Logic
Bram
What is Logic?
What is logic?
• Logic is the study of valid reasoning.
• That is, logic tries to establish criteria to
decide whether some piece of reasoning is
valid or invalid.
• OK, so then what do we mean by ‘valid
reasoning’?
Reasoning
• When we reason, we infer something (Y)
from something else (X).
• That is, when we reason, we go like: “Well,
if such-and-such-and-so (X) is the case,
then this-and-that-and-the-other-thing (Y)
must also be the case ”
• X and Y are thus things-that-can-be-thecase-or-not-be-the-case
Reasoning Diagram:
XY
Good vs Bad Reasoning
• What is the purpose of reasoning? Well, through
reasoning, we try to gain new knowledge.
• ‘Good’ reasoning is a piece of reasoning that
successfully fulfills this purpose, i.e. that indeed
gives us a new piece of knowledge.
• ‘Bad’ reasoning is a piece of reasoning that, for
some reason or other, is unsuccessful in this
purpose.
• What can go wrong?
– Reasoning is invalid
– Reasoning is unsound
– Reasoning is circular
Valid Reasoning
• While in every piece of reasoning something is
believed to follow from something else, this may
in fact not be so.
• Example: “If I win the lottery, then I’m happy.
However, I did not win the lottery. Therefore, I
am not happy.”
• A piece of reasoning in which Y is believed to
follow from X is valid if Y does indeed follow
from X. Otherwise, the reasoning is said to be
invalid.
Sound Reasoning
• Not all valid reasoning is good reasoning.
• Example: “If I win the lottery, then I’ll be poor.
So, since I did win the lottery, I am poor.”
• This piece of reasoning is valid, but not very
good, since part of what it assumed is absurd (‘If
I win the lottery, I’ll be poor.’ Huh??) (also, I did
not win the lottery  )
• A piece of reasoning where Y is believed to
follow from X is sound if a) it is valid, and b) X is
true (or at least acceptable/plausible).
Truth and Implication
• Logic studies the validity of reasoning.
• Logic does not study soundness.
• Therefore, logic alone cannot tell us
whether an argument is good. Hence,
logic alone is not a guide to truth.
• Instead, logic can tell us, assuming certain
things to be true, what else will be true as
well. Thus, logic is a guide to implication.
Arguments
• A piece of reasoning consists of a
sequence of statements, some of which
are claimed to follow from previous ones.
That is, some are claimed to be inferred
from others.
• Example: “Either the housemaid or the
butler killed Mr. X. However, if the
housemaid would have done it, the alarm
would have gone off, and the alarm did not
go off. Therefore, the butler did it.”
Arguments, Premises and
Conclusion
• In logic, pieces of reasoning are analyzed using
the notion of an argument
• An argument consists of any number of
premises, and one conclusion
• Again, in logic, we are merely interested in
whether the conclusion follows from the
premises: we are not interested in whether those
premises are true or acceptable.
• If you want to study all aspects of good
reasoning, take my class Methods of Reasoning.
Deductive Validity vs
Inductive Validity
• An argument is said to be deductively valid if,
assuming the premises to be true, the
conclusion must be true as well.
• An argument is said to be inductively valid if,
assuming the premises to be true, the
conclusion is likely to be true as well.
• For now, we will limit ourselves to deductive
validity only!
• If you want to study non-deductive reasoning,
take my Methods of Reasoning class.
Argument Forms
• “If I win the lottery, then I am poor. I win the
lottery. Hence, I am poor.”
• This argument has the following abstract
structure or form: “If P then Q. P. Hence, Q”
• Any argument of the above form is valid,
including “If flubbers are gook, then trugs are
brig. Flubbers are gook. Hence, trugs are brig.”!
• Hence, we can look at the abstract form of an
argument, and tell whether it is valid without
even knowing what the argument is about!!
Formal Logic
• Formal logic studies the validity of
arguments by looking at the abstract form
of arguments.
• Formal logic always works in 2 steps:
– Step 1: Use certain symbols to express the
abstract form of premises and conclusion.
– Step 2: Use a certain procedure to figure out
whether the conclusion follows from the
premises based on their symbolized form
alone.
Example Step 1: Symbolization
• Use symbols to represent simple propositions:
– H: The housemaid did it
– B: The butler did it
– A: The alarm went off
• Use further symbols to represent complex
claims:
– H  B: The housemaid or the butler did it
– HA: If the housemaid did it, the alarm would go off
– ~A: The alarm did not go off
Example Step 2: Evaluation
• One possible technique is to transform the
symbolic representations using basic rules that
reflect elementary valid inferences:
1. H  B
A.
2. HA
A.
3.
A
A.
4.
H
2, 3 MT
5. B
1, 4 DS
Since every step along the way
is an instance of an obviously
valid inference, the conclusion
does indeed follow from the
premises. So, valid argument!
Propositional Logic
• Propositional Logic studies validity at the level of
simple and compound propositions.
• Simple proposition: An expression that has a
truth value (a claim or a statement). E.g. “John is
tall”
• Compound proposition: An expression that
combines simple propositions using truthfunctional connectives like ‘and’, ‘or’, ‘not’, and ‘if
… then’. E.g. “John is tall and Mary is smart”
Predicate Logic
• Predicate Logic extends Propositional
Logic by adding individuals, predicates,
and quantifiers
• Individuals: ‘John’, ‘Mary’
• Predicates: ‘tall’, ‘smart’
• Quantifiers: ‘all’, ‘some’
Just to Put Things in
Perspective
All arguments
All deductive arguments
All deductive arguments that
can be analyzed using the formal
logics we cover in class
(And I’m probably
optimistic here!)
Uses of Formal Logic
• Evaluation/Checking:
– Formal logic can be used to evaluate the validity of
arguments.
• Clarification/Specification:
– Formal logic can be used to express things in a
precise and unambiguous way.
• Demonstration/Proof:
– Formal logic can be used to figure out what follows
from a set of assumptions.
• Computation/Automated Reasoning:
– Formal logic can be used for machine reasoning.
Logic, Computers, and AI
• Formal logic has many connections to
computers:
– Computation: Formal logic played a crucial role in the
development of the notion of ‘computation’ (See my
class PHIL 4420 Computability and Logic)
– Circuit Design: Formal logic can be used for circuit
design (See CSCI 2500 Computer Organization)
– Artificial Intelligence: Formal logic is central to many
AI applications (See CSCI 4150 Artificial Intelligence)
Boolean Connectives
Propositional Logic
• Propositional Logic is the logic involving
complex claims as constructed from
atomic claims and connectives.
• Propositional Logic is not as powerful as
Predicate Logic, but it has some powerful
applications already.
Truth-Functional Connectives and
Boolean Connectives
• Connectives are usually called truth-functional
connectives:
– This is because the truth value of a complex claim
that has been constructed using a truth-functional
connective is considered to be a function of the truth
values of the claims that are being connected by that
connective.
– This is also why propositional logic is also called
truth-functional logic.
• For now, we will focus on three connectives:
and, or, not; these are called the Boolean
connectives.
Negation
• The claim “a is not to the right of b” is a complex
claim. It consists of the atomic claim “a is to the
right of b” and the truth-functional connective
“not”.
• We will call the above statement a negation.
• To express negations, we use the symbol ‘’
• ‘’ should be put in front of what you want to be
negated.
• If we symbolize the atomic claim “a is to the right
of b” as P, then the original claim will be
symbolized as: P
Truth-Table for Negation
• ‘’ is truth-functional, since the truth-value
of a negation is the exact opposite of the
truth-value of the statement it negates.
