INTRO LOGIC
DAY 23
Derivations in PL
2
1
Overview



+
Exam 1
Exam 2
Exam 3
Exam 4
6 derivations
Exam 5
Exam 6
Sentential Logic Translations (+)
Sentential Logic Derivations
Predicate Logic Translations
Predicate Logic Derivations
@ 15 points
+ 10 free points
very similar to Exam 3
very similar to Exam 4
+
+
2
Rule Sheet
Intro Logic
provided
on exams
&I
&O


––––––
 &


––––––
& 
I
O


–––––– ––––––
    
available
on course
web page
(textbook)
I
 

––––––

 

––––––

O

 
–––––––


 
–––––––
 
I see CD
DN
O
O
 : 






CD
(  )
––––––––––
 & 
 : 

 : 

D
Rep

––

O
DD

––
(ID)
 : 
()



D
O

–––––



––––

D
(  )
––––––––––
  


–––––––

DN

–––––

(  )
–––––––––


 O
 
––––––– –––––––
    


–––––––

I
keep this
in front of
you when
doing
homework
SENTENTIAL LOGIC
Rules of Derivation
&O
&D
 &  &
( & )
 : &
–––––– ––––––
–––––––––




  




 : 

 : 

ID
 :



 :

 : 

PREDICATE LOGIC
In the following,  is any variable; [] is any formula in which  occurs free;
[o] results by substituting o for every free occurrence of , where o is any old name.
[n] results by substituting n for every free occurrence of , where n is any new name.
A name counts as old if it occurs in a line that is neither boxed nor cancelled;
otherwise it counts as new.
I see UD
O
O
UD
[]

 : []
––––––
––––––
new  : [n]
[o]
old



I
O
[o]
old
––––––
[]
[]
––––––
[n]
D
O
new
[]
––––––
[]
don’t
make
up
your
own
rules
(ID)
 : []
[]
 : 

4
Sentential Logic Rules






DD
ID
CD
D
&D
etc.






&I
&O
vO
O
O
etc.
5
Predicate Logic Rules
Universal Derivation
Universal-Out
Tilde-Universal-Out
UD
O
O
day 2
day 1
day 3
Existential-In
Existential-Out
Tilde-Existential-Out
I
O
O
day 1
day 2
day 3
6
Rules Already Introduced – Day 1
O

–––––


–––––

I
 is an OLD name
(more about this later)
7
Rules to be Introduced Today
Universal Derivation
UD
Existential-Out
O
8
Example 1
every F is H ; everyone is F / everyone is H
(1) x(Fx  Hx)
Pr
(2) xFx
Pr
(3) : xHx
??
(3) : Ha & Hb & Hc & ……… &.&.&.D
(a)
(?)
(b)
(?)
(c)
(?)
…
: Ha
??
: Hb
??
: Hc
??
…
what
is
??
ultimately
??
involved
??
in
??
??
??
…
showing
a
universal
9
Example 1a
(1) x(Fx  Hx)
Pr
(2) xFx
Pr
(3) : xHx
(4) : Ha
??
DD
(5)
Fa  Ha
1,
O
(6)
Fa
2,
O
(7)
Ha
5,6, O
one down,
a zillion to go!
10
Example 1b
(1) x(Fx  Hx)
Pr
(2) xFx
Pr
(3) : xHx
(4) : Hb
??
DD
(5)
Fb  Hb
1,
O
(6)
Fb
2,
O
(7)
Hb
5,6, O
two down,
a zillion to go!
11
Example 1c
(1) x(Fx  Hx)
Pr
(2) xFx
Pr
(3) : xHx
(4) : Hc
??
DD
(5)
Fc  Hc
1,
O
(6)
Fc
2,
O
(7)
Hc
5,6, O
three down,
a zillion to go!
12
But wait – the derivations all look alike!
(4)
(5)
(6)
(7)
: Ha
Fa  Ha
(4)
(5)
(6)
(7)
: Hb
Fb  Hb
(4)
(5)
(6)
(7)
: Hc
Fc  Hc
Fa
Ha
Fb
Hb
Fc
Hc
DD
1,O
2,O
5,6,O
DD
1,O
2,O
5,6,O
DD
1,O
2,O
5,6,O
13
The Universal-Derivation Strategy
So, all we need to do is
do one derivation with one name (say, ‘a’)
and then argue that
all the other derivations will look the same.
To ensure this,
we must ensure that the name is general,
which we can do by making sure
the name we select is NEW.
a name counts as NEW precisely if
it occurs nowhere in the derivation
unboxed or uncancelled
14
The Universal-Derivation Rule (UD)
: 
UD
: 
??
 is any variable
 is any (official) formula
 replaces 
°
 must be a NEW name,
°
i.e., one that is occurs
nowhere in the derivation
unboxed or uncancelled
°
°
15
Comparison with Universal-Out
O
UD

