is not valid. In the following table, the premises are both true in the second row while the conclusion is false, and that is enough to establish invalidity:

2.3 Normal Forms

In this section we introduce disjunctive and conjunctive normal forms for formulas. Each formula of propositional logic has an equivalent formula in disjunctive normal form, and an equivalent formula in conjunctive normal form. The normal forms are used to standardize (normalize) the forms of logical formulas for various reasons, such as allowing the use of a computational proof technique known as resolution (Robinson 1965). We will use normal forms to prove functional completeness in Section 2.5. More importantly, the normal forms will allow us to make some semantic connections among classical logic, three-valued logics, and fuzzy logics.

We’ll begin with disjunctive normal form. First we define literals to include all atomic formulas and their negations: A, ¬A, B, ¬B, . . . . Next we define what a phrase is:

1.A literal is a phrase

2.If P and Q are phrases, so is (P Q)

A phrase is either a single literal, or a conjunction of literals: A, ¬A, B, ¬B, A A, A ¬B, (D ¬E) F, and so forth. Finally, we define disjunctive normal form:

1.Every phrase is in disjunctive normal form

2.If P and Q are in disjunctive normal form, so is (P Q)

So a formula is in disjunctive normal form if it contains at most the three connectives ¬, , and , such that negations only appear in front of atomic formulas and no subformula that is a conjunct contains a disjunction. Examples are A, A ¬B,

A ¬B, (A ¬B) B, (A ¬B) B, and (C ¬E) (D (E F)).

We claimed that each formula of classical propositional logic is equivalent to a formula (at least one) that is in disjunctive normal form. To prove this, we show how to transform any formula of classical propositional logic into a formula in disjunctive normal form, where each step of the transformation produces an equivalent formula. The transformation uses the following equivalences:

(Proof that these forms are equivalent is left as an exercise.) We’ll explain the transformation process using the formula

¬(¬(P → Q) (¬R → (S ↔ T)))

First we use the Implication equivalences to eliminate all conditionals and biconditionals, producing the formula

¬(¬(¬P Q) (¬¬R ((¬S T) (¬T S))))

Next we use DeMorgan’s Laws to move negations deeper into the formula until all negations appear in front of atomic formulas. In our example we use the second DeMorgan Law first to obtain

¬¬(¬P Q) ¬(¬¬R ((¬S T) (¬T S)))

and then

¬(¬¬P ¬Q) (¬¬¬R ¬((¬S T) (¬T S))

Next we can use the first DeMorgan Law twice to obtain

(¬¬¬P ¬¬Q) (¬¬¬R (¬(¬S T) ¬(¬T S))) and then the second Law twice more to obtain

(¬¬¬P ¬¬Q) (¬¬¬R ((¬¬S ¬T) (¬¬T ¬S)))

Now Double Negation eliminates all double negations:

(¬P Q) (¬R ((S ¬T) (T ¬S))).

We’re almost done, but note that this is a conjunction, and some of the conjuncts are disjunctions, so the formula is not yet in disjunctive normal form (if all of the conjuncts were themselves conjunctions, it would be in disjunctive normal form). We can use the first Distribution equivalence to convert the overall formula into a disjunction, with ¬P in place of P, Q in place of Q, and (¬R ((S ¬T) (T ¬S))) in place of R:

(¬P (¬R ((S ¬T) (T ¬S)))) (Q (¬R ((S ¬T) (T ¬S)))).

4These two laws are named after the nineteenth-century mathematician Augustus DeMorgan, who stated these laws set-theoretically. According to one historian, these laws did not actually originate with DeMorgan but “were known in scholastic times and probably even in antiquity” (Delong 1970, p. 106).

Next we can apply the second Distribution equivalence to both occurrences of the subformula (¬R ((S ¬T) (T ¬S)), with ¬R in place of P, (S ¬T) in place of Q, and (T ¬S) in place of R, to obtain

(¬P ((¬R (S ¬T)) (¬R (T ¬S)))) (Q ((¬R (S ¬T)) (¬R (T ¬S))))

This formula is in disjunctive normal form, and it is equivalent to the original formula

¬(¬(P → Q) (¬R → (S ↔ T)))

because each step of the transformation produces an equivalent formula.

