Symbolic Logic I



Symbolic Logic I

Statement Forms and Dominant Operators

Every compound WFF has one operator that governs what we will call the form of the statement. There are exactly five (5) basic statement forms, one for each of the five truth-functional connectives that we have studied. The five statement forms are:

Conjunction p ( q

Disjunction p ( q

Negation ~p

Conditional p ( q

Biconditional p ( q

in which 'p' and 'q' are sentential variables (place holders for simple statements). Every WFF, no matter how complex it appears, corresponds to exactly one of these five forms. We obtain a WFF from a statement form by replacing all of the sentential variables with WFF's (either simple statements or compound ones), being careful to replace one sentential variable with exactly one WFF. Thus, (P ( Q) ( (R ( S) is a conjunction obtained by replacing 'p' with '(P ( Q)' and 'q' with '(R ( S)'. Note that in this conjunction one conjunct is a conditional , i.e. (P ( Q),' while the other is a disjunction, '(R ( S)'.

The operator (the connective) which determines the form of a statement is called the dominant operator. The dominant operator in a formula is the one with the fewest parentheses surrounding it (zero being maximally few, as in the example above).

Exercises

Identify the statement form of each of the following WFF's.

1) (P ( (Q ( S)) ( R

2) (P ( Q) ( (P ( (~R ( S))

3) [~(P ( Q) ( (~R ( (S ( T))]

4) ~[(P ( Q) ( (~R ( (S ( T))]

5) [(P ( Q) ( R] ( ~(S ( T)

6) {[(P ( Q) ( (R ( ~S)] ( [(~S ( T) ( (~P ( R)]}

7) (P ( ~(Q ( ~R))

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