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Transcript
```Computing Fundamentals 1
Lecture 2
Lecturer: Patrick Browne
http://www.comp.dit.ie/pbrowne/
Room K308
Based on Chapter 2.
A Logical approach to Discrete Math
By David Gries and Fred B. Schneider
Logic
• ‘Formal logic’ is not a phrase that attracts. Everybody,
perhaps, would like to be logical — but not too logical,
in case they become unfeeling like Star Trek’s Mr
Spock... Hardly anybody wants to be formal.,. We’d
rather be informal, casual, easy, and logical only up to
a point.
• But formal logic is hot stuff, because it is the machinery
in the engine room of computing. Computers do very
simple formal logical reasoning, and can’t do anything
else. The programming languages that drive those
computers are formal logics. The protocols that drive
the internet and the grid are formal logics too.
Computers do what they do as well as they do because
we know quite a lot about formal logic, and they keep
falling over because we don’t yet know quite enough.
From: Proof and Disproof in Formal Logic by Richard
Boolean Expressions
• Boolean constants: true and false.
• Boolean variables can only have values true
or false.
• Boolean operators: ,, ,  , , ˅, ˄.
• Expressions are of type Boolean (or sort Bool
in CafeOBJ, or type bool in Python).
Truth table for negation () and
identity
(id)
There are as many unary truth-functions as there are different ways to
fill in a two row column in a truth-table. For unary connectives, there
are 2^(2^1)=4, for binary connectives, 2^(2^2)=16, for ternary
connectives, 2^(2^3), and so on)
Unary Ops
id
not

Argument
true
true true
false false
false
true false true
false
Unary Op. Truth Table
• The previous TT provides a complete
enumeration of the Boolean operations with
one argument (unary operators). The first and
last operation results do not depend on the
argument, they have no names. The identity
(id) operation returns the argument, and
negation or not operation () returns the
constant that is not the argument. The truth
table for Boolean operations with two
arguments is shown in the next slide.
Truth table for two argument ops
Operations
˅
O
R


I
M
P
L
I
E
S

=
E
Q
U
A
L
˄
A
N
D
N
A
N
D

=
N
O
R
Arg1
Arg2
t
t
t
t
t
t
t
t
t
t
f
f
f
f
f
f
f
f
t
f
t
t
t
t
f
f
f
f
t
t
t
t
f
f
f
f
f
t
t
t
f
f
t
t
f
f
t
t
f
f
t
t
f
f
f
f
t
f
t
f
t
f
t
f
t
f
t
f
t
f
t
f
A look at implication
•
•
1.
2.
3.
4.
While the TTs for ˅ and ˄ are symmetric and fairly intuitive the TT for
implication requires some discussion.
All States
True States
"4 is odd  4 is even" !!!
T premise implies a T conclusion, therefore T  T is T;
T premise cannot imply a F conclusion, therefore T  F is F
and 4. You can conclude anything from a false assumption,
so F  anything is T.
The value of marked conclusions (some states of Q) doesn’t affect the
value of entire proposition
Implies() in terms of OR(˅)
• implies:
• e  s
• OR
• e ˅
s
• Use the on an line truth table generator to
evaluate the above expressions.
http://turner.faculty.swau.edu/mathematics/materialslibrary/truth/
8
Implication with constants T/F and variables
P/Q in P  Q
A look at implication
• Logicians claim that the TT for implication
is consistent.
• P is stronger than Q, the implication is true
only once when P is true, twice when Q is
true
• Q is weaker than P
•
All States
True States
Strong and Weak assertions
• Definition: If P and Q are two assertions, then
P is stronger than Q if P  Q. If P is stronger
than Q, then Q is weaker than P.
• Example: i<0 (P) is stronger than i<1(Q)
because i<0  i<1. Fewer states satisfy
i<0 than satisfy i<1.
• Stronger means more selective or specific.
• Weaker means more frequent or general.
