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LOGIC MAIN PAGE

LOGICAL OPERATORS

Sample sentences
AND operator
IF/THEN operator
NOT operator
OR operator
XOR operator

RULES OF LOGIC

Chain Rule
Conjunctive Addition
Contrapositive
DeMorgan's Law
Disjunctive Addition
Disjunctive Inference
Disjunctive Infer. (XOR)
Double Negation
Modus Ponens
Modus Tollens
Mutual Exclusion
Simplification

VALIDITY PROOFS

2-step
3-step
4-step
5-step or more
Bad Argument

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