• We can express this using a truth table:
P P
T F
F T
Conjunction
• The claim “a is to the right of b, and a is in
front of b” is called a conjunction.
• The two claims that are being conjuncted
in a conjunction are called its conjuncts.
• To express conjunctions, we will use the
symbol ‘’
• ‘’ should be put between the two claims.
• Thus, the above statement can be
symbolized as: P  Q
Truth-Table for Conjunction
• ‘’ is truth-functional, since a conjunction is
true when both conjuncts are true, and it is
false otherwise.
• Again, we can show this using a truth
table:
P
T
T
F
F
Q PQ
T
T
F
F
T
F
F
F
Disjunction
• The claim “a is to the right of b, or a is in
front of b” is called a disjunction.
• The two claims that are being disjuncted in
a disjunction are called its disjuncts.
• To express disjunctions, we will use the
symbol ‘’
• ‘’ should be put between the two claims.
• Thus, the above statement can be
symbolized as: P  Q
Truth-Table for Disjunction
• ‘’ is truth-functional, since a disjunction is
true when at least one of its disjuncts is
true, and it is false otherwise.
• Again, we can show this using a truth
table:
P
T
T
F
F
Q PQ
T
T
F
T
T
T
F
F
Combining Complex Claims:
Parentheses
• Using the truth-functional connectives, we
can combine complex claims to make
even more complex claims.
• We are going to use parentheses to
indicate the exact order in which claims
are being combined.
• Example: (P  Q)  (R  S) is a
conjunction of two disjunctions.
Parentheses and Ambiguity
• An ambiguous statements is a statement whose
meaning is not clear due to its syntax. Example :
”P or Q and R”
• In formal systems, an expression like P  Q  R
is simply not allowed and considered
unsyntactical.
• Claims in our formal language are therefore
never ambiguous.
• One important application of the use of formal
languages is exactly this: to avoid ambiguities!
Exclusive Disjunction vs
Inclusive Disjunction
• Notice that the disjunction as defined by ‘’
is considered to be true if both disjuncts
are true. This is called an inclusive
disjunction.
• However, when I say “a natural number is
either even or odd”, I mean to make a
claim that would be considered false if a
number turned out to be both even and
odd. Thus, I am trying to express an
exclusive disjunction.
How to express Exclusive
Disjunctions
• We could define a separate symbol for exclusive
disjunctions, but we are not going to do that.
• Fortunately, exclusive disjunctions can be
expressed using the symbols we already have:
(PQ)  (PQ)
P
T
T
F
F
Q (P  Q)  (PQ)
T
T F F T
F
T T T F
T
T T T F
F
F F T F
!
Conditionals
The Material Conditional
• Let us define the binary truth-functional
connective ‘’ according to the truth-table
below.
• The expression P  Q is called a conditional. In
here, P is the antecedent, and Q the
consequent.
P
T
T
F
F
Q PQ
T
T
F
F
T
T
F
T
‘If … then …’ Statements
• The conditional is used to capture ‘if … then …’ statements.
• However, the match isn’t perfect. For example, we don’t want to
say that the claim “If grass is green then elephants are big” is
true just because grass is green and elephants are big, nor that
any ‘if … then’ statement is automatically true once the ‘if’ part is
false or the ‘then part true. The problem is that most English
‘if…then’ expressions aren’t meant to make a claim that is truthfunctional in nature.
• Still, any ‘if … then …’ statement will be false if the ‘if’ part is
true, but the ‘then’ part false, and the conditional captures at
least this important truth-functional aspect of any ‘if … then …’
statement.
• So, while we will from now on refer to the conditional as an ‘if …
then’ statement, we must be careful about the use of this, just as
care must be taken when applying Newtonian physics to some
situation.
Necessary and Sufficient
Conditions
• Conditionals can be used to express necessary
and sufficient conditions:
• Sufficient Condition: Something (P) is a sufficient
condition for something else (Q) iff P being the
case guarantees Q being the case. Hence, if we
know that P is true, we know that Q is true: P 
Q
• Necessary Condition: Something (P) is a
necessary condition for something else (Q) iff P
being the case is required for Q being the case.
Thus, while P may be true without Q being true,
we do know that if Q is true, P is true: Q  P
‘If’ vs ‘Only if’
• Sufficient conditions are expressed in
English using ‘if’, while necessary
conditions are expressed using ‘only if’.
• Thus:
– ‘If P then Q’: P  Q
– ‘P if Q’: Q  P
– ‘P only if Q’: P  Q
– ‘Only if P, Q’: Q  P
‘If and only if’ and the
Material Biconditional
• A statement of the form ‘P if and only if Q’ (or ‘P
iff Q’) is short for ‘P if Q, and P only if Q’. Hence,
we could translate this as (P  Q)  (Q  P).
However, since this is a common expression, we
define a new connective ‘’:
P
T
T
F
F
Q PQ
T
T
F
F
T
F
F
T
Logical Properties
Truth Tables
• Truth-tables can be used for:
– defining the truth-conditions of truth-functional
connectives
– evaluating the truth-conditions of any complex
statement
Tautologies
• A tautology is a statement that is
necessarily true.
• Example: P  P
P P  P
T TT F T
F F T TF
Contradictions
• A contradiction is a statement that is
necessarily false.
• Example: P  P
P P  P
T TF F T
F F F TF
Contingencies
• A contingency is a statement that can be
true as well as false
• Example: P
P P
T T
F F
Equivalences
• Two statements are equivalent if they have
the exact same truth-conditions.
• Example: P and P
P P P
T T TF T
F F F TF
Contradictories
• Two statements are contradictories if one
of them is false whenever the other one is
true and vice versa.
• Example: P and P
P P P
T T FT
F F TF
Implication
• One statement implies a second statement
if it is impossible for the second statement
to be false whenever the first statement is
true.
• Example: P implies P  Q
P
T
T
F
F
Q
T
F
T
F
P
T
T
F
F
PQ
T
T
T
F
Consistency
• A set of statements is consistent if it is
possible for all of them to be true at the
same time.
• Example: {P, P  Q, Q}
P
T
T
F
F
Q
T
F
T
F
P
T
T
F
F
PQ
T
T
T
F
Q
F
T
F
T
Consequence
• A statement is a consequence of a set of
statements if it is impossible for the
statement to be false while each statement
in the set of statements is true.
• Example: P is a consequence of {PQ, Q}
P
T
T
F
F
Q
T
F
T
F
PQ
T
T
T
F
Q
F
T
F
T
P
T
T
F
F
Validity
• An argument is valid if it is impossible for
the conclusion to be false whenever all of
its premises are true.
• Example: P  Q, Q  P
P
T
T
F
F
Q
T
F
T
F
PQ
T
T
T
F
Q
F
T
F
T
P
T
T
F
F
Implication, Consequence, Validity
• The notions of implication, consequence, and
validity are very closely related:
• A statement  implies a statement  if and only if
 is a consequence of the set of statements {}
• For implication and consequence we use the
symbol ‘’:
– If statement  implies statement  we write   
– If statement  is a consequence of a set of
statements {1, …, n}, we write {1, …, n}  
• An argument consisting of premises 1, …, n
and conclusion  is valid iff {1, …, n}  
• The terms implication, consequence and validity
can therefore be used interchangeably.
Summary
•
Logical properties of a single statement:
– Tautology: cannot be false
– Contradiction: cannot be true
– Contingent: can be true and can be false
•
Logical properties of 2 statements:
–
–
–
–
–
•
Equivalent: always the same truth-value
Contradictory: always opposite truth-values
Contrary: cannot be both true
Subcontrary: cannot be both false
Implication: implied statement cannot be false while the implying statement is
true
Logical properties of a set of statements:
– Consistent: can be all true at the same time
•
Logical properties of a set of statements in relation to a single statement:
– Consequence: statement cannot be false if all of the statements of the set are
true
•
Logical properties of an argument:
– Valid: conclusion cannot be false if all premises are true
But Wait!
• Consider the statement ‘a=a’
• This statement is a tautology, since it
is always true.
