1.5 Abbreviating WFFs

The two interpretations are illustrated below in this highly scientific diagram: Cartoon showing a human poking a dog using a stick, and a human poking a dog that has a stick in its mouth.

Cartoon showing a human poking a dog using a stick, and a human poking a dog that has a stick in its mouth.

Similarly, even if we agreed that \(\land\) means “and”, \(\lor\) means “or”, \(\to\) means “implies”, formulas like

\[ P\land Q\lor R \qquad\qquad P\to Q\to R \qquad\qquad \lnot \,S\lor T \] leave open the question of whether the first formula should be interpreted as “(\(P\) and \(Q\)) or \(R\)” or “\(P\) and (\(Q\) or \(R\))” and whether the second formula should be interpreted as “(\(P\) implies \(Q\)) implies \(R\)” or “\(P\) implies (\(Q\) implies \(R\))”, or whether the third equation means “(not \(S\)) or \(T\)” or “not (\(S\) or \(T\))”. The answer matters, because we can use truth tables to verify that the possibilities are not logically equivalent.

Our official definition of Well-Formed Formulas requires parentheses around every conjunction, disjunction, and implication. This solves the problem (for logicians), because the above formulas are simply not considered well-formed. Instead, we are required to write whichever of \[ ((P\land Q)\lor R) \qquad\qquad (P\land (Q\lor R)) \] \[ ((P\to Q)\to R) \qquad\qquad (P\to (Q\to R)) \] \[ (\lnot S \ \lor\ T) \qquad\qquad \lnot (S\lor T) \] we intended to say. By requiring parentheses, every WFF is either unambiguously a propositional variable (e.g., \(P\)), a propositional constant (e.g., \(\bot\)), a negation, and conjunction, and disjunction, or an implication, and it’s unambiguous what subformulas are intended.

Example

The WFF \(((P\land Q)\lor(R\to S))\) can only be interpreted as the disjunction of \((P\land Q)\) and \((R\to S)\)
The WFF \(\lnot(P\to Q)\) can only be interpreted as the negation of \(P\to Q\)
The WFF \((\lnot P\ \to\ (Q\lor R))\) can only be interpreted as saying that \(\lnot P\) implies \(Q\lor R\)

This lack of ambiguity is incredibly convenient for the study of logical systems. However, when we humans are working with large formulas one must admit it is tedious to have to write \[ (((((P\land Q)\land R)\land S)\land T)\to((U\lor V)\lor W)) \] instead of just \[ P\land Q\land R\land S\land T \ \ \to \ \ U\lor V\lor W. \] Therefore, we will make the following compromise.

Definition

When writing WFFs, we can write abbreviated forms, and use the following rules to figure out which WFF we are referring to:

Precedence: \(\lnot\) groups most tightly, followed in order by \(\land\), \(\lor\), and \(\to\)
Associativity: \(\land\) and \(\lor\) are left-associative, but \(\to\) is right-associative
Other: we allow ourselves to write \(A\leftrightarrow B\) instead of \(((A\to B)\land(B\to A))\) (where \(A\) and \(B\) here represent arbitrary WFFs). The \(\leftrightarrow\) operator has lowest precedence

Example

\(P\land Q\lor R\) abbreviates \(\ ((P\land Q)\lor R)\) since \(\land\) groups tighter than \(\lor\)
\(P\to Q\to R\) abbreviates \(\ (P\to (Q\to R))\) since \(\to\) is right-associative.
\(\lnot \,S\lor T\) abbreviates \(\ (\lnot S\lor T)\) since \(\lnot\) groups tighter than \(\lor\)
\(P\land Q\leftrightarrow R\) abbreviates the WFF \((((P\land Q)\to R)\land(R\to(P\land Q)))\) since \(\land\) binds tighter than \(\leftrightarrow\).