Binary Op Truth Table(1)
• The TT on slide 5 provide a complete enumeration of the
Boolean operations with two arguments (binary operators).
Eight of the infix ops have names.
• b = c “b equals c”
• b  c “b differs c” (AKA exclusive or, XOR)
• b  c “b equivales c” or b and c are equivalent. The Boolean
expression b  c is evaluate exactly as b=c, except that 
can be used only when b and c are Boolean expressions (= can
be used for arithmetic equality e.g. 2 = 1 + 1).
• b  c “the inequivalence of b and c”, again strictly for
Boolean operands.
Binary Op Truth Table(2)
• b ˄ c conjunction or AND, &, or &&
• b ˅ c disjunction or OR, ||, +
• bc implication. “b implies c”, “if b then c”.
b is called the antecedent
c is called the consequent
• bc consequence. “b follows from c”
b is called the consequent
c is called the antecedent
Binary Op Truth Table(2)
• Alternative symbols for equivalence:
“x if and only if y”
iff

<->

• The last symbol is used in the course text. The should not be
confused with the CafeOBJ equality operation ==.
• CafeOBJ’s BOOL module uses iff for equivalence.
• Check the definition in CafeOBJ (note spaces):
• open BOOL .
• show op (_ iff _)
Evaluating Truth Tables
• In addition to defining Boolean operators TT can be used to
compute the value of a Boolean expression in every state.
They offer a decision procedure.
• The TT on the following slide gives the value of :
p ˅ (q ˄ r)
• The precedence of operators are described at:
http://www.cs.cornell.edu/Info/People/gries/Logic/Prec.html
• A precedence specifies the order of evaluation of various
operations (lower the number the higher the precedence)
• Note the precedence rules may differ in CafeOBJ. This can be
check with the show op or show module command.
Table of Precedence from A Logical approach to
Discrete Math
By David Gries and Fred B. Schneider
Because of the higher precedence of * over + in
mathematics 2 + 5 * 3 = 17 (and not 21) . Most
programming languages use precedence levels that
conform to the order commonly used in mathematics
for arithmetical operations, comparisons, Boolean
operations.
Evaluating Truth Tables
p
q
r
t
t
t
t
f
f
f
f
t
t
f
f
t
t
f
f
t
f
t
f
t
f
t
f
r
f
t
f
t
f
t
f
t
q ˄ r
p
f
t
f
f
f
t
f
f
t
t
t
t
f
t
f
f
˅ (q ˄ r)
Truth Table Generators
• b  ( b ˅ c)
• b  (b ˅ c)
•
http://turner.faculty.swau.edu/mathematics/materialslibrary/truth/
Truth Table Example
• Is b  (b ˅ c) a tautology? True in
all cases.
• One solution is to draw TT of
• b  (b ˅ c)
b c b (b ˅ c) b  (b ˅ c)
t t
f
t
t
t f
f
t
t
f t
t
t
t
f f
t
f
f
Table Example
Is b  ( b ˅ c) satisfiable? True in at
least one case.
•b
•
•
•
•
t
t
f
f
c
t
f
t
f
b ( b ˅ c) (b  ( b ˅ c))
f
f
t
t
t
t
t
f
f
f
t
f
Equality versus Equivalence
• The Boolean expression a  b is evaluated exactly
the same as a = b, except that  can only be used
when b and c are Boolean expressions.
• They each have different precedence indicating which
operation is done 1st , 2nd , 3rd , etc.:
= (arithmetic) higher precedence (done before  )
 (Boolean) lower precedence (done last)
• Conjunctional and Associative:
– Conjunctional: b = c < d abbreviation for (b = c) ˄
(c < d)
– Associative: a  b  c, (a  b)  c, or
a  (b  c)
Satisfiability and validity
• A Boolean expression P is satisfied in a state if its
value is true in that state; P is satisfiable if there is a
state in which it satisfied; and P is valid if it satisfied
in every state. A valid Boolean expression is called a
tautology.
