Introduction to Logic
(‘Methodenkurs Logik’)
Winter term 2014-2015
Mondays 2:30--4:00pm @ 2321.HS 3H
INSTRUCTOR
Todor Koev ([email protected])
Propositional logic: wrap-up
 Interpretation rules for propositional logic
 Logical equivalences
 Truth functions
Interpretation rules for PL
 Truth tables are an elegant tool for determining the truth
value of a complex formula in terms of the truth values of its
parts.
 We can encode the same information by providing
interpretation rules for formulas based on the interpretation
function  , a function from formulas to truth values.
(i)
(ii)
(iii)
(iv)
(v)
   1
     1
     1
     1
     1
iff
iff
iff
iff
iff
   0
   1 and    1
   1 or    1
   0 or    1
    
Remarks
    1 stands for “ is true”,    0 stands for “ is false”.
 “iff” is shorthand for “if and only if”. This is the
metalanguage counterpart to the logical connective of
equivalence/biconditional (  ).
 The negation rule can also be stated as follows:
(i')    0
iff
   1
 The implication rule can more transparently be stated like
this:
(iv')      0
iff
   1 and    0
Example 1
 Sample derivation of the truth conditions of ( p  q )  p :
( p  q )  p  1
iff  p  q   1 or p   1
iff ( p   1 and q   1) or p   1
iff ( p   1 and q   1) or  p   0
(iii)
(ii)
(i)
 This result is confirmed by the truth table for ( p  q )  p :
p
1
1
0
0
q p  q p ( p  q )  p
1
1
0
1
0
0
0
0
1
0
1
1
0
0
1
1
Example 2
 Sample derivation of the truth conditions of ( p  ( p  q )) :
( p  ( p  q ))  1
iff p  ( p  q )  0
iff p   1 and  p  q   0
iff  p   0 and  p  q   0
iff  p   0 and ( p   1 and q   0 )
iff  p   0 and  p   1 and q   0
iff never, because p cannot be both true and false!
(i)
(iv)
(i)
(iv)
(remove parens)
 So: ( p  ( p  q )) is always false, i.e. it is a contradiction!
 In other words: The final column of its truth table only has 0s in it.
 What about p  ( p  q ) ? A contradiction, a tautology, a contingency?
Logical equivalences
 Two formulas of propositional logic are logically equivalent
iff they are true under the same truth value assignment to
atomic sentences.
 Example: p  q and q  p are logically equivalent because
both formulas are true iff p is true and q is true.
 To express logical equivalence between two formulas  and
 , we write:    (that is,  is true iff  is true).
 While  is strictly speaking a part of the metalanguage, you
can read  as  :    iff    is a tautology. (This follows
from the way  is defined).
 Logical equivalences/Tautologies are also called logical laws.
Sample logical equivalences 1
 Double negation:   
o If you negate a formula twice, you are back at the original formula.
o The law of double negation ‘almost’ works in English but not quite. It
is often explored to trigger pragmatic inferences.
(1) Max is not an unwise person.
Implication: Max did something stupid.
o Also, in many dialects or languages two negations do not cancel each
other out.
(2) I ain’t got no money.
(African American English)
Means: I don’t have any money.
Sample logical equivalences 2
 Contraposition:       
o Recall the inferences previously discussed in class: (3) and (4) say the
same thing.

(3) If it’s raining, the streets are wet.
(4) If the streets aren’t wet, it’s not raining. 
pq
q  p
o You can use truth tables to check that        is not a law
of logic! So, from (3) we cannot conclude “If it’s not raining, the
streets are dry.”
Sample logical equivalences 3
 De Morgan’s laws:
(   )    
(   )    
o Also works in language: (5) and (6) are equivalent.

( p  q )
(5) It’s false that he is young and rich.

p  q
(6) He is not young or he is not rich.
o The De Morgan’s laws allow us to express conjunction in terms of
disjunction and vice versa.
o The De Morgan’s laws are at the heart of the fact that Boolean operators
(negation, conjunction, and disjunction) have found such wide use in
modern electronics.
Sample logical equivalences 4
 “Back-and-forth” laws: Allow us to go back and forth between
conjunction/disjunction and implication.
    (   )  (   )
    (   )    
    (   )
      
 Excluded middle: A formula is either true or false.
  
 Ex falso sequitur quodlibet: If you start with falsity, you can derive whatever
(follows from the truth table/interpretation rule for implication).
(   )  
Truth functions
 So far, logical connectives were interpreted as part of bigger
formulas, e.g. p or p  q .
 But we can as well look at connectives as truth functions, i.e.
functions from truth values to truth values.
 For example:
o Negation: (1)  0 , (0)  1
o Conjunction: (1,1)  1, (1,0)  (0,1)  (0,0)  0
o Disjunction: …
 What about adding more connectives/truth functions, e.g.:
o Identity: id (1)  1, id (0)  0
 Q1: Does logic ‘need’ more connectives?
 Q2: Does language ‘need’ more connectives?
How many truth functions?
 There are exactly 4 unary (=one-place) truth functions:
negation, identity, and two more:
1
0
f1
f2
f3
f4
0
1
1
0
1
1
0
0
f 1  , f 2  id
 There are exactly 16 binary (=two-place) truth functions:
conjunction, disjunction, implication, equivalence, and 12
more… (List them all!)
2n
2 n-ary truth functions (truth functions
of n propositional variables/truth values).
 In general, there are
Functional completeness
 A set of connectives is called functionally complete if it can express any
truth function.
 The set , , , ,  is functionally complete (see textbook for a proof).
 No new connectives are needed as far as logic is concerned!
 In fact, the Boolean set , ,  is functionally complete. Here is why:
o Any subformula of the form    can be substituted with
(   )  (   ) .
o Any subformula of the form    can be substituted with
(   ) or    .
o There are only , ,  left in the remaining formula!
o Example: p  (q  r ) is equivalent to p  (q  r ) , which in turn is
equivalent to p  ( q  r ) .
 Moreover, both ,  and ,  are functionally complete. Why?
Just one connective?
 The binary connective  is called the Peirce arrow (after Charles Peirce)
and is defined as follows:
   
1
1
0
0
1
0
1
0
0
0
0
1
 Surprise:  turns out to be functionally complete!
o     
o (   )  (   )    
o We already know that ,  are functionally complete.
Does language ‘need’ more connectives?
 Propositional connectives are somewhat arbitrary: they are there for
convenience and fairly closely mirror natural languages.
 Ad-hoc connectives can always be defined, if needed, e.g. exclusive ‘or’.
(7) You can have an apple or you can have a pear.
Implication: You cannot have both.


 
1
1
0
0
1
0
1
0
0
1
1
0
 However:
o Preferring exclusive ‘or’ over classical ‘or’ invalidates the De
Morgan’s laws.
o Exclusive ‘or’ might be due to classical ‘or’ plus pragmatics.
What you should know
 Understand the syntax of propositional logic, be able to
draw construction trees.
 Understand how truth tables work, be able to provide truth
tables for complex formulas.
 Understand interpretation rules, be able to derive truth
conditions for formulas.
 For next time, please read Set Theory (=an appendix to
Chierchia & McConnell-Ginet 2000)!