• However, since this statement does
not involve any truth-functions,
propositional logic considers this an
atomic statement, and symbolizes it
as ‘P’. But ‘P’ is a contingency.
• What is going on??
Truth-Functional Tautologies
• What is going on is that truth-tables only capture
the truth-functional aspects of sentences.
• So, a statement may be a tautology for reasons
other than truth-functional reasons. ‘a=a’ is an
example.
• A statement that is a tautology because of truthfunctional considerations is called a truthfunctional tautology.
• Notice that while every truth-functional tautology
is a tautology, not every tautology is a truthfunctional tautology (again, ‘a=a’ is a tautology,
but not a truth-functional tautology)
Normal Forms and
Expressive Completeness
Logically Equivalent Statements
• To express that two statements P and Q
are logically equivalent, we will write:
PQ
• ‘’ is not a symbol of F!!
Some Important Equivalences
• Double Negation:
– PP
• DeMorgan:
– (P  Q)  P  Q
– (P  Q)  P  Q
• Distribution:
–
–
–
–
P  (Q  R)  (P  Q)  (P  R)
P  (Q  R)  (P  Q)  (P  R)
(Q  R)  P  (Q  P)  (R  P)
(Q  R)  P  (Q  P)  (R  P)
More Equivalences
• Commutation:
– PQQP
– PQQP
• Association:
– P  (Q  R)  (P  Q)  R
– P  (Q  R)  (P  Q)  R
• Idempotence:
– PPP
– PPP
• Subsumption:
– P  (P  Q)  P
– P  (P  Q)  P
Even More Equivalences
• Implication:
– P  Q  P  Q
– (P  Q)  P  Q
• Transposition:
– P  Q  Q  P
• Exportation:
– P  (Q  R)  (P  Q)  R
• Absorption:
– P  Q  P  (P  Q)
• Equivalence:
– P  Q  (P  Q)  (Q  P)
– P  Q  (P  Q)  (P  Q)
Simplifying Statements I
• Using the principle of substitution of logical
equivalents, and using the logical equivalences
that we saw before (Double Negation,
Association, Commutation, Idempotence,
DeMorgan, Distribution, and Subsumption), we
can often simplify statements.
• Example:
(A  B)  A  (Commutation)
(B  A)  A  (Association)
B  (A  A)  (Idempotence)
BA
Generalized Conjunctions and
Generalized Disjunctions
• Recall the Association equivalences:
– P  (Q  R)  (P  Q)  R
– P  (Q  R)  (P  Q)  R
• Because of this, we’ll allow to drop brackets:
– PQR
– PQR
• Thus we can generalize conjunctions and
disjunctions
– A generalized conjunction (disjunction) can have any
number of conjuncts (disjuncts)
Simplifying Statements II
• The conjuncts (disjuncts) of a generalized
conjunction (disjunction) can be switched
around in any way you want. This really
helps with simplifying statements.
Example:
C  (A  (B  C))  (Distribution)
C  (A  B)  (A  C)  (Subsumption)
C  (A  B)
‘⊤’ and ‘⊥’
• A generalized conjunction is false if it has
at least one false conjunct, otherwise it is
true.
– So, a generalized conjunction with 0
conjuncts cannot have a false conjunct, and
hence cannot be false. Therefore, it is a
tautology! We will write this as ‘⊤’.
• A generalized disjunction is true if it has at
least one true disjunct, otherwise it is
false.
– Hence, a generalized disjunction with 0
disjuncts can never be true, and is therefore a
contradiction! We will write this as ‘⊥’.
Some equivalences involving ‘⊤’
and ‘⊥’
•
•
•
•
•
•
•
•
⊥  ⊤
⊤  ⊥
P⊥⊥
P⊤⊤
P⊤P
P⊥P
P  P  ⊥
P  P  ⊤
Simplifying Statements III
• Using ‘⊤’ and ‘⊥’, we can simplify
statements even more. Example:
(A  (B  (A  B))  (DeMorgan)
A  (B  (A  B))  (Double Neg.)
A  B  (A  B)  (Distribution)
(A  B  A)  (A  B  B) 
⊥⊥
⊥
Negation Normal Form
• Literals: Atomic Sentences or negations thereof.
• Negation Normal Form: An expression built up
with ‘’, ‘’, and literals.
• Using repeated DeMorgan and Double
Negation, we can transform any truth-functional
expression built up with ‘’, ‘’, and ‘’ into an
expression that is in Negation Normal Form.
• Example:
((A  B)  C)  (DeMorgan)
(A  B)  C  (Double Neg, DeM)
(A  B)  C
Disjunctive Normal Form
• Disjunctive Normal Form: A disjunction of
conjunctions of literals.
• Using repeated distribution of  over , any
statement in Negation Normal Form can be
written in Disjunctive Normal Form.
• Example:
(AB)  (CD)  (Distribution)
[(AB)C]  [(AB)D]  (Distribution (2x))
(AC)  (BC)  (AD)  (BD)
Conjunctive Normal Form
• Conjunctive Normal Form: A conjunction of
disjunctions of literals.
• Using repeated distribution of  over , any
statement in Negation Normal Form can be
written in Conjunctive Normal Form.
• Example:
(AB)  (CD)  (Distribution)
[(AB)  C]  [(AB)  D]  (Distribution (2x))
(AC)  (BC)  (AD)  (BD)
Truth-Functional Connectives
• So far, we have seen one unary truth-functional
connective (‘’), and two binary truth-functional
connectives (‘’, ‘’).
• Later, we will see two more binary connectives
(‘’, ‘’)
• However, there are many more truth-functional
connectives possible:
– First of all, a connective can take any number of
arguments: 3 (ternary), 4, 5, etc.
– Second, there are unary and binary connectives other
than the ones listed above.
Unary Connectives
• What other unary connectives are there
besides ‘’?
• Thinking about this in terms of truth tables,
we see that there are 4 different unary
connectives:
P
*P
P
*P
P
*P
P
*P
T
F
T
T
T
F
T
F
T
F
F
T
T
F
F
F
Binary Connectives
• The truth table below shows that there are
24 = 16 binary connectives:
P Q
P*Q
T T
T/F
T F
T/F
F T
T/F
F F
T/F
In general:
n sentences 
2n truth value combinations
(i.e. 2n rows in truth table) 
2n
2 different n-ary connectives!
Expressing other connectives
using ‘and’, ‘or’, and ‘not’
• We saw that we can express the exclusive
disjunction using ‘and’, ‘or’, and ‘not’.
• Q: Can we express all other connectives as
well?
• A: Yes! We can generalize from this example:
P Q
P*Q
T T
F
Step 1:
T F
T
 PQ
F T
T
 PQ
F F
F
Step 2:
 (PQ)  (PQ)
Truth-Functional Expressive
Completeness
• Since I can express any truth function using ‘’,
‘’, and ‘’, we say that the set of operators {,
, } is (truth-functionally) expressively
complete.
• DeMorgan Laws:
– (P  Q)  P  Q
– (P  Q)  P  Q
• Hence, by the principle of substitution of logical
equivalents, since {, , } is expressively
complete, the sets {, } and {, } are
expressively complete as well!
Applications: Computer
Hardware and Software
• The results that we have seen on the
previous slides have important
applications in both computer hardware
and software:
– Digital Circuits
– Machine Reasoning
Logic and Computer Circuitry
1’s and 0’s
• All what modern digital computers do is
transform strings of 1’s and 0’s, called bitstrings.
• Information is represented using 1’s and 0’s, and
information is processed through the
manipulation of those bitstrings.
• The 1’s and 0’s can be physically realized using
any kind of physical dichotomy. We can
therefore use pure mechanics (levers, pulleys,
punchcards, etc.), electronics, optics, DNA,
quantum physics, toilet paper and pennies, or
just about anything else to physically implement
the 1’s and 0’s.
Logic Gates
• To process information, bitstrings need to be
manipulated.
• Thus, depending on whatever way the 1’s and
0’s are physically implemented, there needs to
be a device to change those physical
representations.
• But, we are not going to be interested in the
physical nature of these devices, since this is
just an issue of implementation.
• Rather, we are going to think of these devices as
logic gates: thingamabobs that transform
bitstrings into other bitstrings.
‘And’, ‘Or’, and ‘Not’ Gates
In1
In2
In1
In2
In
Out
Out
Out
Out
In1
In2
1
1
1
1
0
0
0
1
0
0
0
0
In1
In2
1
1
1
1
0
1
0
1
1
0
0
0
In
Out
1
0
0
1
Out
Representing Numbers
• We normally represent numbers using the
decimal system. That is, we take 10 as our
base number to represent numbers.
• Example:
53627
7*100 =
7*1 =
7
2*101 =
2*10 =
20
6*102 = 6*100 = 600
3*103 = 3*1000 = 3000
5*104 = 5*10000 = 50000
53627
Binary Numbers
• Binary numbers take 2 as their base.
• Example:
10110
0*20 = 0*1 = 0
1*21 = 1*2 = 2
1*22 = 1*4 = 4
0*23 = 0*8 = 0
1*24 = 1*16 = 16
22
Adding Binary Numbers
111
1 1
11010110
10110011
110001001
Computing Binary Addition
• To compute the addition of two binary
numbers, we need to implement the
following architecture:
In10
In20
+
Out0
Carry1
In11
In21
+
Out1
Carry2
In12
In22
+
Out2

