PHIL 220. Introduction to Logic
Truth tables provide a decision method for propositional validity. They enable us to decide in a finite number of steps whether a given argument in propositional logic is valid:
If the argument is valid, there is no row of the truth table on which the premises come out true and the conclusion false.
If the argument is invalid, there is at least one row in which the premises come out true and the conclusion is false.
There is a counterexample to the validity of the argument.
Unfortunately, truth tables are not helpful when it comes to more complex formal languages for which we will have resort to a different strategy.
The other strategy for propositional validity involves proofs:
If an argument is valid, then there is a proof of the conclusion from the premises, which consists of a chain of simple inferences governed by sanctioned rules of inference associated with each connective.
We break down the argument into two simple steps:
from 1 and 3:
interest rates are high and economic growth is slow.
from 2 and the above:
inflation will come down.
One step proceeds from premises 1 and 3 and is sanctioned by the semantic behavior of and:
given \(\varphi\) and \(\psi\), we may infer the conjunction \(\varphi\) and \(\psi\).
The other step proceeds from premise 2 and the outcome of the prior step and is sanctioned by the behavior of if … then: given a conditional
if \(\varphi\), then \(\psi\)
and
\(\varphi\),
we are free to conclude \(\psi\).
\(p\): Interest rates are high.
\(q\): Economic growth is slow.
\(r\): Inflation will come down.
Lines 1, 2, and 3 are premises
Line 4 is obtained from lines 1 and 3 by \(I\wedge\):
from \(\varphi\) and \(\psi\), infer \(\varphi \wedge \psi\)
Line 5 is obtained from lines 2 and 4 by the rule of inference \(E\to\):
from if \(\varphi\), then \(\psi\) and \(\varphi\) infer \(\psi\)
Line 5 completes the proof because the formula is exactly the conclusion we are after.
A proof in propositional logic is a finite sequence of formulas each of which is either
an assumption, or
the outcome of an application of a sanctioned rule of inference to prior formulas.
The last formula is the conclusion.
If an argument of propositional logic is valid, then there is a proof of its conclusion from its premises.
Each connective comes with at least two rules of inference:
an introduction rule:
when you may write a formula with that connective as main connective
an elimination rule:
when you may exploit one that is already available from a prior line
Before we look at them, there is one structural rule to our disposal.
There is just one structural rule.
You may repeat a formula \(\varphi\) on a line if it is available on a prior line of the derivation.
The simplest proof you will ever write:
\[ p \vdash p \]
Two rules:
one introduction rule explains when you may write a conjunction \(\varphi \wedge \psi\) on the basis of prior lines.
one elimination rule explains how to exploit a conjunction \(\varphi \wedge \psi\) from a prior line.
You may write \(\varphi \wedge \psi\) if \(\varphi\) and \(\psi\) are both available at prior lines.
\[ p, q \vdash p \wedge q \]
\[ p, q, r \vdash (p \wedge r) \wedge q \]
You may write either conjunct of a conjunction available at a prior line.
\[ p, q \wedge r \vdash p \wedge r \]
\[ (p\wedge q) \wedge (r \wedge t) \vdash (p \wedge r) \wedge (q \wedge t) \]
Three derivations, given only their assumptions.
We will now provide introduction and elimination rules for the conditional and disjunction.