2Review of Classical Propositional Logic

and to indicate grouping among connectives of the same priority. So, for example,

¬ (J C) symbolizes It’s not true that John or Christy is a mathematician (i.e., Neither John nor Christy is a mathematician), and the parentheses are necessary in (J C) P to indicate that the formula means: Either John and Christy are both mathematicians or Christy is a philosopher rather than John is a mathematician and Christy is either a mathematician or a philosopher. The latter sentence would be symbolized as J (C P). We will always require parentheses to indicate the order of evaluation for the binary connectives. For example, a set of parentheses is required to indicate the order of evaluation in J C P even though (J C) P (one way of placing the parentheses) is equivalent to J (C P).

The rules for forming formulas in the language of classical propositional logic are as follows:

1.Every uppercase roman letter, with or without an integer subscript, is a formula.

2.If P is a formula, so is ¬P.

3.If P and Q are formulas, so are (P Q), (P Q), (P → Q), and (P ↔ Q).2

We call single (possibly subscripted) roman letters A, B, and so on, atomic formulas, while formulas formed using one or more connectives are called compound formulas. The connectives introduced in clauses 2 and 3 are the main connectives of the formulas so formed. For example, because the formula ((A B) C) is formed from the formulas (A B) and C by clause 3, the main connective of ((A B) C) is the second disjunction. By convention, we will drop outermost parentheses in compound formulas—thus ((A B) C) may be written as (A B) C.

A compound formula is named after the operation symbolized by its main connective: a compound formula whose main connective is ¬ is called a negation, and so on. Context will always make it clear whether we are talking about the operation itself or a formula. The immediate subformulas P and Q of a conjunction P Q are called its conjuncts, and the immediate subformulas P and Q of a disjunction P Q are called its disjuncts. P is the antecedent of the conditional P → Q and Q is its consequent. (At times we will also refer to connectives by the name of the operation they symbolize; e.g., we will call ¬ a negation.)

2.2 Semantics of Classical Propositional Logic

The five logical connectives that we have introduced are truth-functional connectives: the truth-values of formulas formed with these connectives are determined by (i.e., are a function of ) the truth-values of the constituent formulas.

2We use boldface letters P, Q, . . . to stand for arbitrary formulas of the language. This will make it clear when we are talking about a particular formula (we will use nonboldface letters) and when we are talking about formulas generally (we will use boldface letters). In this context the boldface letters are called metavariables.

The truth-functional operations in classical propositional logic are captured by the following truth-tables:

where T and F stand for true and false, respectively. The information in these tables can be used to determine the truth-values of arbitrary formulas on truth-value assignments. A truth-value assignment is an assignment of truth-values to atomic formulas of the language, and in classical logic true and false are the only truthvalues. If we have a truth-value assignment on which J is false and C is true, then ¬J is true, ¬C is false, J C is false, J C is true, J → C is true, C → J is false, and C ↔ J is false.

More generally, a truth-table can be used to display the values that a formula will have on all truth-value assignments. Here’s a truth-table for the formula

¬J (C R):

All combinations of truth-values that C, J, and R can have are listed to the left of the vertical bar. For each of these combinations we list the value of ¬J (C R) and of each of its subformulas to the right of the vertical bar. The truth-value for an atomic subformula is written immediately below the atomic subformula, and the truth-value for a compound subformula (as well as for the formula as a whole) is written under its main connective. Thus, the first F in the first row—under the negation—tells us that the subformula ¬J is false on any truth-value assignment that assigns T to C, J, and R, the F under the conjunction tells us that the entire formula is false on such assignment, while the T under the disjunction tells us that

C R is true. The semantics of a language consists in its meaning (or interpretation), and in the case of classical propositional logic the semantics consists of bivalent truth-value assignments and the definitions of the truth-functional operations that can be used to construct a truth-table for any formula.

Logicians sometimes single out some connectives as primitive and introduce the other connectives as defined ones. One common way of doing this is to take ¬ and as primitive and then to define the others as follows:

P Q =def ¬(¬P ¬Q)

P → Q =def ¬(P ¬Q)

P ↔ Q =def ¬(P ¬Q) ¬(¬P Q).

The symbol =def means: is defined as. Simple reasoning confirms that these definitions are correct. For example, a formula ¬(¬P ¬Q) is true when ¬P ¬Q is false, and ¬P ¬Q is false when either when one or both of ¬P, ¬Q are false, that is, when one or both of P, Q are true—and that is exactly when a disjunction P Q is true. So the preceding definition for disjunction is correct. Alternatively, we can verify the correctness with a truth-table:

The column of truth-values under the disjunction is identical to the column under the second formula’s main connective—the first negation—so the formulas are equivalent. The correctness of the definitions for the conditional and biconditional can be verified similarly.

Another common (and similar) way of dividing the connectives into primitive and defined ones is to take ¬ and as primitive and then to introduce the others as:

P Q =def ¬(¬P ¬Q)

P → Q =def ¬P Q

P ↔ Q =def ¬(¬P ¬Q) ¬ (P Q).

The two connectives ¬ and → can also be taken as primitive, as will be confirmed in an exercise.

Formulas in the language of classical propositional logic that are true on all truth-value assignments are called tautologies, and formulas that are false on all truth-value assignments are called contradictions. The truth-values appearing in the truth-table in the column under a formula’s main connective will indicate whether that formula is a tautology, a contradiction, or neither. A symbolic version

The columns under the main connectives of the two formulas, the conditional connective in each case, are identical. On the other hand, the formulas C → J and J → C are not equivalent:

The formulas have different truth-values on truth-value assignments represented by the second and third rows—truth-value assignments on which C and J have different values. We note as a special case that all tautologies are equivalent to one another because they are all true on every truth-value assignment, and all contradictions are equivalent to one another for a similar reason.

A set (pronounced gamma) of formulas entails a formula P if, whenever all of the formulas in the set are true, P is true as well (that is, there is no truth-value assignment on which all the formulas in are true and P is false). An argument consists of one or more formulas, the premises, and an additional formula, the conclusion. We say that an argument is valid if the set consisting of its premises entails the argument’s conclusion. We also say that the conclusion of a valid argument follows from the premises.

The components of an argument are traditionally displayed by writing the premises, one per line, followed by a separator line and then the conclusion, for example,

J → C

J

C

This argument is valid, as is shown by the fact that every row in the following truth-table in which both premises are true also has the conclusion true (there is only one such row in this case, the first one):

On the other hand, the argument

J → C

¬J

¬C