2 Bit and Carry Adder
Carryn
In1n
In2n
Outn
+
Carryn+1
In1n
In2n Carryn
Outn
Carryn+1
1
1
1
1
1
1
1
0
0
1
1
0
1
0
1
1
0
0
1
0
0
1
1
0
1
0
1
0
1
0
0
0
1
1
0
0
0
0
0
0
Circuitry for the 2 Bit and Carry
Adder (output bit)
In1n
In2n
Carryn
Outn
Circuitry for the 2 Bit and Carry
Adder (carry bit)
In1n
In2n
Carryn
Carryn+1
Simplifying
• While the Disjunctive Normal Form
provides us with a working circuitry (and
thus guarantees us of one!), this circuitry
may not be the most efficient one.
• Carryn+1 = (In1n  In2n)  (In1n  Carryn) 
(In2n  Carryn)
• Outn = (In1n XOR In2n) XOR Carryn
– where P XOR Q = (P  Q)  (P  Q)
An Interesting Trade-Off
• To keep production costs down:
– Use as few gates as possible
– Use as few different kinds of gates as
possible
• However, there is a trade-off between
these two objectives: The fewer the
number of kinds of gates one uses, the
more gates of those kinds are needed.
Example of the Trade-Off
• The Disjunctive Normal Form tells us that we
can build any circuit using only 3 kinds of gates.
• With more types of gates (e.g. the XOR), we
could have saved on the total number of gates.
• On the other hand, because of the DeMorgan’s
Law, we know that we can express any
expression using only ‘and’ and ‘not’. Thus, we
can also try and cut down on the number of
types of gates, but this will mean an increase in
the number of gates.
The NAND
• Let us define the binary truth-functional
connective ‘NAND’ according to the truth-table
below.
• Obviously, P NAND Q  (P  Q) (hence the
name!)
P
T
T
F
F
Q P NAND Q
T
F
F
T
T
T
F
T
Expressive Completeness of the
NAND
• The NAND has a very interesting property, in
that it can express any truth-functional
connective, i.e. {NAND} is expressively
complete!
• Proof: We already know that we can express
every truth-functional connective using only 
and . Furthermore:
– P NAND P  (P  P)  P
– (P NAND P) NAND (Q NAND Q)  ((P NAND P) 
(Q NAND Q))  (P  Q)  P  Q
• In other words, we can build circuitry using only
one kind of logic gate!! Of course, the drawback
is that we need many of those gates.
The Miniac
• Behold! The world’s most powerful computer
that fits in the palm of your hand: a penny!!
• Instructions: Ask any question with a yes or no
answer. Flip the coin. Tails means ‘yes’ and
heads means ‘no’. To see whether the Miniac’s
answer is correct or incorrect, flip the coin a
second time, asking: “Is your answer to this
question just as correct as your answer to the
previous question?”
• Question: How does this work?
Formal Proofs
Demonstrating Invalidity
• To demonstrate invalidity one has to show
that it is possible for all premises to be true
and the conclusion to be false all at the
same time.
• One way to do this is to come up with a
possible scenario (or possible world) in
which all premises are true and the
conclusion false (Tarski’s World). This is
called a counterexample.
Demonstrating Validity
• To demonstrate validity, we have to show that there is no
possible way for all premises to be true and the
conclusion false all at the same time.
• Showing a scenario in which all premises are true, and in
which the conclusion is true as well, does not
demonstrate validity, b/c there may still be a different
scenario in which all premises are true and the
conclusion false.
• Of course, we could try and generate all possible worlds,
but this method is either impractical (b/c there are too
many possible worlds), or simply impossible (b/c there
are infinitely many possible worlds).
Proofs
• OK, so what do we do? Well, we can do what we
do in everyday reasoning: we start with the
premises, and we gradually work our way to the
conclusion: “Either the housemaid or the butler
killed Mr. X. Now, we know that if the housemaid
would have done it, the alarm would have gone
off. But, the alarm did not go off. Therefore, the
housemaid did not do it. So, since it was either
the housemaid or the butler, it must have been
the butler.”
Intermediate Results
• The previous argument had 3 premises:
– 1. Either the housemaid or the butler did it.
– 2. If the housemaid did it, the alarm would have gone
off.
– 3. The alarm did not go off.
• The conclusion was: The butler did it.
• We combined premises 2 and 3 to get an
intermediate result: The housemaid did not do it.
• We then combined the intermediate result with
premise 1 to get the conclusion.
• We use intermediate conclusions because
without them, the inference from the premises to
the conclusion may not be obvious, but with
them, each of the steps does become obvious.
‘Obvious’
• In formal proofs, we try and formalize this stepby-step inference process, where each inference
is obvious.
• OK, but ‘obvious’ is a bit of a vague term, as
what is obvious to some, may not be obvious to
others. So, what are going to count as ‘obvious’?
• We are going to play it safe: In formal proofs, we
are only going to allow steps that are about as
obvious as we can get. Thus, we are only going
to allow ‘baby inferences’.
• So, in formal proofs, bigger inferences, which
may still be obvious to many (if not all of us), will
still have to be broken up into smaller ones!
Inference Rules
• Formal systems of logic come with a finite
set of inference rules that reflect ‘baby
inferences’.
• There are many formal systems of logic,
each with their own set of inference rules:
– The nature of the inference rules depends on
the symbols that the system uses to express
statements.
– However, even if two systems use the same
symbols, they may still have different
inference rules.
F:
A ‘Fitch’-style Deductive System
• The formal system that our book uses is called
F.
• F has 2 inference rules for each connective:
– Introduction: A rule to infer a statement with that
connective as its main connective
– Elimination: A rule to infer something from a
statement with that connective as its main connective.
• Formal systems with these two types of
inference rules are called ‘Fitch’-style systems.
• Warning: While Fitch-style systems are
mathematically elegant, they are not always very
user-friendly. In particular, it does not contain
inference rules that reflect some ‘obviously’ valid
inferences!
The Structure of Proofs in F
• A formal proof in F will look like this:
n
P1

