Lecture Notes, 29 Sep 2010
Last Update: 29 September 2010
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§1.5: Rules of Inference
- Artificial intelligence (AI) is the branch of computer science
concerned with computational theories of cognition.
- Knowledge representation and reasoning (KRR) is the branch
of AI concerned with how to:
- represent information (what we've done so
- in a formal language that can be handled by a
- & reason with that information
(as represented in that language)
- Reasoning involves constructing & verifying
"arguments" or "proofs"…
- …which are used:
- to answer questions
- & to give reasons for the answers
- E.g.: How can you pass this course?
(Or: Why didn't I pass this course?)
- If you get a passing grade in all assignments,
then you will pass the course. (premise 1)
- You get a passing grade. (premise 2)
- ∴ You pass. (conclusion)
This is called an "argument"
(in the sense of a legal argument,
not a yelling match)
Let p = "You get a passing grade in all assignments."
Let q = "You (will) pass the course".
Then the form of this argument can be shown by the following
sequence of propositions:
(p → q)
This "argument form" is also called a "rule of inference"
Consider a truth table for this argument:
|prem 2||conc||prem 1|
Note that row 1 is the only row where both premises are T.
In that row, the conclusion is also T.
- The name of this rule of inference (or basic argument form) is
modus ponens (MP) (Latin for "method of affirmation (of the
- When is an argument a "good" argument?
- There are 2 separate conditions for goodness:
- logical goodness
- factual goodness
An argument from premises
P1, …, Pn
to conclusion C
a sequence of propositions 〈P1, …, Pn, C〉,
where C is alleged to follow logically from
An argument (or: an argument form) is (semantically)
valid (= "logically good")
it is truth-preserving.
i.e., if all of its premises are T,
then its conclusion must be T
or: it is impossible for all of its premises to be
simultaneously T, yet its conclusion is F.
A proof isdef a valid argument.
- How can you tell if an argument is truth-preserving?
- An argument is truth-preserving (or: is a proof)
it is an argument in which each proposition
Pi or conclusion C is either:
- a tautology (hence T)
- a "premise"
- (premises can be logically contingent
but they are assumed to be T)
- or follows validly from previous
propositions in the sequence
by one or more rules of inference
- (and—because they are valid argument forms—each rule of inference is itself
- Note the recursion:
- Base Case = propositions assumed to be T
- Recursive Case = "sub-proofs" based on valid
(truth-preserving) rules of inference
- Another rule of inference: Hypothetical Syllogism (HS)
(A → B)
(B → C)
∴ (A → C)
You should construct a truth table to convince yourself that this is
truth-preserving, hence valid.
- "syllogism" is just an old-fashioned word for
- This can also be called "transitivity of →"
- Here is a more complex e.g. of an argument:
If you pass all assignments, then you pass the course.
If you pass the course, then you'll be successful in life.
You pass the course.
∴ You'll be successful in life.
Is this valid?
We can prove validity by syntactically deriving all
- First, let's symbolize the argument:
Let p = You pass all assignments.
Let q = You pass the course.
Let r = You'll be successful in life.
P1: (p → q)
P2: (q → r)
C : ∴ r
- Proof of validity:
(compare this to a computer program with line numbers
|| (p → q)
||:|| Prem P1|
|| (q → r)
||:|| Prem P2|
|| (p → r)
||:|| From lines 1,2; by HS|
||:|| Prem P3|
||:|| From lines 3,4; by MP|
- Alternative proof:
(Just as you can write more than one program to accomplish
the same task
(have the same input-output behavior),
you can also write more than one proof of an argument.)
- (p → q) : P1
- p : P3
- q : 1,2; MP
- (q → r) : P2
- r : 3,4; MP
- But is this really a good argument?
Text copyright © 2010 by William J. Rapaport
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