This series of steps will convert any formula to an equivalent formula in disjunctive normal form—we first get rid of conditional and biconditional connectives, then we use the DeMorgan Laws to move negations in as far as they will go, and we use Double Negation to eliminate all double negations. At this point all negations occur in front of atomic formulas, and the binary connectives are either conjunctions or disjunctions. The Distribution laws enable us to convert the result to disjunctive normal form by replacing each conjunction with a disjunctive conjunct into a disjunction of conjunctive disjuncts. In Section 2.5 we will see another way to produce formulas in disjunctive normal form.

In contrast, formulas in conjunctive normal form are conjunctions of disjunctions, rather than disjunctions of conjunctions. We define a clause as:

1.A literal is a clause

2.If P and Q are clauses, so is (P Q)

and conjunctive normal form as:

1.Every clause is in conjunctive normal form

2.If P and Q are in conjunctive normal form, so is (P Q)

Any formula of classical propositional logic can be converted to an equivalent formula in conjunctive normal form by following the same steps as we did for disjunctive normal form, but using the following Distribution equivalences at the end:

Let us call a pair of literals complementary if one is the negation of the other; for instance, S and ¬S are a complementary pair of literals. We now state two important general results:

Result 2.1: A clause C of classical propositional logic is a tautology if and only if C contains a complementary pair of literals.

Proof: If a clause C is a tautology then C must be a disjunction (since no literal is a tautology), and on each truth-value assignment at least one disjunct is true. But then C must contain a complementary pair of literals, because

German logician Gottlob Frege (Frege 1879). We have chosen the Frege-Lukasiewicz system—which we will designate as CLA (for classical propositional logic axiomatic system)—because some simple modifications will serve to axiomatize three-valued and fuzzy logics that were also developed by Lukasiewicz. CLA contains three axiom schemata:

CL1. P→ (Q → P)

CL2. (P → (Q → R)) → ((P → Q) → (P→ R))

CL3. (¬P → ¬Q) → (Q → P)

and the single inference rule MP, which is short for the rule’s traditional name, Modus Ponens:

MP (Modus Ponens). From P and P→ Q, infer Q.

An axiom schema stands for infinitely many axioms, namely, all formulas that have the overall form exemplified by the schema. We call such formulas instances of the axiom schema. We can define an instance of an axiom schema to be any formula that results from uniform substitution of formulas of the language (not necessarily distinct) for each of the letters P, Q, and R. By uniform substitution we mean that in a given instance, the same formula must be substituted for every occurrence of P, and similarly for Q and R. So, for example, the following formulas are all instances of the axiom schema CL1:

Note that each of the axiom schemata has the form of a tautology. For example, here is the truth-table for CL2:

It is left as an exercise to verify that the other two axiom schemata also have tautologous forms. In addition, the rule Modus Ponens is a truth-preserving rule—a rule that when applied to true formulas results in a true formula: if P and P → Q are both true, then Q must be true as well.

A derivation is a sequence of formulas each of which is designated as an assumption, or is an instance of an axiom schema, or can be derived from earlier formulas in the sequence using the derivation rule MP. In addition, the formulas designated as assumptions must begin the derivation. (This last stipulation is not theoretically necessary but will make rules in later chapters easy to state, with no loss of derivational power.) Here is an example of a derivation annotated in a standard way:

We have derived the conclusion M from the assumptions A and (B → A) → (A → M). We have numbered each of the formulas in the sequence constituting the derivation and have added annotations. Each line of the displayed derivation is annotated to indicate the justification for entering that formula in the derivation: either as an assumption (lines 1 and 2), as an instance of an axiom schema (line 3; the annotation indicates that axiom schema CL1 has been used, with A substituted for P and B for Q), or by virtue of being derived from earlier formulas (lines 4–6; each annotation indicates to which formulas the rule MP was applied). When a formula appears in a derivation that begins with a set of assumptions, we say that the formula is derived from those assumptions. So each of the formulas on lines 1–6 is derived from the assumptions on lines 1 and 2. If a formula can be derived from a set of assumptions, we also say that the formula is derivable from any set containing those assumptions. So the preceding derivation shows that the formula M is derivable from any set that contains both A and (B → A) → (A → M). As another example, the following derivation shows that A → B is derivable from ¬A:

Some formulas, namely, the tautologies of classical logic, can be derived without using any assumptions. For example, A → A is a tautology and we can derive it without any assumptions as follows:

A derivation that does not contain any assumptions is called a proof, and a formula is called a theorem if there is a proof ending with that formula. The proof just given establishes that A → A is a theorem. We may call the proof a proof of that theorem.