• If P cannot be satisfied (e.g. P= (A ˄ A)) then P
is always false for all values of A. If P is unsatisfiable
• A Boolean expression that is neither a tautology nor a
contradiction is called a contingency or satisfiable .
Boolean Expressions and
Propositions
• Boolean expressions are constructed from the
constants true and false, Boolean variables,
which can be associated (only) with the values
true and false , and the Boolean operators such
as (, , , ˅, ˄). The constants true and false
are often called Boolean values, and a Boolean
expression is often said to be of type Boolean.
• A proposition is a statement that can be
interpreted as being either true or false.
Propositions can be translated into a Boolean
expressions.
Boolean Expressions and
Propositions
Proposition
• It is raining and 5 is greater than 3.
• It is raining or 5 is greater than 3.
Boolean Expression
• r ˄ g
• r ˅ g
• It is not raining.
• r
• Whether or not it is raining, I am
going swimming.
• (r ˅ r)  s
A Boolean variable that can denote a proposition is sometimes called a
propositional variable. The operator greater than (<) is a predicate (covered in
Later). Predicates are not directly representable as propositions. Greater than can
be regarded as a function of two numerical arguments and a Boolean result.
Propositional Calculus
• Propositional calculus is a method of
calculating with Boolean expressions that
involve propositional variables. Gries and
Schneider call their propositional calculus
equational logic E.
Satisfiable and Valid1
•
•
•
•
The value of a Boolean expression is its state
which can be either true or false. State is a list
of variables with associated values.
pq is satisfiable because it is satisfied if (p,
true) is in state S1. It is not valid in the state
S2: (p,false), (q,false).
pp  p is valid in every state.
A formula is valid if and only if it is true under
every interpretation (or state).
Modelling English propositions
• A proposition can be interpreted as being either
true or false. For example:
• We wish to translate English propositions to
Boolean expressions because:
– English is ambiguous, computers require logical
clarity.
– We can automate, analyse, reason about, simplify,
and prove properties of Boolean expressions.
– Rules of logic provide effective computer based way
to simulate human reasoning (in English).
Modelling English propositions
• Issues involving inclusive vs. exclusive or are
encountered when translating natural language to
symbolic logic. We need to decide which "or" the natural
language means.
• Example:
1. I worked hard or I played the piano.
2. If I worked hard, then I will get a bonus.
3. I did not get a bonus.
Converting English
• Lets use: R: its raining, S: I’m going
swimming.
• Whether or not it is raining, I am going
swimming: (R ˅ R)  S
• If it is raining then I will not go swimming:
R  ¬S
• I will go swimming only if it is not raining:
S  ¬R (see later)
Converting English
• If we write p -> q for the sentence “We will
true, but “you help us”(q) is false.
Logic and reality
• Two ways to say the same thing?
– To stay dry it is sufficient to have an umbrella.
– If you have an umbrella, then you will stay
dry: hu  sd
• We will assume common sense meanings and
avoid complex interpretation issues.
Coventry and Garrod1
Modelling English propositions
• Let Boolean (or propositional) variable p
denote the proposition:
• p : Henry VIII had one son and Cleopatra had two.
• The proposition p contains two subpropositions:
• x: Henry VIII had one son.
• y: Cleopatra had two (sons).
•x ˄ y
Modelling English propositions
• Negated Disjunction: English sentences
containing neither and nor are treated as
negated disjunction (i.e. a negative version of
English or).
• Example: [S1] John neither washed the car nor
went to the cinema.
• Let W = washed car, C = went to cinema.
• S1 can be written as ¬(W ˅ C) .
Translating into a Boolean
Expression
• 1. Introduce a Boolean variable to denote
sub-propositions.
• 2. Replace these sub-propositions by their
corresponding Boolean variables.
• 3. Translate the result of step 2 into a
Boolean expression, using obvious
translations of English words into
operators. See table on next slide.
