This document discusses rules of inference used in propositional logic and quantified statements. It defines rules like modus ponens, modus tollens, and universal instantiation that allow valid logical arguments to be constructed by moving from premises to a conclusion. An example argument is broken down step-by-step to demonstrate applying rules of inference like simplification, modus tollens, and modus ponens to derive a conclusion from a set of premises. Finally, inference rules for quantified statements including universal and existential generalization and instantiation are introduced.

1. Rules of Inference

2. Proofs in mathematics are valid arguments
An argument is a sequence of statements that end in a conclusion
By valid we mean the conclusion must follow from the truth of the preceding
statements or premises
We use rules of inference to construct valid arguments

3. If you listen you will hear what I’m saying
You are listening
Therefore, you hear what I am saying
Valid Arguments in Propositional Logic
Is this a valid argument?
Let p represent the statement “you listen”
Let q represent the statement “you hear what I am saying”
The argument has the form:
q
p
q
p



4. Valid Arguments in Propositional Logic
q
p
q
p


q
p
q
p 

 )
)
(( is a tautology (always true)
1
1
1
1
1
1
0
0
0
1
1
0
1
1
0
1
0
1
0
0
)
)
((
)
( q
p
q
p
p
q
p
q
p
q
p 





This is another way of saying that
therefore


5. Valid Arguments in Propositional Logic
When we replace statements/propositions with propositional variables
we have an argument form.
Defn:
An argument (in propositional logic) is a sequence of propositions.
All but the final proposition are called premises.
The last proposition is the conclusion
The argument is valid iff the truth of all premises implies the conclusion is true
An argument form is a sequence of compound propositions

6. Valid Arguments in Propositional Logic
The argument form with premises
q
p
p
p n 


 )
( 2
1 
and conclusion q
is valid when
n
p
p
p ,
,
, 2
1 
is a tautology
We prove that an argument form is valid by using the laws of inference
But we could use a truth table. Why not?

7. Rules of Inference for Propositional Logic
q
p
q
p


modus ponens
aka
law of detachment
modus ponens (Latin) translates to “mode that affirms”
The 1st law

8. Rules of Inference for Propositional Logic
q
p
q
p


modus ponens
If it’s a nice day we’ll go to the beach. Assume the hypothesis
“it’s a nice day” is true. Then by modus ponens it follows that
“we’ll go to the beach”.

9. The rules of inference Page 66
Resolution
)
(
)]
(
)
[(
n
Conjunctio
)
(
))
(
)
((
tion
Simplifica
)
(
Addition
)
(
syllogism
e
Disjunctiv
)
)
((
syllogism
al
Hypothetic
)
(
)]
(
)
[(
tollen
Modus
)]
(
[
ponens
Modus
)]
(
[
Name
Tautology
inference
of
Rule
r
p
r
p
q
p
r
q
r
p
q
p
q
p
q
p
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
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
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




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


10. You might think of this as some sort of game.
You are given some statement, and you want to see if it is a
valid argument and true
You translate the statement into argument form using propositional
variables, and make sure you have the premises right, and clear what
is the conclusion
You then want to get from premises/hypotheses (A) to the conclusion (B)
using the rules of inference.
So, get from A to B using as “moves” the rules of inference
Another view on what we are doing

11. Using the rules of inference to build arguments An example
It is not sunny this afternoon and it is colder than yesterday.
If we go swimming it is sunny.
If we do not go swimming then we will take a canoe trip.
If we take a canoe trip then we will be home by sunset.
We will be home by sunset

12. Using the rules of inference to build arguments An example
1. It is not sunny this afternoon and it is colder than yesterday.
2. If we go swimming it is sunny.
3. If we do not go swimming then we will take a canoe trip.
4. If we take a canoe trip then we will be home by sunset.
5. We will be home by sunset
)
conclusion
(the
sunset
by
home
be
will
We
trip
canoe
a
take
will
We
swimming
go
We
yesterday
n
colder tha
is
It
afternoon
sunny this
is
It
t
s
r
q
p
t
t
s
s
r
p
r
q
p
.
5
.
4
.
3
.
2
.
1






propositions hypotheses

13. Hypothesis
.
1
Reason
Step
q
p 

(1)
using
tion
Simplifica
.
2
Hypothesis
.
1
Reason
Step
p
q
p



Hypothesis
.
3
(1)
using
tion
Simplifica
.
2
Hypothesis
.
1
Reason
Step
p
r
p
q
p




