoted by “x y”.

Example 2.36: page 91.

• Note: x p(x) x p(x), but x p(x) does not logically imply x p(x).

Example 2.38: page 93

• (the truth value of a quantified statement may depend on the universe prescribed).

Example 2.39: page 94.

Table 2.21 summarize and extend some results for quantifiers.

Definition 2.6:

• Let p(x), q(x) be open statements defined for a given universe. The open statements p(x) and q(x) are called (logically) equivalent, and we write x [p(x) q(x)] when the biconditional p(a) q(a) is true for each replacement a from the universe (that is, p(a) q(a) for each a in the universe).

• If the implication p(a) q(a) is true for each a in the universe (that is, p(a) q(a) for each a in the universe), then we write x [p(x) q(x)] and say that p(x) logically implies q(x).

Definition 2.7:

• For open statements p(x), q(x) – defined for a prescribed universe – and the universally quantified statement x [p(x) q(x)] we define:– The contrapositive of x [p(x) q(x)] to be x [q(x)

p(x)]. – The converse of x [p(x) q(x)] to be x [q(x) p

(x)]. – The inverse of x [p(x) q(x)] to be x [p(x) q

(x)].

Example 2.40: page 95~96.

Example 2.41: page 96~97.

Example 2.42: page 97

• (the existential quantifier x does not distribute over the logical connective ).

Table 2.22:

• Logical equivalences and logical implications for quantified statements in one variable.

Example 2.43: page 98.

Table 2.23: Rules for negating statements with one quantifier.

Example 2.45: page 100.

Example 2.46: page 101.

Example 2.47: page 101.

Example 2.48: page 101

2.5 Quantifiers, Definitions, and the Proofs of Theorems

• In this section we shall combine some of the ideas we have already studied in the prior two sections.

• The Rule of Universal Specification: If an open statement becomes true for all replacements by the members in a given universe, then that open statement is true for each specific individual member in that universe.

• (A bit more symbolically – if p(x) is an open statement for a given universe, and if x p(x) is true, then p(a) is true for each a in the universe.)

Example 2.53: page 111.

The Rule of Universal Generalization:

• If an open statement p(x) is proved to be true when x is replaced by any arbitrarily chosen element c from our universe, then the universally quantified statement x p(x) is true.

• Furthermore, the rule extends beyond a single variable. So if, for example, we have an open statement q(x,y) that is proved to be true when x and y are replaced by arbitrarily chosen elements from the same universe, or their own respective universes, then the universally quantified statement x y q(x, y) [or, x, y q(x, y)] is true. Similar results hold for the cases of three or more variables.

Example 2.54: page 115.

Example 2.55: page 116.

Example 2.56: page 116~117.

• The results of Example 2.54 and especially Example 2.56 lead us to believe that we can use universally quantified statements and the rules of inference – including the Rules of Universally Specification and Universal Generalization – to formalize and prove a variety of arguments and, hopefully, theorems.

Example:

Definition 2.8,

Example 2.57 

Theorem 2.3

Theorem 2.4

Theorem 2.5