Translation of English Words
and, but
˄
or
˅
not

it is not the case that

if p then q
p  q
Translation
Boolean variable English Sentence
x: Henry VIII has one son
z: I’ll eat my hat
w: 1 is prime
Translation
Proposition
translation
Henry VIII has one son or I’ll
eat my hat
x ˅ z
Henry VIII has one son and 1
is not prime
x ˄ w
If 1 is prime and Cleopatra
I’ll eat my hat
w ˄ y  z
Translating Or
• We must distinguish between:
• “inclusive or” (or, ˅)
• “exclusive or” (xor, b  c, b  c )
– sometimes written ‘||’ or ⊕
•
•
•
•
Would you like coffee or tea?
Would you like sugar or milk?
Is it raining or not raining? Yes!
Island of Liars and Truth tellers?
Translating
Can you express the above statement using just one logical connective?
This example is from Logic Made Easy by Deborah J. Bennett
Implication
• “If a then b” becomes ab
• “if you don’t eat your spinach I will scold
you.” becomes:
• es  sy
• In English the following have same
meaning:
– “if you don’t eat your spinach then I will scold
you.”
– “eat your spinach or I will scold you.”
Implication
• Two logically equivalent propositions:
• e  s
• e ˅ s
Contrapositive
• Contrapositive:
• e  s
• s  e
Implies in terms of disjunction
• implies:
• e  s
• OR
• e ˅
s
• Use the on line truth table generator to
evaluate the expression in this lecture.
http://turner.faculty.swau.edu/math
ematics/materialslibrary/truth/
44
Implication
• In English the antecedent and the
consequence can be switched, and the
then dropped:
• If you pay me then I will go.
• I will go if you give me money.
• There is a subtle difference between
• ‘If’ meaning ‘in case’ or ‘in the event of’
• ‘If’ meaning ‘when’
Implication
• There is a subtle difference between
– ‘If’ meaning ‘in case’ or ‘in the event of’
– ‘If’ meaning ‘when’
• “When the sun is shining, I feel happy”
• Also, in logic a consequential relation is not
necessary.
• If I drink too much, then I get sick.
• If pigs can fly, then 2+2=5.
• Logic usually needs a sensible interpretation to be
useful.
Implication And Equivalence
• The English “if” should be sometimes
regarded as an equivalence.
• “If two sides of a triangle are equal, the
triangle is isosceles.”
• Which as a definition could be stated:
• t : two sides of the triangle are equal
• Is : the triangle is isosceles.
• t  is
Necessity
• A necessary condition for the occurrence of a specified
event is a circumstance in whose absence the event
cannot occur. Oxygen is necessary for combustion or
oxygen is necessary for human life. A necessary
implication can be used to represent sub-set-of relation
(AKA subsumption or “is kind of” or “is-a”).
• Example being a student implies being a person.
• Student Û Person
All members of B are members of A
• Does “B is a kind of A” mean “All Bs are
As”?
• All members of a subclass can be inferred
to be members of its superclass.
• Given an object of type A it cannot be
inferred that it is a B.
• So B implies A but not visa versa.
Question
• On Computing Fundamentals 1 it is
necessary to pass the exam to pass the
module.
• What is sufficient to pass the module?
Sufficient
• A sufficient condition for the occurrence of
an event is a circumstance in whose
presence the event must occur.
• If a conditional statement (aÛb) is true
then its antecedent is said to provide a
sufficient condition for its consequent.
• In contrast the consequent is the
necessary condition of antecedent.
Necessity & Sufficiency
• Two ways to say the same thing?
– To stay dry it is sufficient to wear a raincoat.
– If you wear a rain coat, then you will stay dry.
• wr  sd
Necessity & Sufficiency(2)
• Two ways to say the same thing?
– To stay dry it is necessary to wear a raincoat.
– You will stay dry only if you wear a raincoat.
• sd  wr
Necessity & Sufficiency(3)
• “x is sufficient for y” means
•x y
• “x is necessary for y” means
•y x
• “x is necessary and sufficient for y”
means
• (x  y) ˄ (y  x)
• Note: care is needed when ‘only’ is used.