–––––

: 
OLD name
NEW name
a name counts as OLD
precisely if it occurs
somewhere in the derivation
unboxed and uncancelled
: 
a name counts as NEW
precisely if it occurs
nowhere in the derivation
unboxed or uncancelled
16
Example 2
every F is H / if everyone is F, then everyone is H
(1) x(Fx  Hx)
Pr
(2) : xFx  xHx
CD
(4)
xFx
: xHx
As
UD
(5)
: Ha
DD
(3)
a new
(6)
Fa  Ha
1,
O
a old
(7)
Fa
3,
O
a old
(8)
Ha
6,7, O
17
Example 3
every F is G ; every G is H / every F is H
(1)
(2)
x(Fx  Gx)
x(Gx  Hx)
Pr
Pr
(3)
: x(Fx  Hx)
: Fa  Ha
UD
CD
(4)
(5)
Fa
As
(6)
: Ha
DD
a new
(7)
Fa  Ga
1,
O
a old
(8)
Ga  Ha
2,
O
a old
(9)
Ga
5,7,
O
(10)
Ha
8,9,
O
18
Example 4
everyone R’s everyone / everyone is R’ed by everyone
(1)
xyRxy
Pr
(2)
: xyRyx
UD
a new
(3)
: yRya
: Rba
UD
DD
b new
(4)
(5)
yRby
1,
(6)
Rba
5,
O b old
O a old
19
Existential-Out (O)
any variable (z, y, x, w …)
any formula

–––––

 replaces 
any NEW name (a, b, c, d, …)
20
Comparison with Universal-Out
O

––––––


––––––

OLD name
NEW name
a name counts as OLD
precisely if it occurs
somewhere
unboxed and uncancelled
O
a name counts as NEW
precisely if it occurs
nowhere
unboxed or uncancelled
21
Example 5
every F is un-H / no F is H
(1) x(Fx  Hx)
Pr
(2) : x(Fx & Hx)
D
(3)
x(Fx & Hx)
As
(4)
: 
DD
(5)
Fa & Ha
3,
O new
(6)
Fa  Ha
1,
O
(7)
Fa
(8)
Ha
5,
&O
(9)
Ha
6,7, O
(10)

8,9, I
old
22
Example 6
some F is not H / not every F is H
(1) x(Fx & Hx)
Pr
(2) : x(Fx  Hx)
(3) x(Fx  Hx)
ID
As
(4)
: 
DD
(5)
Fa & Ha
1,
O
new
(6)
Fa  Ha
3,
O
old
(7)
Fa
(8)
Ha
5,
&O
(9)
Ha
6,7, O
(10)

8,9, I
23
Example 7
every F is G ; some F is H / some G is H
(1) x(Fx  Gx)
Pr
(2) x(Fx & Hx)
(3) : x(Gx & Hx)
Pr
DD
(4)
Fa & Ha
2,
O
new
(5)
Fa  Ga
1,
O
old
(6)
Fa
(7)
Ha
4,
&O
(8)
Ga
5,6, O
(9)
Ga & Ha
7,8, &I
(10) x(Gx & Hx)
9,
I
24
Example 8
if anyone is F then everyone is H
/ if someone is F, then everyone is H
(1) x(Fx  xHx)
Pr
(2) : xFx  xHx
(3) xFx
CD
As
(4)
: xHx
UD
(5)
: Ha
DD
new
(6)
Fb
3,
O
new
(7)
Fb  xHx
1,
O
old
(8)
xHx
6,7, O
(9)
Ha
8,
O
old
25
Example 9
if someone is F, then everyone is H
/ if anyone is F then everyone is H
(1) xFx  xHx
Pr
(2) : x(Fx  yHy)
(3) : Fa  yHy
UD
CD
(4)
Fa
As
(5)
: yHy
UD
(6)
: Hb
DD
new
new
(7)
xFx
4,
I
(8)
xHx
1,7, O
(9)
Hb
8,
O old
26
Example 10 (a fragment)
someone R’s someone
??missing premises??
/ everyone R’s everyone
(1) xyRxy
Pr
(2) ???
(3) : xyRxy
Pr
UD
new
new
(4)
: yRay
UD
(5)
: Rab
??
(6)
yRcy
(7)
Rcd
1, O new
6, O new
(8)
??
??
27
THE END
28