Hedman. A First Course in Logic, 2004 (Oxford)

• |= F means every assignment models F . That is, F is a tautology.

So although |= has multiple interpretations, in context it is not ambiguous. The notion of consequence is closely related to the notion of “implies” dis-

cussed in Section 1.1. A formula G is a consequence of a formula F if and only if “F implies G” is always true. We restate this as the following proposition.

Proposition 1.19 For any formulas F and G, G is a consequence of F if and only if F → G is a tautology.

Proof We show that F → G is not a tautology if and only if G is not a consequence of F .

By the definition of “tautology,” F → G is not a tautology if and only if there exists an assignment A such that A |= ¬(F → G).

By the definition of “→,” A |= ¬(F → G) if and only if A |= ¬(¬F G). By the semantics of propositional logic, A |= ¬(¬F G) if and only if both

A |= F and A |= ¬G.

Finally, by the definition of “consequence,” there exists an assignment A such that A |= F and A |= ¬G if and only if G is not a consequence of F .

Suppose we want to determine whether or not formula G is a consequence of a formula F . We refer to this as the consequence problem. By Proposition 1.19 this can be rephrased as a validity problem (since G is a consequence of F if and only if G → F is valid). Such problems can be resolved by computing a truth table. If the truth values for F → G are all 1s, then G is a consequence of F . Otherwise, it is not. In particular, if F is a contradiction, then G is a consequence of F regardless of G.

Example 1.20 Let F and G be formulas. Each of the following can easily be verified by computing a truth table.

(F G) |= F

F |= (F G)

(F ¬F ) |= G

Definition 1.21 If both G is a consequence of F and F is a consequence of G, then we say F and G are equivalent. We denote this by F ≡ G.

It follows from Proposition 1.19 that two formulas F and G are equivalent if and only if F ↔ G is a tautology. So we can determine whether two formulas F and G are equivalent by computing a truth table. Each of the equivalences in the following examples can easily be verified in this manner.

Example 1.22 For all formulas F and G, (F G) ≡ (G F ) and (F G) ≡ (G F ).

Example 1.25 (Distributivity rules) The following two equivalences exhibit the distributivity rules for and . For all formulas F , G, and H,

(F (G H)) ≡ ((F G) (F H)) and

(F (G H)) ≡ ((F G) (F H)).

Example 1.26 (DeMorgan’s rules) For all formulas F and G

¬(F G) ≡ (¬F ¬G), and

¬(F G) ≡ (¬F ¬G).

The equivalences in the previous examples are basic. Note that we refer to some of these equivalences as “rules.” Each of these holds true for arbitrary formulas. From these basic equivalences, more elaborate equivalences can be created.

Example 1.27 Using the equivalences in the previous examples, we show that ((C D) A) ((C D) B) (E ¬E) ≡ (A B) (C D). Let L denote the formula on the left in this equivalence. Note that (E ¬E) is a tautology. By Example 1.23, L is equivalent to ((C D) A) ((C D) B). According to the second distributivity rule in Example 1.25, this is equivalent to (C D) (A B) (viewing (C D) as the formula F in that rule).

By Example 1.22, this is equivalent to (A B) (C D) which is the formula on the right in our equivalence.

Using the basic rules in Examples 1.22–1.26, we were able to verify that ((C D) A) ((C D) B) (E ¬E) ≡ (A B) (C D). This is itself a rule, holding for any formulas A, B, C, D, and E. Alternatively, we could have verified this equivalence by computing a truth table. Such a truth table would have had 25 = 32 rows. The previously established rules provided a more e cient method of verification.

Likewise, we could state “rules for consequence” that would allow us to show that one formula is a consequence of another without having to compute truth tables. In the next section, we exploit this idea and introduce the notion of formal proof. Formal proofs allow us to “derive” formulas from sets of formulas. The following definition extends the notion of consequence to this setting.

Table 1.6 More rules for derivations

There are a lot of rules. Note the organization of the list. It begins with a couple of rules that are quite intuitive: Assumption and Monotonicity. There follow a few similarly named rules for various symbols of propositional logic. The four rules that conclude the list reflect conventions (C1) and (C2). In addition to the rules in Table 1.5, we have the rules in Table 1.6 regarding the definitions of , →, and ↔.

Our list of rules is both too big and too small. It is too big in the sense that some of these rules are redundant. For example, since can be expressed in terms of ¬ and , -Symmetry follows from -Symmetry (see Exercise 1.13). We could pare these redundant rules from our list. In fact, we really need only

more succinct by using additional rules. For example, note that in the previous proof we repeatedly introduced the symbol → only to eliminate it. Instead, we could have separately proved the following rule.

Premise: F (¬F G), F F Conclusion: F G

The proof in Example 1.31 essentially repeats the above argument three times. We can treat this four-line proof as a subroutine. Had we established this rule prior to Example 1.31, we could have referred to it three times to make the proof more concise. For lack of a better name, we christen this rule -Modus Ponens. “Modus ponens” is a standard name for the rule we call →-Elimination. As the archaic name suggests, this rule has been around for a while. It is found in what can be considered the origin of formal proofs: Euclid’s Elements. The next five examples establish some other rules that facilitate the construction of proofs.

Example 1.32 (Tautology rule) This rule states that, for any formula G, (¬G G) can be derived from any set of formulas.

Premise: None Conclusion: F (¬G G)

2.F (G → G) →-Introduction applied to 1

3.F (¬G G) →-Definition applied to 2

Example 1.33 (Contradiction rule) This rule states that any formula G can be derived from the contradiction F ¬F .

Premise: F (F ¬F )

Conclusion: F G

Example 1.34 (Contrapositive) This is the rule of logic that states that if p implies q, then ¬q implies ¬p.

Premise: F {F } G Conclusion: F {¬G} ¬F

Example 1.35 (Proof by cases) This rule provides a useful way to structure a proof.

Premise: F {F } G, F {¬F } G

Conclusion: F G

Example 1.36 (Proof by contradiction) Another way to structure a proof is by contradiction. As the following proof indicates, proof by contradiction is (in this context) essentially the contrapositive of proof by cases.

Premise: F {F } G, F {F } ¬G

Conclusion: F ¬F

These are now established rules that may henceforth be used to justify statements in proofs. By “established” we mean that these rules are true, provided that the rules in Table 1.5 are true. We must prove that this is the case. We must prove that our proof system is sound: that if G can be derived from F , then G is in fact a consequence of F .

Theorem 1.37 (Soundness) If F G, then F |= G.

Proof If F G, then there is a formal proof concluding with F G. Each line of the proof contains a statement of the form X Y which is justified by one of the rules in Tables 1.5 or 1.6. We want to show that for each line of the proof, if X Y , then X |= Y . This can be accomplished by verifying each rule in the tables one-by-one. We demonstrate this by verifying three rules: Assumption,-Elimination, and →-Introduction. The verification of the remaining rules are left as an exercise.

We begin with the first rule of Table 1.5: Assumption. The conclusion of this rule is that F G. We must show, under the premise of this rule, that F |= G. But this is clear since the premise states that G is in F (if A models F then A must model G).

Refer next to -Elimination. This rule states that if F (F G), then F F . We must show that if F |= (F G), then F |= F . That is, we must show that F is a consequence of (F G). This is verified by the following truth table.