(3)
and
(2)
using
tollens
Modus
.
4
Hypothesis
.
3
(1)
using
tion
Simplifica
.
2
Hypothesis
.
1
Reason
Step
r
p
r
p
q
p





Hypothesis
.
5
(3)
and
(2)
using
tollens
Modus
.
4
Hypothesis
.
3
(1)
using
tion
Simplifica
.
2
Hypothesis
.
1
Reason
Step
s
r
r
p
r
p
q
p







(5)
and
(4)
using
ponens
Modus
.
6
Hypothesis
.
5
(3)
and
(2)
using
tollens
Modus
.
4
Hypothesis
.
3
(1)
using
tion
Simplifica
.
2
Hypothesis
.
1
Reason
Step
s
s
r
r
p
r
p
q
p







Using the rules of inference to build arguments An example
)
conclusion
(the
sunset
by
home
be
will
We
trip
canoe
a
take
will
We
swimming
go
We
yesterday
n
colder tha
is
It
afternoon
sunny this
is
It
t
s
r
q
p
t
t
s
s
r
p
r
q
p
.
5
.
4
.
3
.
2
.
1






Resolution
)
(
)]
(
)
[(
n
Conjunctio
)
(
))
(
)
((
tion
Simplifica
)
(
Addition
)
(
syllogism
e
Disjunctiv
)
)
((
syllogism
al
Hypothetic
)
(
)]
(
)
[(
tollen
Modus
)]
(
[
ponens
Modus
)]
(
[
Name
Tautology
inference
of
Rule
r
p
r
p
q
p
r
q
r
p
q
p
q
p
q
p
q
p
q
p
p
q
p
p
q
p
q
p
p
q
p
p
q
p
q
p
q
p
q
p
r
p
r
q
q
p
r
p
r
q
q
p
p
q
p
q
p
q
p
q
q
q
p
p
q
p
q
p



















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
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



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



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




















Hypothesis
.
7
(5)
and
(4)
using
ponens
Modus
.
6
Hypothesis
.
5
(3)
and
(2)
using
tollens
Modus
.
4
Hypothesis
.
3
(1)
using
tion
Simplifica
.
2
Hypothesis
.
1
Reason
Step
t
s
s
s
r
r
p
r
p
q
p








(7)
and
(6)
using
ponens
Modus
.
8
Hypothesis
.
7
(5)
and
(4)
using
ponens
Modus
.
6
Hypothesis
.
5
(3)
and
(2)
using
tollens
Modus
.
4
Hypothesis
.
3
(1)
using
tion
Simplifica
.
2
Hypothesis
.
1
Reason
Step
t
t
s
s
s
r
r
p
r
p
q
p









14. Using the resolution rule (an example)
1. Anna is skiing or it is not snowing.
2. It is snowing or Bart is playing hockey.
3. Consequently Anna is skiing or Bart is playing hockey.
We want to show that (3) follows from (1) and (2)

15. Using the resolution rule (an example)
1. Anna is skiing or it is not snowing.
2. It is snowing or Bart is playing hockey.
3. Consequently Anna is skiing or Bart is playing hockey.
snowing
is
it
hockey
playing
is
Bart
skiing
is
Anna
r
q
p
propositions
q
r
r
p



.
2
.
1
hypotheses
r
q
r
p
q
p





Consequently Anna is skiing or Bart is playing hockey
Resolution rule

16. Rules of Inference & Quantified Statements
All men are Mortal
John is a man
Therefore John is Mortal
Above is an example of a rule called “Universal Instantiation”.
We conclude P(c) is true, where c is a particular/named element
in the domain of discourse, given the premise )
(x
P
x


17. Rules of Inference & Quantified Statements
tion
generalisa
l
Existentia
)
(
c
element
some
for
P(c)
ion
instantiat
l
Existentia
c
element
some
for
)
(
P(x)
x
tion
generalisa
Universal
)
(
c
arbitrary
an
for
P(c)
ion
instantiat
Universal
)
(
P(x)
x
Name
Inference
of
Rule
x
P
x
c
P
x
P
x
c
P