Necessity & Sufficiency(3)
• "If X then Y", or "X is enough for Y",
can also be understood as saying that
X is a sufficient condition for Y.
• Bob eats dessert if it is apple pie.
– (If it‟s apple pie, then Bob will eat it, AÛB)
• Bob eats dessert only if it is apple pie.
– (If Bob eats dessert, then it must be apple pie;
BÛA)
Sequences of equivalences
• Operator ≡ is associative, which means that b ≡
c ≡ d, (b ≡ c) ≡ d, and b ≡ (c ≡ d) are
all equivalent. The Boolean expression
• P0 ≡ Pi ≡ ... ≡ Pn
• is true exactly when an even number of the Pi
are false.
Sequences of equivalences
• Without any additional formal manipulation
it can be determined that
false ≡ false ≡ false ≡ true
is false, because an odd number of its
equivalents are false.
The following is true.
false ≡ false ≡ false ≡ false ≡ true
Bob’s Eyesight
• Logical equivalence () is associative.
• Bob can use no eyes
• Bob can use one eye
• Bob can use two eyes
• Generalizes to “Bob can use j eyes”
• Bob0  Bob1  Bob2
• Is this always true?
• Draw truth table.
• What if we say “exactly n eyes”
Equality (=) and Equivalence(≡)
• Note that equality = can be used with or
non-Boolean data (e.g. integers) and is
conjunctional rather than associative.
Using equivalance
• None or both of p and q is true.
• ¬p ≡ ¬q ˅ p ≡ q
Examples
Examples
• Whether or not it is raining, I am going
swimming.
A logical equivalence
Examples(*)
The English use of Unless1
• ‘Unless’ generally means ‘or’. ‘I will go out unless it rains’
translates to ‘I will go out ˅ it will rain’. (Note the extra
‘will’.)
• You could also use ‘￢(it will rain)  I will
go out’.
• But you may think that ‘I will go out unless it rains’
implies that if it does rain then I won’t go out. This is a
stronger reading of ‘unless’. If you take that view, you’d
translate it as ‘I will go out ⇔ ￢(it will
rain)’. This is a bit like ‘exclusive or’.
• Deciding whether to take the strong or weak reading is
done by individual context, and can be impossible:
English is not always precise. You must decide what the
correct logical translation is.
The English use of Unless
• English is sometimes difficult to translate:
• “I will go out unless it rains” could be
¬(rains)  I will go out
or
¬(rains) ⇔ I will go out
or
(rains) ˅ I will go out
The English use of But1
• ‘But’ can mean ‘and’.
• ‘I will go out, but it is raining’ translates to
• ‘I will go out ˄ it is raining’.
Translating
• If the bill was sent, then you will be paid
and the bill was sent.
• B = Bill was sent
• P = You will be paid
• (B->P) ˄ B
Translating
• If Mary is the author of the book, then the
book is fiction, but the book is non-fiction
implies that Mary did not write the book.
• M = Mary is Author
• F = Book is Fiction
•
Translating
Translating from English Example
Translating from English Example
• Premise Indicators
– For
– Since
– Because
– Assuming that
Translating from English Example
• Conclusion Indicators
– Therefore
– Thus
– Hence
– So
Expressions in CafeOBJ
• These convert to in the BOOL module of
CafeOBJ.
• red not(b) implies (b or c) .
• red not(b) iff (b or c) .
Prove the following
Prove the following
Prove the following
Prove the following
Expressions in TT and CafeOBJ
What results does CafeOBJ give for the following?
open BOOL .
-- set trace whole on .
vars a b c : Bool .
red not(b)
iff (b or c) .
red not(b)
iff b or c .
red not(b) implies (b or c) .
red not(b) or c iff b implies
c .
In some case we get true in others we get a term.
