Equivalence, Exportation and Tautology (Equiv, Exp, Taut)

The last three rules of replacement handle the biconditional, conditionals with conjunctions inside, and repeated parts. With them you have all ten, completing the list begun in Rules of Replacement and continued in Transposition and Material Implication (Trans, Impl).

Material equivalence, exportation and tautology.
Rule	Form	Form
Equiv	$p \leftrightarrow q$	$(p \to q) \land (q \to p)$
Equiv	$p \leftrightarrow q$	$(p \land q) \lor (\neg p \land \neg q)$
Exp	$(p \land q) \to r$	$p \to (q \to r)$
Taut	$p$	$p \lor p$
Taut	$p$	$p \land p$

Key idea

Equiv unpacks a biconditional into two conditionals, or into "both or neither". Exp moves the first conjunct of an antecedent out in front: $(p \land q) \to r$ :: $p \to (q \to r)$. Taut removes or adds a repeated part joined to itself.

Why it works

A biconditional is true when both sides match, which is exactly when each side implies the other, or when both are true or both are false. Exportation reflects how conditions stack: "if you have a ticket and it is Saturday, you can enter" says the same as "if you have a ticket, then if it is Saturday, you can enter". Taut is obvious once said aloud: "$p$ or $p$" and "$p$ and $p$" both just say $p$.

Exportation: the first two columns match; the third does not.

Worked examples

Worked example: Equiv to get a conditional

From $p \leftrightarrow q$ and $p$, prove $q$.

Line 3: $(p \to q) \land (q \to p)$ by Equiv 1.
Line 4: $p \to q$ by Simp 3.
Line 5: $q$ by MP 4, 2.

Answer: Three lines.

Worked example: Exportation

From $(r \land s) \to t$ and $r$, prove $s \to t$.

Line 3: $r \to (s \to t)$ by Exp 1.
Line 4: $s \to t$ by MP 3, 2.

Answer: Two lines.

Worked example: Taut after CD

From $(p \to q) \land (r \to q)$ and $p \lor r$, prove $q$.

Line 3: $q \lor q$ by CD 1, 2.
Line 4: $q$ by Taut 3.

Answer: CD, then Taut.

Common mistake

Exporting into $(p \to q) \to r$. That is not equivalent: the third column above shows rows where $(p \land q) \to r$ is true and $(p \to q) \to r$ is false (for instance $p$ F, $q$ F, $r$ F). Exportation keeps $r$ at the very end and moves only $p$ out in front: $p \to (q \to r)$.

Each of these rules earns its place in proofs. Equiv is the only way to use a biconditional: no rule of inference cites one directly. Exp lets MP reach a consequent when you have the conjuncts of the antecedent on separate lines (with Exp you need only one at a time). Taut tidies up after CD when both conditionals lead to the same place.

A biconditional you want to use: Equiv first.
An antecedent that is a conjunction: consider Exp.
A disjunction or conjunction of a formula with itself: Taut.

That completes the ten rules of replacement: DM, Com, Assoc, Dist, DN, Trans, Impl, Equiv, Exp and Taut. With the eight rules of inference, that makes eighteen. The next lesson puts them all to work together.

Equivalence, Exportation and Tautology (Equiv, Exp, Taut)

Worked examples

Practice

Builds on

Leads to

Sources and further reading