Pn
n+1
I1
1
Justification 1

n+m
Im
Justification m
n+m+1
C
Justification m+1
Justification
• In a formal proof, you have to indicate from which
premises or intermediate results you infer the new
statement. Thus, each step needs to have its own
justification.
• Inference rules may need any number of statements
from which the new statement is inferred (though with
too many statements, the rule may no longer be
considered ‘obvious’).
• Most inference rules require one or two statements.
• Some inference rules require no statements at all. This is
when the inferred statement is unconditionally true.
• To help refer to previous statements, we are going to
number the statements.
 Elim
• Conjunction Elimination ( Elim) allows
one to infer any conjunct from a
conjunction.
P1  P2  …  Pn

Pi
 Intro
• Conjunction Introduction ( Intro) allows
one to conjunct any number of previously
established statements in any order.
P1

Pn
P1  P2  …  Pn
 Intro
• Disjunction Introduction ( Intro) allows
one to construct any disjunction using a
previous result as one of its disjuncts.
Pi

P1  …  Pi  …  Pn
 Elim
• Negation Elimination ( Elim) allows one
to infer P from  P:
P

P
Do we Have Free Will?
• Either determinism is true or not. Now, if
determinism is true, then my actions
cannot be otherwise from what they are,
i.e. I don’t any freedom to exert my will. On
the other hand, if indeterminism is true,
then my actions are partly determined by
pure randomness, so there is no such
thing as a will that is in total control of my
actions. Either way, I don’t have free will.
Proof by Cases
• The proof we just saw follows a certain pattern:
‘Either P is the case or Q is the case. However, if
P is the case, then S is the case, and if Q is the
case, then S is the case as well. Either way, S is
therefore the case. Hence, S is the case.’
• This pattern of reasoning is called Proof by
Cases
• Obviously, the above pattern can be generalized
to disjunctions with any number of disjuncts.
• However, a very common form is to start with:
‘Either P is the case or P is not the case’.
Is Space Continuous?
• Suppose space is continuous.
• Then between any two (different) points A and B
there exist infinitely many other points.
• Thus, in order to move from any point A to any
other point B, you have to completely go through
a sequence of infinitely many points. But, you
can never reach the end of an infinite sequence.
Hence, motion is impossible.
• But, things do move.
• Contradiction!
• So, space is not continuous. (thanks to Zeno!)
Proof by Contradiction
• The proof we just saw relied on the
following pattern: ‘Assuming P to be the
case, then I get some kind of impossibility
or contradiction. Hence, contrary to my
assumption, P cannot be the case.’
• This pattern of reasoning is called Proof by
Contradiction (or Indirect Proof or
Reductio ad Absurdum or simply
Reductio).
Proof by Cases and
Proofs by Contradiction
• Proof by cases and proof by contradiction are
two important proof techniques that we would
like to formalize.
• But, these proof techniques do not work by
inferring some statement from some other
statement(s).
• Rather, they work by pointing to the fact that I
am able to infer something from something else.
Subproofs
• At any time during a proof, a subproof may be
started by making an additional assumption
which can then be used to draw further
inferences.
• The subproof may be ended at any time. When it
is ended, the individual statements from the
subproof can no longer be used to infer others.
• Subproofs demonstrate that certain statements
can be inferred when an additional assumption
is made, and this result can be used in the proof
itself. That is, the subproof as a whole can be
used to infer other statements.
Formalizing Proof by Cases using
Subproofs and ‘’
• Using subproofs, we can now formalize the
Proof by Cases technique:
– You have a disjunction P1  …  Pi  …  Pn
• These are the possible cases
– You start a subproof for each of the possible disjuncts
• This is going through each of the cases (what if P1 is the
case?; what if P2 is the case?, etc.)
– You infer the same statement (Q) in all subproofs
• This show that in all cases, the same thing (‘Q’) can be
inferred
– You now point to the initial disjunction and all the
relevant subproofs to conclude Q
 Elim
• Disjunction Elimination ( Elim) is the
formal counterpart of Proof by Cases:
P1  …  Pi  …  Pn
P1