Another theorem is ¬A → (A → B). Before presenting a proof, we draw the reader’s attention to the relation between this theorem and the fact that (as we have shown) A → B is derivable from ¬A, namely: the consequent and the antecedent of the theorem are the two formulas in question. It turns out that whenever a formula Q is derivable from a formula P in CLA there is a corresponding theorem P → Q in CLA, and vice versa. This general fact is known as the Deduction Theorem:6

Result 2.5 (The Deduction Theorem for Classical Propositional Logic): Q is derivable from P in CLA if and only if P → Q is a theorem in CLA.

Proof: The if part—if P → Q is a theorem then Q is derivable from P—is easy to establish. If P → Q is a theorem then there is a proof of P → Q in CLA. We can add P to this proof as an assumption and then use Modus Ponens to derive Q from P and P → Q.

For the only if part we will explain how to convert any derivation of Q from the assumption P into a proof of P → Q, thus establishing the theoremhood of the latter. (We’ll then illustrate the method by converting our derivation of A → B from ¬A into a proof of ¬A → (A → B).) The strategy is to show how, given a derivation of Q from the assumption P—a derivation consisting of the sequence P, R1, R2, . . . , Rn–1, Rn where Rn is Q—we can produce a derivation in which each of the formulas P → P, P → R1, P → R2, . . . , P → Rn–1, P → Rn occurs as a theorem. The last theorem in this sequence is the desired theorem for this result. The formulas will all be theorems because the new derivation will not include the assumption P or any other assumption.

We begin the derivation by deriving the first new formula P → P exactly as we derived A → A on the previous page, using P in place of A. Now, each of R1, R2, . . . , Rn–1, Rn either was an instance of an axiom schema or followed from previous formulas in the derivation by MP. For each one of these formulas Ri, if Ri is an instance of an axiom schema we can add the following lines to derive P → Ri in the new derivation:

If Ri followed by MP from earlier formulas Rk and Rk → Ri in the sequence R1, R2, . . . , Ri–1, Rn, then we already have P → Rk and P → (Rk → Ri) in the new derivation, say, on lines m and n, and three additional lines will derive P → Ri:

6There is also a semantic version of the Deduction Theorem: the set {P} entails Q if and only if P → Q is a tautology. This is easy to prove directly, but we also note that the semantic version follows from Result 2.5 and the fact that CLA is sound and complete for classical propositional logic (see p. 26).

mP → Rk

. . .

nP → (Rk → Ri)

. . .

Using the method described in the proof of the Deduction Theorem and the earlier derivation

weconstruct the following derivation establishing the theoremhood of ¬A → (A → B) (to the left of the lines containing the conditionals whose consequents are formulas from the earlier derivation we write the line numbers from that derivation):

Shorter derivations are certainly possible—for example, in this case we did not need to derive the formula on line 11 since it already appears on line 6; our point here was to illustrate the mechanical conversion procedure introduced in the proof of the Deduction Theorem, a procedure that shows that we can always convert a derivation of Q from P into a proof of P → Q.

A derivation system is said to be sound for classical propositional logic if (a) all theorems are tautologies and (b) whenever a formula P is derivable from a set of formulas, then P is also entailed by the set . The system CLA is sound. For example, we have already noted that the theorem A → A is a tautology of classical logic, and so is the theorem ¬A → (A → B). The arguments corresponding to our other two derivations:

A

(B → A) → (A → M) M

and

A

¬A → ¬B

are both valid (as can be confirmed by truth-tables). CLA is a sound derivation system precisely because the axioms are tautologies and the single rule is truthpreserving.

