natural deduction: conditional and disjunction

PHIL 220. Introduction to Logic

Gabriel Uzquiano

natural deduction: conditional and disjunction

We have introduction and elimination rules for conjunction. We now look at two more connectives: conditional and disjunction.

where we left off

  • \(R\): repetition \(\checkmark\)
  • I\(\wedge\): conjunction introduction \(\checkmark\)
  • E\(\wedge\): conjunction elimination \(\checkmark\)
  • E\(\to\): conditional elimination (modus ponens) to do
  • I\(\to\): conditional introduction (conditional proof) to do
  • I\(\vee\): disjunction introduction to do
  • E\(\vee\): disjunction elimination (proof by cases) to do

the conditional

Two rules:

  • one elimination rule explains how to exploit a conditional \(\varphi \to \psi\) from a prior line.

  • one introduction rule explains when you may write a conditional \(\varphi \to \psi\) on the basis of prior lines.

elimination rule: E\(\to\)

  1. \(\varphi \to \psi\)
  2. \(\varphi\)
  1. \(\psi\)

You may write \(\psi\) if \(\varphi \to \psi\) and \(\varphi\) are both available at prior lines. This is what we have called modus ponens in the past.

how to apply E\(\to\)

  1. \((p \wedge q) \to t\)
  2. \(p\)
  3. \(q\)
  1. \(t\)

\[ (p \wedge q) \to t, p, q \vdash t \]

one more example

  1. \(p \to (q \to r)\)
  2. \(p \wedge q\)
  1. \(r\)

\[ p \to (q \to r), p \wedge q \vdash r \]

practice

introduction rule: I\(\to\)

You may write \(\varphi \to \psi\) if you can prove \(\psi\) from the assumption that \(\varphi\).

Once you complete the subproof, bracket the lines of that subproof and never appeal to them again.

rationale for I\(\to\)

  1. If interest rates are now lowered below 2% and the economy is near full employment, then inflation will rise above the 2% target.
  2. The economy is near full employment.
  1. If interest rates are lowered below 2%, then inflation will rise above the 2% target.

How would you argue for that conclusion given that interest rates are not lowered below 2%?

  1. Assume that interest rates are now lowered below 2%.
  2. Reason within that assumption that
  • interest rates are now lowered below 2% and the economy is near full employment.
  • inflation will rise above the 2% target.
  1. Exit the scope of the assumption to conclude:
  • if interest rates are lowered below 2%, then inflation will rise above the 2% target.

how to apply I\(\to\)

  1. \(p \to (q \wedge r)\)
  1. \(p \to r\)

\[ p \to (q \wedge r) \vdash p \to r \]

  • line 2 opens the subproof
  • line 5 closes it and discharges the assumption.

another application

  1. \(p \to r\)
  2. \(q \to t\)
  1. \((p \wedge q) \to (r \wedge t)\)

\[ p \to r, q \to t \vdash (p \wedge q) \to (r \wedge t) \]

practice

disjunction

Two rules:

  • one introduction rule to explain when to write a disjunction \(\varphi \vee \psi\) on the basis of prior lines.

  • one elimination rule to explain how to exploit a disjunction \(\varphi \vee \psi\) from a prior line

introduction rule: I\(\vee\)

  1. \(\varphi\)
  1. \(\varphi \vee \psi\)

You may write \(\varphi \vee \psi\) once either disjunct is available at a prior line.

how to apply I\(\vee\)

  1. \(p\)
  2. \(q\)
  1. \((p \vee r) \wedge (r \vee q)\)

\[ p, q \vdash (p \vee r) \wedge (r \vee q) \]

another application

  1. \(p\)
  2. \((p \vee q) \to r\)
  1. \(r\)

\[ p, (p \vee q) \to r \vdash r \]

practice

elimination rule: E\(\vee\)

  1. \(\varphi \vee \psi\)
  2. \(\varphi \to \chi\)
  3. \(\psi \to \chi\)
  1. \(\chi\)

You may write \(\chi\) on the basis of \(\varphi \vee \psi\) in the presence of two conditionals:

  • \(\varphi \to \chi\)

  • \(\psi \to \chi\)

You will often have to establish the conditionals by indepedendent applications of I\(\to\)

how to apply E\(\vee\)

  1. \(p \to r\)
  2. \(q \to r\)
  3. \(r \to t\)
  4. \(p \vee q\)
  1. \(t\)

two more applications

\[ (p \wedge q) \vee (r \wedge p) \vdash p \]

\[ (p \wedge q) \vee (r \wedge p) \vdash q \vee r \]

worked example

All four rules, on an argument from natural language.

a natural language argument

The British are coming by land or they are coming by sea. Paul Revere will light one lamp if the British are coming by land. However, Paul Revere will light two lamps if they are coming by sea. So, Paul Revere will light one lamp or he will light two lamps.

in premise-conclusion form

  1. The British are coming by land or they are coming by sea.
  2. If the British are coming by land, then Paul Revere will light one lamp.
  3. If the British are coming by sea, then Paul Revere will light two lamps.
  1. Paul Revere will light one lamp or he will light two lamps.

translation

\(p\): The British are coming by land.

\(q\): The British are coming by sea.

\(r\): Paul Revere will light one lamp.

\(s\): Paul Revere will light two lamps.

  1. \(p \vee q\)
  2. \(p \to r\)
  3. \(q \to s\)
  1. \(r \vee s\)

The argument is valid because there is a proof of the conclusion from its premises.

four rules, one system

  • E\(\to\): from \(\varphi \to \psi\) and \(\varphi\), infer \(\psi\)
  • I\(\to\): derive \(\psi\) from assumed \(\varphi\), discharge to get \(\varphi \to \psi\)
  • I\(\vee\): from \(\varphi\) (or \(\psi\)), infer \(\varphi \vee \psi\)
  • E\(\vee\): from \(\varphi \vee \psi\), \(\varphi \to \chi\), \(\psi \to \chi\), infer \(\chi\)