S

Pn

S
S
Subproofs and Scope
• An additional line is used to indicate the start
and end of the subproof.
• The line can also be seen as the scope of the
additional assumption made at the start of the
subproof: every statement within that scope is
inferred from the truth of that assumption and all
previous assumptions.
• The line of the proof itself can be seen in exactly
this way as well. Therefore, there is no real
difference between subproofs and proofs.
Subproofs within Subproofs
• Within any subproof, another subproof can
be started.
• Subproofs within subproofs must be ended
before the original subproof is ended.
• The general rule is: one can use as
justification all and only statements that is
either one of the assumptions whose
scope one is working in, or some
statement inferred from those.
Formalizing Proof by Contradiction
using subproofs and ‘’
• We can now formalize the Proof by
Contradiction
– Start a subproof, and assume P
• All proofs by contradiction start by assuming
something, and this is the opposite of what you
want to prove!
– In the subproof, derive 
• This shows that assuming P leads to a
contradiction
– Point to the subproof, and conclude P
 Intro
• Negation Introduction ( Intro) is the
formal counterpart of Proof by
Contradiction:
P


P
 Intro
•  Introduction ( Intro) allows one to infer
 from a pair of statements P and P:
P

P


More on contradictions
• Theorem: For any statement P:   P
• Proof: It is impossible for  to be true, so it
is impossible for  to be true and P to be
false, and hence for any P:   P
• In other words: Anything is a logical
consequence from a logical contradiction!
 Elim
•  Elimination ( Elim) allows one to infer
any statement P from :


P
 Elim
• Conditional Elimination ( Elim) allows
one to infer the consequent of a
conditional, given the truth of its
antecedent:
PQ

P

Q
This pattern is better known as
Modus Ponens
Conditional Proof
• We have seen two uses of subproofs: for Proofs
by Contradiction, and for Proofs by Cases.
• A third use for subproofs is to do a Conditional
Proof.
• A Conditional Proof infers some kind of
conditional P  Q from a given set of
statements by making P an extra assumption,
and trying to infer Q from the given statements
and the additional assumption P.
 Intro
• Conditional Introduction ( Intro) is the
formal counterpart of Conditional Proof:
P

Q
PQ
Modus Tollens
Pattern:
Proof:

1.


2.


3.

4.

1,3  Elim
5.

2,4  Intro
6. 
3-5  Intro
Disjunctive Syllogism
Pattern:
Proof:

1.


2.


3.

4.
5.


6.

7.

2,3  Intro
4  Elim
1,3-5,6-6  Elim
Fitch
• Fitch is the program that allows the user to
construct formal proofs in F.
• Fitch has a number of additional features:
– Checks whether a rule is applied correctly
– Allows shortcuts that are not allowed in F
– Provides CON rules
Metalogic
Logic and Metalogic
• Metalogic is the study of logic. That is, where logic
has no specific subject matter (logic can be applied in
any field where reasoning takes place), the subject
matter of metalogic is logic itself.
• Metalogic makes claims about logical properties and
relationships. For example: “A statement is a
tautology if and only if its negation is a contradiction”
is a metalogical claim.
• Of course, the paradox of metalogic is that it needs
logic to support the claims it makes about logic!
Hmmm…
A Central Metalogical Result:
Consequence as a Central Notion
• Many interesting logical properties can be
expressed in terms of logical consequence. For
example:
– Tautology: A statement  is a tautology iff {}  
– Contradiction: A statement  is a contradiction iff
⊥
– Equivalence: Two statements  and  are
equivalent iff    and   
– Inconsistency: A set of statements {1, …, n} is
logically inconsistent iff {1, …, n}  
Consequence and Formal Proof
• Since formal proofs can be used to demonstrate consequence,
and since consequence can be used to demonstrate other
logical properties, formal proofs can be used to demonstrate
these other logical properties:
– Tautology:
• To prove that something is a tautology, derive that statement from an
empty set of premises.
– Contradiction:
• To prove that a statement is a contradiction, derive  from that statement
as a premise.
– Equivalence:
• To prove that two statements P and Q are equivalent, do two proofs:
– First, assume P as a premise, and derive Q.
– Second, assume Q as a premise, and derive P.
– Inconsistency:
• To prove that a set of statements is inconsistent, assume all those
statements as premises, and derive .
Talking about Proofs …
• How do we know if a formal proof does what it is
supposed to do? That is, if I can derive a sentence 
from a set of sentences {1, …, n}, does that really
mean that  is a truth-functional consequence of {1,
…, n}?
• Notice that this is a metalogical questions: it asks
something about formal proofs in relation to a logical
property. But of course, we want to settle this
question through the use of a rigorous proof, i.e. we
want to prove something about formal proofs!?!
Modus Bogus
• In order to demonstrate that the question on the
previous slide is an interesting and meaningful
question, consider the following rule:
PQ

P

Q
Obviously, a formal proof system that would
contain this rule would be able to ‘prove’
things that just don’t follow!
TF
• Recall:  is a truth-functional consequence of {1,
…, n} iff according to truth-functional properties it
is impossible for  to be false if each i is true.
• Let us use the symbol TF to indicate truthfunctional consequence:  TF  iff  is a truthfunctional consequence of .
• Remember: If  TF  then   , but not vice
versa. E.g. LeftOf(a,b)  RightOf(b,a), but not
LeftOf(a,b) TF RightOf(b,a).
Truth-functional Provability
• Let us define truth-functional provability with
regard to some formal deductive logic system
S (e.g F) as follows: Q is truth-functionally
provable from a set of premises {P1, …, Pn} in
the system S iff there exists a formal proof in S
going from P1, …, Pn as premises and Q as the
conclusion using the rules for ‘’, ‘’, ‘’,
‘’, ‘’, and ‘’ (or any other truthfunctional connective defined by S).
TF(S)
• Let us use the symbol TF(S) to indicate truthfunctional provability in S:  TF(S)  iff  is
truth-functionally provable from  in the
system S.
• The subscript TF(S) indicates that we restrict
our proofs to the truth-functional rules of S.
Two Very Important Properties
• For every deductive system of formal logic S
we can define the following 2 properties:
– 1. Truth-Functional Deductive Soundness: A
system S is truth-functionally deductively sound iff
for any  and :
• if  TF(S)  then  TF 
– 2. Truth-Functional Deductive Completeness: A
system S is truth-functionally deductively
complete iff for any  and  :
• if  TF  then  TF(S) 
F is Sound
• Q: Is F truth-functionally deductively sound?
• A: Yes!
• If you want the full proof, take ‘Intermediate
Logic’.
A Trivially Sound System
• Let S be a logic system that has no inference
rules. Then, trivially, all inference rules of S
are sound. Hence S is deductively sound as
well.
• In other words, it is trivial to make a
deductively sound logic system: just don’t
define any inference rules!
Deductive Completeness
• Q: Why is truth-functional deductive completeness
important?
• A: If a logic system S is not complete, then for certain
 and ,  TF  but not   TF(S) . So, although
 is a truth-functional consequence of , the rules do
not allow one to prove  from !
• Q: How is that possible?
• A: Easy. While P TF P, the system S will not
be able to prove this. Hence, S is not a truthfunctionally deductively complete system!
Proving Deductive Completeness
• Proving completeness for some logic system S can be
very difficult. This is not hard to understand: one
needs to prove that for every  and  such that  
, there exists a proof in S going from  to . But,
there are an infinite number of pairs  and  such
that   , and the proofs don’t seem to follow any
kind of systematic pattern.
• Note: If we already know a certain system S’ to be
complete, we can try to prove S to be complete by
demonstrating how S can prove anything that S’ is
able to prove.
F is Complete
• Q: Is F truth-functionally deductively
complete?
• A: Yes, but the proof for this is rather
complicated and outside the scope of this
class. Again, if you’re interested, take
‘Intermediate Logic’.
An Algorithm for
Producing Formal Proofs
• Restricting ourselves to , , , and  only:
– Start a proof by contradiction (assume negation of
conclusion). Let  be the set of statements you have. Then
go through the following loop to obtain  (keep adding
results to ):
•
•
•
•
•
•
•
If  : stop
If    and   : get  by  Intro
If   : get  by  Elim
If (1  …  n )  : get 1  …  n by DeMorgan pattern
If (1  …  n )  : get 1  …  n by DeMorgan pattern
If 1  …  n  : get 1, … , n by  Elim
If 1  …  n  : set up subproof for each 1 and derive  from
 {1} / {1  …  n} with this same method. Then get  with
 Elim
Hokus Ponens
• The logic system B (‘Bram’) contains only one
inference rule, called Hokus Ponens:

P
Woohoo!
One line proofs!
This system is going to make me famous!
(Only one small problem …)
Automated Theorem Proving
Systematic Procedures
• A systematic procedure is a procedure that
follows a certain step-by-step algorithm to
perform a certain task.
• Examples are cookbook recipes and computer
programs.
• A systematic procedure will either stop after a
finite amount of time, or never stop (e.g.
because it goes into an infinite loop).
• The truth-table method is a systematic
procedure.
• The method of formal proof is not a systematic
procedure (though it can be used to make one).
Positive Tests, Negative Tests,
and Full Tests
• A positive test is a systematic procedure that tries to
figure out whether certain things have a certain property.
• A negative test is a systematic procedure that tries to
figure out whether certain things do not have a certain
property.
• A full test is a systematic procedure that tries to figure
out whether certain things do or do not have a certain
property.
• The truth table method is a full test. It can answer that
something is or is not a logical consequence of
something else.
• The formal proof method, even if it were systematic, is
not a full test, but only a positive test. It can answer that
something is a logical consequence of something else,
but it never concludes that it isn’t a logical consequence.
Soundness and Completeness for
Tests and Decision Procedures
• A test (positive, negative, or full) is sound
iff:
– if the test claims that something has (not) a
certain property, then it has (not) that property.
• A test (positive, negative, or full) is
complete iff:
– If something has (not) a certain property, then
the test claims that it has (not) that property.
Decision Procedures
• A full test that is both sound and complete is
called a decision procedure.
• The truth-table method is a decision procedure
for truth-functional consequence.
• The Taut Con ‘rule’ (mechanism!) is also a
decision procedure for truth-functional
consequence.
• Questions: Is there a decision procedure for TF
consequence that is more efficient than the
truth-table method. In fact, how does Taut Con
work?
Truth-Trees
Logical Possibility
• All logically interesting claims can be reduced to
questions about logical possibility:
– Logical Consistency: Is it possible for all statements
to be true?
– Logical Validity: Is it possible for all premises to be
true and the conclusion false?
– Logical Consequence: Is it possible for the implying
statements to be true and the implied statement to be
false?
– Logical Equivalence: Is it possible for the two
statements to have a different truth value?
– Logical Tautology: Is it possible for the statement to
be false?
Truth Table Method
• The truth table method systematically exhausts
all possible truth value combinations of the
statements involved.
• In the truth-table we look for a row that reflects a
certain possibility, and that will tell us the answer
to whatever question we had (e.g. if there is no
row where statement is false, then it is not
possible for that statement to be false, and
hence it is a tautology).
Drawback and Room for
Solution
• A drawback of the truth table method is that the
number of rows grows exponentially.
• Fortunately, there is room for a solution to this
problem. Since all we are interested in, is the
existence of a specific combination of truth
values of the statements involved, all we need to
find is one example of such a case. Once we
have found such a case, there is no need to
exhaust all other cases.
Simple Solution: Stopping Early
• One solution to the problem of big truth tables is
therefore to simply stop once you have found a
row that represents the combination of truth
values you are interested in.
• Thus, rather than working out a truth table
column by column, you may want to do it row by
row, so that you can stop as soon as you have
found a row of the kind you are looking for.
• A big drawback of this approach is that if no row
of the kind you are looking for exists, then you
have to complete the whole truth table after all.
A More Focused Search
• Consider the following argument:
P  (Q  R)
R  Q
R
• We are interested in whether all premises can be
true and the conclusion false:
– In order for the conclusion to be false, R must be false.
– In order for the second premise to be true while R is
false, Q must be false.
– In order for the first premise to be true while Q and R
are both false, P must be false.
The Short Truth Table Method
• The Short Truth Table Method assigns truth
values to the involved atomic and complex
statements in order to try and obtain a certain
combination of truth values:
P  (Q  R)
F T F FF
RQ
F TTF
/
R
F
• The Short Truth Table Method thus tries to
generate one row of the truth table that has
the combination of truth values you are
interested in.
Short Truth Table Method and
Indirect Proof
• As you assign truth values to certain statements,
the truth values of other statements can be
forced.
• If you are forced to make a statement both true
and false, then you know that the combination of
truth values you are looking for does not exist:
P  (Q  P)
TF TF F
 Contradiction, so the
statement is a tautology!
• The short truth table method is therefore a kind
of indirect proof (proof by contradiction), except
that you don’t always get a contradiction.
Drawback of the Short Truth
Table Method
• A drawback of the short truth table method is
that you are not always forced to assign any
further truth values:
R  (Q  P)
T T
(Q  R)  P
T
T
RP
T
Q
T
• At this point, you can choose to assign certain
truth values, but if your choice does not lead
to the row you are looking for, then you need
to try a different option, and the short truth
table method has no tools to do go through all
of your options in a systematic way.
Truth Trees
• The obvious solution to the drawback of the
short truth table method is to incorporate tools to
systematically keep track of multiple options.
• One method that does so is the truth tree
method:
– The truth tree method tests for the consistency of a
set of statements and, as such, can be used to
determine validity, tautologies, equivalence, etc.
– Like the short table method, it infers which other
statements are forced to be true under this
assumption.