Monoticity
What results does CafeOBJ give for the following?
open BOOL .
vars p q r : Bool .
red (p implies q) implies ((p or r) implies (q or r)) .
red (p implies q) implies ((p and r) implies (q and r)) .
red (p implies q) implies ((r implies p) implies (r implies q)) .
Superman Example
• If Superman were able and willing to
prevent evil, he would do so. If Superman
were unable to prevent evil, he would be
ineffective; if he were unwilling to prevent
evil, he would be malevolent. Superman
does not prevent evil. If Superman exists,
he is neither ineffective nor malevolent.
Therefore Superman does not exist.
Superman Example
a: Superman is able to prevent evil.
w: Superman is willing to prevent evil.
i: Superman is ineffective.
m : Superman is malevolent.
p: Superman prevents evil.
e: Superman exists.
82
Superman Example
83
Superman Proof: Gries &
Schnieder
• Proof using G&S E logic to manipulate Boolean Expressions.
84
Superman Proof using CafeOBJ
Propositional Logic.
mod! SUPERMAN {
ops a w i m p e : -> Bool
ops aw ai nw pe ee : -> Bool
eq [F0] : aw = (a and w) implies p .
eq [F1a] : ai = not a implies i .
eq [F1b] : nw = not w implies m .
eq [F2] :
pe = not p .
eq [F3] :
ee = e implies (not i) and (not m) . }
open SUPERMAN .
red (aw and ai and nw and pe and ee) implies (not e)
This returns true using CafeOBJ’s equational logic that
includes propositional logic.
Superman Proof using CafeOBJ Theorem
Prover Propositional Logic.
mod! SUPERMAN {
-- the basic facts
ops a w i m p e : -> Bool
ax [F0] : (a & w) -> p .
ax [F1] : (~(a) -> i) & (~(w) -> m) .
ax [F2] : ~(p) .
ax [F3] : (e -> (~(i) & ~(m))) .
}
This returns true using CafeOBJ’s equational logic that
includes propositional logic.
Superman Proof truth table.
******How Many rows in TT? *********
Alternative Formalization in
CafeOBJ
This is the notation used by the CafeOBJ
theorem prover.
mod! SUPERMAN {
-- the basic facts
ops a w i m p e : -> Bool
ax [F0] : (a & w) -> p .
ax [F1] : (~(a) -> i) & (~(w) -> m) .
ax [F2] : ~(p) .
ax [F3] : (e -> (~(i) & ~(m))) .
}
88
CafeOBJ set up for proof
• open PORTIA1
• -- The goal is to prove that the portrait
• option reset .
• goal(gc) . -- We are only interested goal
line
• flag(auto, on) .
• flag(very-verbose,on) .
• flag (universal-symmetry,on)
• param(max-proofs, 1) .
• resolve .
• close
89
Alternative Proof in CafeOBJ
•
•
The expected output is
** PROOF ________________________________
•
1:[] ~(a) | ~(w) | p
•
2:[] a | i
•
3:[] m | w
•
4:[] ~(p)
•
5:[] ~(e) | ~(i)
•
6:[] ~(e) | ~(m)
•
7:[] e
•
15:[hyper:7,5,2] a
•
16:[hyper:3,1,15] m | p
•
17:[hyper:16,4] m
•
18:[hyper:17,6,7]
• ** ______________________________________
90
Work or Play?
• Write the following statements in
propositional logic:
• I worked hard or I played the piano.
• If I worked hard, then I will get a bonus.
• I did not get a bonus.
• (W ˅ P) ˄ (W ⇒ B) ˄ ¬(B)
Work or Play?
• Construct a TT using:
• http://turner.faculty.swau.edu/mathematics/materialslibrary/truth/
• This expression is only true in one case.
Work or Play?
• Here are the intermediate truth tables for
(W ˅ P) ˄ (W ⇒ B) ˄ ¬(B)
Overall argument
Work or Play?
• The intermediate truth tables for (W ˅ P) ˄ (W ⇒ B) ˄ ¬(B)
.
• The following are useful transformations: ¬(¬(w)) = w.
