and we rewrite the first premise as E → (¬U → S). Here is a derivation:

We stress that these derived axiom schemata and rules are a convenience for constructing derivations; the set of axiom schemata CL1–CL3 alone with the single rule MP form a complete derivation system for classical propositional logic, and so the additional axioms and rules do not add to the power of the system. Nor do they affect its soundness, since they are all derivable within a system that was sound to begin with.

2.5 Functional Completeness

In Section 2.2 we pointed out that we could take one of several pairs of connectives as primitive and define the rest in terms of these. In Section 2.4 we took advantage of this fact: the axiomatic system CLA uses only two connectives, since formulas containing the other connectives can be rewritten using the two connectives ¬ and → that appear in CLA.

There is an important related theoretical issue to which we now turn. First, we formally define the concept of a truth-function: a truth-function is a function that maps truth-values to truth-values. More specifically, a truth-function is always a function of a given finite number of arguments: one, two, three, whatever; it is a function that maps each sequence of the appropriate number of truth-values to a truth-value. The negation truth-function is a truth-function of one argument as specified by the following truth-table template:

T F

F T

This function maps the single truth-value T (more precisely, the single-membered sequence <T>) tothe truth-value F,and it maps the single truth-value F (<F>) to the truth-value T. The conditional truth-function is a truth-function of two arguments:

It maps the sequence <T, F> to the truth-value F, and all other sequences of two truth-values to the truth-value T.

We say that a formula P of propositional logic expresses a truth-function of n arguments if the truth-table for P specifies that truth-function; that is, the values under P’s main connective are the values to which the function maps each sequence of n truth-values listed to the left of the vertical line. So, for example, and by design, ¬P expresses the negation truth-function, and P → Q expresses the conditional truth-function (other formula letters may be used). Other truthfunctions may require more complicated formulas. For example, the neither-nor truth-function, captured in the following truth-table template,

T T F

T F F

F T F

F F T

is expressed by the formula ¬(P Q) or equivalently by ¬P ¬Q.

Here is the theoretical issue: can every classical truth-function be expressed by a formula of classical propositional logic using only the five connectives ¬, ,, →, and/or ↔? The answer is yes. We shall show how, given any truth-function, to construct a formula that expresses exactly that truth-function. To facilitate the proof we shall assume that the function in question has been laid out in a truthtable template as previously, and that n is the number of arguments that the truthfunction operates on.

First, we choose n atomic formulas P1, . . . , Pn, one corresponding to each argument place. These will head the columns to the left of the vertical line in the truthtable template. Next, for each row of the truth-function template we form a corresponding conjunction conjoining the atomic formulas that have the value T in that row along with the negations of the atomic formulas that have the value F—in the terminology of Section 2.3, each such conjunction is a phrase. So, for example, phrases corresponding to the four rows of the neither-nor function template are, respectively, P Q, P ¬Q, ¬P Q, and ¬P ¬Q. Note that each of these phrases is true exactly when P and Q have the truth-values in its corresponding row. Next we form a disjunction of the phrases corresponding to the rows that have T to the right of the vertical line, thus producing a formula in disjunctive normal form. In the case of the neither-nor function there is one such row, the fourth, so we form the

“disjunction” of the single phrase for that row: ¬P ¬Q. (A “disjunction” of a single formula is simply the formula.) This formula expresses the function captured in the neither-nor truth-table template.

As a more complicated example consider the function of three arguments:

Assuming that we have chosen the atomic formulas P, Q, and R, a disjunction of phrases corresponding to the rows with T to the right of the vertical line is

((((P Q) R) ((P Q) ¬R)) ((P ¬Q) R)) ((¬P ¬Q) R). This formula expresses the truth-function specified in the truth-table template, as the reader may easily confirm.

Wemust add two special cases. In one we have a truth-function of one argument, such as,

T T

F T

In this case the phrase corresponding to a row is a single atomic formula or its negation: P for the first row and ¬P for the second. Since both rows contain T to the right of the vertical line, the disjunction P ¬P of these two phrases expresses the truth-function. In the other case there are no Ts to the right of the vertical line. In this case we may simply conjoin the phrase corresponding to the first (or any other) row with its negation. So for the truth-function

T F

F F

we have the formula P ¬P, and for the truth-function

T T F

T F F

F T F

F F F

we have the formula (P Q) ¬(P Q).

Note that there are other formulas—in fact infinitely many other formulas— that express these same truth-functions, so it is important to keep in mind that we are only showing that, given any truth-function, there is at least one formula using the five connectives that expresses it. Now, we’ve claimed that our procedure will

always work—but how do we know this? It’s rather simple. Each phrase corresponding to a row of a truth-function template is true on the truth-value assignments represented by that row that is false on all other truth-value assignments. Thus a disjunction of the phrases corresponding to rows that have a T to the right of the vertical line will be true on the truth-value assignments represented by those rows and false on all other truth-value assignments. In the case where there are no Ts to the right of the vertical line we produce a contradictory formula of the general form P ¬P, which is always false. Conclusion: we’ve specified a way to construct, for any classical truth-function, a formula that exactly expresses that truth-function.

When a set of connectives is sufficient to express every truth-function, we say that the set of connectives is functionally complete. Thus, we have proved

Result 2.6: The set of connectives {¬, , } is functionally complete,

since these are the only connectives we have used in formulas to express any truthfunction. From this it follows that the full set {¬, , , →, ↔} is also functionally complete—since we are not required to use all of the connectives in the candidate formulas expressing the various truth-functions. But since we know that there are three subsets consisting of only two of our connectives that are sufficient to define the others, we may also conclude that those three subsets, {¬, }, {¬, }, and {¬, →}, are truth-functionally complete.9

2.6 Decidability

Classical propositional logic has a desirable property that isn’t shared by all logical systems: its set of tautologies is decidable. A set of formulas is decidable if there is a decision procedure for membership in the set, that is, a mechanical procedure that will, given any formula, correctly decide after a finite number of steps whether that formula is a member of .10

The set of tautologies of classical logic is decidable because there exist mechanical procedures for testing whether a formula is a tautology. We’ve already seen one such procedure: given any formula we can construct a truth-table for that formula and examine the column of truth-values under the formula’s main connective. If that column consists solely of Ts then the formula is a tautology; otherwise it is not. Clearly truth-tables can be constructed mechanically, and just as clearly the construction and examination of the relevant column of truth-values take only a finite number of steps. Similarly, the set of contradictions of classical propositional logic

9These are the only sets consisting of two of the five connectives that are truth-functionally complete. First, we note that we need the negation connective, for without it we can never produce a formula that is false when all of its atomic components are true. Second, we note that negation and the biconditional won’t suffice because, for example, every formula constructed from two atomic formulas using only these two connectives will have an even number of Ts and an even number of Fs in its truth-table (proof is left as an exercise).

10Sets that contain things other than formulas can also be said to be decidable, but that is not our interest here.