Download Many-Valued Logic

Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Many-Valued Logic
Daniel Bonevac
February 27, 2013
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Bivalence
I
Classical logic is bivalent—every sentence is true or false.
I
Aristotle restricted logic to statements, sentences that can be
true or false.
I
Bivalent systems go beyond that to assume that all
statements are actually true or false.
I
Most logicians throughout history have been willing to make
that assumption.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Truth Value Gaps
I
Some, however, think there are good reasons to deny certain
sentences a truth value.
I
Many-valued logics revise classical logic by reinterpreting the
truth-functional connectives themselves, allowing for
additional truth values or truth value gaps.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
History
I
Hugh MacColl (1906), Charles Sanders Peirce (1909), Nikolai
A. Vasilev (1910), Jan Lukasiewicz (1920), and Emil Post
(1921) asked what might happen if bivalence were rejected,
and developed the first many-valued logics.
I
D. A. Bochvar (1939), Kurt Gödel, and Stephen C. Kleene
(1952) devised alternative systems.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Presupposition and Denotation Failure
I
Some sentences have presuppositions:
I
I
I
I
Has Jones stopped beating his wife?
The present King of France is bald.
Bill Clinton regrets his affair with Margaret Thatcher.
Bill Clinton does not regret his affair with Margaret Thatcher.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Presupposition and Denotation Failure
I
The first is a classic example of a complex question, a
question with presuppositions.
I
Every possible answer to it presupposes something—that
Jones has been beating his wife.
I
In general, one sentence presupposes another if its assertion
is infelicitous—that is, inappropriate—unless the latter is true.
I
The other sentences are all inappropriate, since there is no
King of France at present and since Bill Clinton never had an
affair with Margaret Thatcher.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Presupposition and Denotation Failure
I
Are these sentences true or false?
I
Philosophers and linguists disagree, but, on one view of the
matter, they are neither.
I
Suppose someone were to ask you, “Is the present King of
France bald?” It seems you can say neither “Yes, he is” nor
“No, he isn’t,” for there is no King of France.
I
That suggests that (2) is neither true nor false.
I
Similarly, suppose someone asks, “Does Bill Clinton regret his
affair with Margaret Thatcher?”
I
You can’t answer with “Yes, he does,” or “No, he doesn’t.”
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Vagueness
I
Some expressions appear to be vague.
I
Is Word and Object long? Is Austin large? Is turquoise blue?
I
Long, large, and blue seem to have no very definite
boundaries.
I
Some books are definitely long: War and Peace, for example,
or Remembrance of Things Past.
I
Others, such as Goodnight Moon, are definitely not long.
I
But many books fall in between. It’s hard to say whether they
are long or not.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Vagueness
I
Similarly, New York City is definitely large, and Salt Flat,
Texas, is definitely not.
I
But it’s hard to say whether Pittsburgh, Austin, Tucson, and
Hartford are large or not.
I
The sky, on a clear day, is definitely blue; the desert, on a
clear day, is definitely not blue.
I
But it’s not so easy to say whether turquoise is blue or not.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Vagueness
I
Now, consider the sentences
I
Word and Object is long.
I
Austin is large.
I
Turquoise is blue.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Vagueness
I
Are they true or false? It’s hard to say.
I
Now, we might adopt two attitudes about that.
I
We might be epistemicists and say that they are true or false,
but that we just have a hard time telling which.
I
Or, we might say that they are neither true nor false.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Degrees of Truth
I
Some expressions, such as fairly, sort of, and in between,
suggest that there are intermediate positions between truth
and falsehood, or at any rate that there are degrees of truth
and falsehood.
I
Word and Object is fairly long, but that doesn’t imply that it’s
long.
I
Austin is sort of large, but that doesn’t imply that it’s large.
I
Turquoise is between blue and green, but that doesn’t imply
that it’s both blue and green.
I
Expressions such as fairly, sort of, and in between seem hard
to understand in a logic in which every statement is either true
or false.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Indeterminacy
I
If we think of the world as determining the truth values of
sentences, we might think that some sentences escape
determination for other reasons.
I
Works of fiction, for example, leave many features of their
imaginative ”worlds” undetermined.
I
Consider Sir Arthur Conan Doyle’s Sherlock Holmes stories.
Holmes was a detective is plainly true, in the world of the
stories, at least; Holmes lived with his wife, Edna, in Rio
Linda, is plainly false; Holmes lived in London when he was a
child seems indeterminate.
I
The stories provide no information to corroborate or contradict
such a claim.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Indeterminacy
I
If this problem were confined to fiction, it might not seem very
serious.
I
One could hold that (almost) all sentences about fictional
objects are simply false.
I
But some philosophers hold that, in other kinds of situations
as well, there is no fact of the matter.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Indeterminacy
I
Aristotle worried that sentences about the future, for example,
might lack a truth value.
I
If it is now true that there will be a sea battle tomorrow, he
reasoned, it must already be determined that there will be a
sea battle tomorrow.
I
If it is now false that there will be a sea battle tomorrow, then it
must already be determined that there will not be a sea battle
tomorrow.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Indeterminacy
I
So, if the sentence There will be a sea battle tomorrow is now
either true or false, it must already be determined whether
there will be a sea battle tomorrow.
