Download Examples of Natural Deduction

Examples of Natural
Deduction
D. Mott
ETS, IBM UK
Using “Fitch”
All pets are happy, so what about
“scruffy?”
Premises
Lines of
proof, each
applying a
natural
deduction
rule to
previous
lines
Goal
In detail
Premises
All pets are happy
scruffy is a pet
Goal
scruffy is happy
cites
FORALL
elimination
->
elimination
scruffy is happy
QED
But …
• We can do this already in CE!
• More interesting if we want to infer a new rule
• Given “All pets are happy”
– can we show that if something is unhappy then it is
not a pet?
• if not happy(X) then not pet(X)
• Need to introduce the idea of a subproof first
Subproof to show scruffy is not a pet
All pets are happy
Scruffy is not happy
Is scruffy not a pet?
SUPPOSE scruffy is a pet
New subproof inside main proof
We can show that scruffy is happy
But we have proved an inconsistency
Therefore the supposition is
false, i.e. scruffy is not a pet
Generalising the subproof
SUPPOSE that is some thing,
which we will call “c” that is not
happy
then “c” is not a pet
But what “c” actually is is not
relevant to the proof, so it
works for ALL things
So ANY thing that is not happy
is not a pet, and we can create
a generic rule
In CE…
• We could use negated information, eg:
– it is false that the thing T is a pet
• But in the logic problems I am using terms that
include a negation:
– cannot be wearing
• So this new rule might be:
– if the thing T is a nonpet then the thing T is an
unhappy thing
• This requires an additional step in the proof to
perform the negation, based on the domain
definition of “unhappy” as being the negation of
“happy”, etc
Could we build a similar system in CE?
• In most cases there is a simple correspondance
between the natural deduction rules and
manipulations that could be made on the CE
statements
• But in the use of subproofs, it is necessary to
manipulate the rationale graphs themselves
– given a rationale link between a premise and its
conclusion, we could construct a new rule
– given a rationale link between a premise and an
inconsistency, we could construct the negation of the
premise.
It might look like…