A derivation system is said to be weakly complete for classical propositional logic if every tautology of classical logic is a theorem in the system (the converse of what goes into soundness) and it is said to be strongly complete, or just complete, if in addition whenever a set of formulas entails a formula P, P is also derivable from within the system (again, the converse of soundness). CLA is complete as well as sound for classical propositional logic, and when a system has both of these properties we say that it is adequate for classical propositional logic.7 Simply put, soundness and completeness mean that you can derive everything you should be able to derive (the system is complete), and nothing you shouldn’t (the system is sound).

We point out that there is prima facie (“at first sight”) something fishy about the claim that CLA is complete for classical propositional logic: completeness means that for every entailment or tautology there is a corresponding derivation in the system, but it is unclear that there is any derivation for entailments or tautologies whose validity depends on connectives other than ¬ and →, because no other connectives appear in either the axioms or the derivation rule MP. As an example, M (M → S) is a tautology, but how can it be derived in CLA? The answer is that when we want to use CLA to produce derivations for formulas containing the connectives , , and ↔,

7See Hunter (1971) for several completeness proofs for the system CLA (called PS in Hunter), along with a soundness proof.

At this point it is probably (frighteningly) clear that it can be difficult to design derivations in CLA – and to follow them; that is why we added the previous caution. Why, you may ask, would anyone propose such a terse system, with merely three axioms and a single rule to choose from? Wouldn’t it be better to have more axioms and rules? The reason for terse axiomatic systems is that authors of these systems strive for elegance, usually in the form of a very small set of axioms and rules, which in turn can make it easier to prove things about the system.8

But when we put axiomatic systems to action, terseness ceases to look like a virtue. In order to make derivations less difficult to construct or to follow, it is common to introduce derived axiom schemata and/or derived rules. For example, we can introduce P → P as a derived axiom schema:

CLD1. P → P

The D is short for derived. We are justified in introducing P → P as a derived axiom schema because we have already presented a proof of A → A, and it is clear that this proof can be converted into a proof of any instance of the schema P → P simply by substituting any other formula for A. In general, once we have proved a formula to be a theorem we may present the formula, with metavariables uniformly substituted for the formula letters, as a derived axiom schema. We can then use the derived schema in subsequent derivations, as we do in the following derivation of

B → (C → C):

Using CLD1 we can also construct a short simple derivation showing that A ¬A is a theorem. First, we rewrite the formula as ¬A → ¬A so that it contains only the negation and conditional symbols. Here’s the derivation!

Now let’s return to the derivation of the theorem ¬M → (M → S). The sole purpose of lines 3–11 of the derivation is to derive ¬M → (M → S) from ¬M → (¬S → ¬M) and (¬S → ¬M) → (M → S). Examining these formulas we see that they are instances of a general inference pattern that derives a formula of the form P → R from formulas having the form P → Q and Q → R. We may introduce this inference pattern, commonly called hypothetical syllogism, as a derived rule:

HS (Hypothetical Syllogism). From P → Q and Q → R, infer P → R.

8Claims about a derivation system are part of what is called the metatheory of logic. The claim that an axiomatic system is sound, or that it is complete, is a metatheoretic claim.

To justify HS we first introduce the derived axiom

CLD2. (Q → R) → ((P → Q) → (P → R))

This axiom is justified as follows (compare with lines 3–9 of the previous derivation of M (M → S):

and the following derivation justifies HS (compare lines 3–5 with lines 9–11 of the longer dervivation):

This derivation shows that if we already have P → Q and Q → R, whether or not they are assumptions, we can derive P → R. Using HS, we can seriously shorten the previous derivation of ¬M → (M → S):

Another useful derived rule is

TRAN (Transposition). From P → (Q → R) infer Q → (P → R).

The rule is called Transposition because it transposes antecedents. The following derivation, which uses both CLD2 and HS, justifies TRAN:

As a final example, with the help of our derived axioms and rules we can show that the argument

If the economy is sound, then either the unemployment rate is low or spending is high.

If the unemployment rate is low, most people are well off. If spending is high, most people are well off.

It’s not true that most people are well off. The economy isn’t sound.

is valid. First we symbolize the argument:

E → (U S) U → W

S → W

¬W

¬E