– When nothing is forced, then the tree branches into
the possible options.
Truth Tree Example
(((PQ)R)  (P(QR))) 
(PQ)R 
(P(QR)) 
P
(QR) 
Q
((PQ)R) 
P(QR) 
PQ 
All branches close
R
 the original
P
statement cannot be
Q
false  tautology!
R
(PQ)  R
×
P Q
×
×
P
×
Q  R 
Q
×
R
×
Decomposition Rules for Truth
Trees
P 
P
PQ 
P
Q
PQ 
PQ 
P
Q
P
(PQ) 
P
Q
Q
PQ 
P
Q
P
Q
(PQ) 
P
Q
(PQ) 
P
Q
(PQ) 
P
Q
P
Q
Truth Trees as Decision
Procedures
• The truth tree method can easily be made
into a systematic procedure.
• As such, the truth tree method becomes a
decision procedure for truth-functional
consequence that is, on average, quite a
bit more efficient than the truth-table
method.
Resolution
Running Example
• As a running example, let us consider the
following argument with premises:
– Horned(a)
– Horned(a)  Elusive(a)
– Dangerous(a)  Horned(a)
– (Elusive(a)  Mythical(a))  Rare(a)
– Dangerous(a)  (Elusive(a)  Rare(a))
• And conclusion:
– Rare(a)
Horn Clauses
• A Horn clause is a disjunction of literals
with at most one positive literal (i.e. atomic
statement).
• We can differentiate between 3 types of
Horn Clauses:
–Q
– P1  …  Pn
– P1  …  Pn  Q
– (Q and each of Pi is atomic)
Efficient Check of Consistency
of a Set of Horn Clauses
• Horn Clauses form an important class of
statements, since the consistency of a set
of Horn clauses can be checked in a
systematic and efficient manner. E.g. the
short truth table method will always work:
– 1. Write ‘T’ under all clauses.
– 2. Make all forced moves until:
• A. You get a contradiction: The set is inconsistent.
• B. Nothing is forced anymore: The set is
consistent.
Example
Putting into Horn Clauses:
Horned(a)  H
Horned(a)  Elusive(a)  H  E
Dangerous(a)  Horned(a)  D  H
(Elusive(a)  Mythical(a))  Rare(a)  E  M  R
Dangerous(a)  (Elusive(a)  Rare(a))  D  E  R
Negating Conclusion: Rare(a)  R
Short Truth Table:
H  E D  H
H
T
F T T T T T FT
E  M  R
FTTTFTF
D  E  R
FTTFTTF
R
TF
Horn Clauses in Conditional
Form
• Horn clauses can be written in conditional
form:
–Q
– (P1  …  Pn)  
– (P1  …  Pn)  Q
Algorithm for Horn Clauses in
Conditional Form
• To check the consistency of a set S of
Horn Clauses in conditional form:
– 1. Make a new set of statements T, starting
with the set of all atomic statements Q.
– 2. If there is a statement (P1  …  Pn)   in
S, and each Pi is in T, then you can stop: S is
inconsistent. Otherwise, go to 3.
– 3. If there is a statement (P1  …  Pn)  Q in
S, and each Pi is in T, then add Q to T, and go
to 2. Otherwise stop: S is consistent.
Example
Putting into conditional form Horn:
H  H
H  E  H  E
D  H  H  D
E  M  R  (E  M)  R
D  E  R  (D  E)  R
R  R  
{H}  {H,E}  {H,E,D}  {H,E,D,R}  
Prolog
• The programming language Prolog is based on
Horn clauses.
• A Prolog program consists of 2 types of lines:
– Facts: These are Horn clauses of type Q. In Prolog: P.
– Rules: These are Horn clauses of the form (P1  … 
Pn)  Q. In Prolog: P :- P1 , … , Pn.
• A Prolog program is run by asking whether some
atomic statement Q follows from the facts and
rules. In Prolog: Q?
• The Prolog program will answer ‘Yes’ or ‘No’.
The Prolog Algorithm
• Prolog checks whether Q follows from the facts
or rules as follows:
– 1. Make a set of goals G, starting with Q.
– 2. If G is empty, stop with answer ‘Yes’.
– 3. If a statement P is in G that is a fact, remove P
from G.
– 4. If P is in G and there is a rule P :- P1 , … , Pn, then
remove P from G, and add each Pi to G.
– 5. If you get stuck, try a different rule P :- P1 , … , Pn.
– 6. If all options fail, stop with answer ‘No’.
Prolog Example
Putting into Prolog:
Query: R?
H  H
{R}
H  E  E :- H.
H  D  D :- H.
(E  M)  R  R :- E, M.
{E, M}
{D, E}
{H, M}
{H, E}
{M}
{E}
(D  E)  R  R :- D, E.
{H}
{} ‘Yes’!
Resolution
• While the consistency of a set of Horn Clauses
can be checked systematically and efficiently,
not all statements are Horn Clauses, nor can
they always be transformed into Horn Clauses.
• However, all sentences can be put into
Conjunctive Normal Form (CNF).
• Horn Clauses are a special case of claims in
CNF.
• The algorithms for Horn Clauses are a special
case of the more general method of resolution
as defined over any set of statements in CNF.
The Rule of Resolution
• The Rule of Resolution is defined over
disjunctions of one or more literals:
P1  …  Pi-1  X  Pi+1  …  Pm
Q1  …  Qi-1  X  Qi+1  …  Qn
P1  …  Pi-1  Pi+1  …  Pm 
Q1  …  Qi-1  Qi+1  …  Qn
(each of Pi and Qi are literals; X is atomic)
Clauses
• A clause is a set of literals.
• Assuming a clause to represent a disjunction of
all literals that are in that clause, we can resolve
two clauses as follows:
{P1 , … , Pi-1 , X , Pi+1 , … , Pm}
{Q1 , … , Qi-1 , X , Qi+1 , … , Qn}
{P1 , … , Pi-1 , Pi+1 , … , Pm ,
Q1 , … , Qi-1 , Qi+1 , … , Qn}
(each of Pi and Qi are literals; X is atomic)
The Method of Resolution
• The method of resolution checks whether
some set of statements S is consistent. It
does this as follows:
– 1. Make a set T of clauses representing all
conjuncts of the CNF of each statement in S.
– 2. Resolve any two clauses from T that can be
resolved, and add the result to T.
– 3. If two clauses resolve to the empty set,
stop: the original set of statements was
inconsistent.
Resolution Example
(Elusive(a)  Dangerous(a))  (Elusive(a)  Rare(a))
(Rare(a)  Dangerous(a))  Horned(a)
Horned(a)  Magical(a)
Negate Conclusion
and put into CNF
Magical(a)
(E  D)  (E  R)
(R  D)  H
HM
M
(E  D)  (E  R)
(R  D)  H
H  M
(R  D)  H
(R  H)  (D  H)
Resolution Example (Cont’d)
(E  D)  (E  R)
{E, D}
{E, R}
{E, H}
From CNF to
clauses and resolve
(R  H)  (D  H)
H  M
M
{R, H}
{H, M}
{M}
{D, H}
{H}
{E, H}
{H}
{}
Inconsistent, so valid!
Soundness and Completeness
of Resolution
• The method of Resolution is sound and
complete with regard to truth-functional
consistency in the sense that:
– If the method finds a set of statements to be
inconsistent, then that set of statements is
indeed inconsistent (soundness).
– If a set of statements is inconsistent, then the
method can find that set of statements to be
inconsistent by deriving the empty clause
(completeness).
Algorithms for Resolution and
ATP’s
• Algorithms for resolution will differ in the
order in which clauses get resolved.
• Many ATP’s are based on some such
algorithm:
– Snark
– Otter
– Taut Con in Fitch
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