Expressing or as ⇒
The contrapositive rule
Work or Play?
•
•
•
•
•
•
Given
I worked hard or I played the piano.
If I worked hard, then I will get a bonus.
I did not get a bonus.
Can we conclude : I played the piano.
In logical notation is the following a valid
argument?
• (W ˅ P) ˄ (W ⇒ B) ˄ ¬(B)
Therefore or
•
P
∴
Work or Play?
• As propositions in CafeOBJ
mod WORK {
ops w p b : -> Bool
ops e1 e2 e3 : -> Bool
eq [e1] : e1 = w or p .
eq [e2] : e2 = w implies b .
eq [e3] : e3 = not b .
}
open WORK .
red (e1 and e2 and e3) implies p .
Valid Argument
• The following is a valid argument
– [P1] If Pats’s program is less than 10 lines
long then it is correct.
– [P2] Pat’s program is not correct.
– [C1] Therefore Pat’s program is more than 10
lines long.
• The first two lines are the premisses and
the last the conclusion. Recall the
contrapositive rule (P ⇒ C)  (¬P ⇒ ¬L)
Invalid argument
• The following is an invalid argument
– [P1] If I am wealthy then I give away lots of
money
– [P2] I give away lots of money
– [C1] Therefore I am wealthy
• The reasoning is not valid because from
the premisses you cannot derive the
conclusion. (W⇒M)  (M⇒W)
Valid argument: Modus Ponens
• An example a very general valid argument
is Modus Ponens (from Latin mode that
affirms)
Turnstile |=
• In propositional logic the double turnstile is
symbol that is placed between propositions and
the conclusion.
• Sequent expressions assert that an entailment
relation holds between the set of propositions
on the left and the conclusion on right-hand
sides of the double turnstile.
• A |= B can be read “A entails B”
• There is no interpretation which makes A true
and B false. As a specification we could say A
satisfies B.
Single Turnstile |• The intuitive meaning of A |- B is that under the
assumption of A the conclusion of B is provable.
It denotes syntactic consequence
• Soundness: A|-B ⇒ A|=B
• Completeness: A|=B ⇒ A|-B
• Soundness prevents proving things that aren't
true when we interpret them.
• Completeness means that everything we know
to be true on interpretation, we must be able to
prove.
Double turnstile |= in relation to 
P1, ..., Pn |= Z iff
|= (P1 ˄ … ˄ Pn)  Z
• Recall the work, piano, bonus example.
(w˅p) ,(w  b) ,(~b) |= p iff
|= (w˅p) ˄(w  b) ˄(~b)  p
Difference between valid
arguments and implies.
• An argument is valid iff it is necessary that
under all interpretations (valuations in
propositional logic), in which the premises are
true the conclusion is true as well: P1,...,Pn |= Z
• P1,...,Pn |= Z if and only if the statement of the
form P1 and ... and Pn implies Z is necessarily
true (a tautology): |= (P1 & … & Pn)  Z
103
Model |= Theory
• A formula is satisfiable if it is possible to
find an interpretation (model) that makes
the formula true.
• A |= B read “A satisfies B”.
• A valid argument is one in which , if the
premises are true, then the conclusion
must be true.
Building And-Or in CafeOBJ
-- No variables
-- We turn of implicit import of built-in BOOL .
set include BOOL off
-- This allows us to develop our own Boolean operations
mod* BOOLEAN1 {
signature {
[ Boolean ]
op false : -> Boolean
op true : -> Boolean
op -_ : Boolean -> Boolean
op _\/_ : Boolean Boolean -> Boolean
op _/\_ : Boolean Boolean -> Boolean }
axioms {
eq - false = true .
eq - true = false .
eq false \/ false = false .
eq false \/ true = true .
eq true \/ false = true .
eq true \/ true = true . }}
eof
-- Check these reductions?
open BOOLEAN1
var A : Boolean
red true \/ false .
red true \/ A .
red A \/ true .
books in a library
```
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