LOGIC MAIN PAGE

LOGICAL OPERATORS

RULES OF LOGIC

VALIDITY PROOFS

Chain Rule

The Chain Rule is a rule of inference pertaining to the IF/THEN operator.

The Chain Rule is used to combine two conditionals of the form p -> q and q -> r into p -> r.

Imagine we are given two conditionals, p -> q and q -> r:

p -> q: "If Jane leaves home late, she will miss her train."

Now let's consider the second conditional, q -> r. Note that it contains the same letter that we used in the first conditional, namely q. This means that q has to remain the same as in the first conditional.

q -> r: "If Jane misses her train, she will be late for work."

Given the two conditionals, It is perfectly natural for us to say: "If Jane leaves home late, she will be late for work." That is how the Chain Rule works. Formally, we would write

p -> q: "If Jane leaves home late, she will miss her train."
q -> r: "If Jane misses her train, she will be late for work."
----------
p -> r: "If Jane leaves home late, she will be late for work."

The given conditionals are above the line of dashes, and the new expression p -> r formed by applying the Chain Rule is below the line.

Other examples of the Chain Rule

W -> M: "If you have a job, you will get money."
M -> F: "If you have money, you can buy food."
----------
W -> F: "If you have a job, you can buy food."

X -> Y: "If you read the book, you're ready for the exam."
Y -> Z: "If you're ready for the exam, you'll pass it."
----------
X -> Z: "If you read the book, you'll pass the exam."

Links to Relevant Problems

These are links to validity proof problems whose solutions contain the Chain Rule.

2-step problem
4-step problem