I
In general, if sentences about the future have truth values, the
future must already be fixed. Or so it has been argued.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Indeterminacy
I
Lukasiewicz and others sought to unify many-valued and
modal logic, thinking of intermediate values as something like
possibility.
I
But this was a dead end. James Dugundji (1940) showed that
none of the Lewis modal logics S1-S5 have finite
characteristic matrices—meaning, among other things, that
they cannot be viewed as many-valued systems with a finite
number of truth values.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Verificationism
I
Some philosophers hold a theory of meaning according to
which the meaning of a sentence is its method of verification.
I
To say that A is true, on this view, is to say that A can be or
has been verified.
I
Some sentences are decidable in the sense that one can
verify whether they are true or false. But many sentences are
not decidable in that sense.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Verificationism
I
Defenders of bivalence treat this as an epistemic problem:
The sentences are either true or false, even if we may not be
able to tell which.
I
But those who identify meaning with verification decline to
assert that they are either true or false.
I
To assume that they are is to venture beyond what we know to
the realm of metaphysical speculation.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Verificationism
I
All these considerations support the idea that some
sentences are neither true nor false.
I
They motivate the development of many-valued logic, logic
with more than two truth values.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Stephen C. Kleene
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Models
I
A logic is a triple < V , D , {fc : c ∈ C } >, where V is a set of
truth values, D ⊆ V a set of designated truth values, and fc a
truth function.
I
An interpretation is a map v from propositional parameters
into V .
I
Validity will be understood as preservation of designated
value: X |= A ⇔ ∀B ∈ X (v (B ) ∈ D ⇒ v (A ) ∈ D ).
I
If V has n members, the logic is n-valued.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
The Strong Kleene System K3
I
The best-known many-valued logic, the strong Kleene system
K3, is a three-valued logic: sentences may be true, false, or
indeterminate in truth value.
I
K3 is a normal logic—it conserves classical logic’s
understanding of truth functions by agreeing with two-valued
logic when the only values involved are 1 and 0.
I
All logical truths in normal many-valued logics, if there are
any, are also logical truths in classical, two-valued logic.
I
The question, then, is how truth functions should handle
indeterminacy.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
K3
I
Suppose we were to retain the obvious classical clauses:
v (¬A ) = 1 ⇔ v (A ) = 0
v (¬A ) = 0 ⇔ v (A ) = 1
v (A ∧ B ) = 1 ⇔ v (A ) = v (B ) = 1
v (A ∧ B ) = 0 ⇔ v (A ) = 0 or v (B ) = 0
v (A ∨ B ) = 1 ⇔ v (A ) = 1 or v (B ) = 1
v (A ∨ B ) = 0 ⇔ v (A ) = v (B ) = 0
v (A ⊃ B ) = 1 ⇔ v (A ) = 0 or v (B ) = 1
v (A ⊃ B ) = 0 ⇔ v (A ) = 1 and v (B ) = 0
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Normal and Uniform
I
This greatly constrains the definitions of the connectives.
I
If we take this to specify the truth functions completely, so that
all other values are i, the resulting logic is K3.
I
K3 is the weakest three-valued logic that is both normal and
uniform.
I
It is normal in the sense that it agrees with two-valued logic on
the values assigned all combinations of 1s and 0s, and it is
uniform in the sense that it maintains that, in defining the
connectives, if a compound has the same value whether a
component is true or false, it also has that value if the
component is indeterminate.
I
Otherwise, it assumes that the value is indeterminate.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
K3 Negation
Table: K3 Negation
I
Daniel Bonevac
A
1
i
0
¬A
0
1
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
K3 Negation
Table: K3 Negation
I
Daniel Bonevac
A
1
i
0
¬A
0
i
1
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
K3 Conjunction
Table: K3 Conjunction
I
1
i
0
Daniel Bonevac
1
1
0
i
0
0
0
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
K3 Conjunction
Table: K3 Conjunction
I
1
i
0
Daniel Bonevac
1
1
i
0
0
0
0
0
0
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
K3 Conjunction
Table: K3 Conjunction
I
1
i
0
Daniel Bonevac
1
1
i
0
i
i
i
0
0
0
0
0
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
K3 Disjunction
Table: K3 Disjunction
I
1
i
0
Daniel Bonevac
1
1
1
i
0
1
0
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
K3 Disjunction
Table: K3 Disjunction
I
1
i
0
Daniel Bonevac
1
1
1
1
i
1
0
1
0
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
K3 Disjunction
Table: K3 Disjunction
I
1
i
0
Daniel Bonevac
1
1
1
1
i
1
i
i
0
1
i
0
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
K3 Conditional
Table: K3 Conditional
I
1
i
0
Daniel Bonevac
1
1
1
i
0
0
1
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
K3 Conditional
Table: K3 Conditional
I
1
i
0
Daniel Bonevac
1
1
1
1
i
0
0
1
1
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
K3 Conditional
Table: K3 Conditional
I
1
i
0
Daniel Bonevac
1
1
1
1
i
i
i
1
0
0
i
1
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Recognize This Man?
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Recognize This Book?l
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Copi’s Introduction to Logic
I
Copi’s system is really K3
I
All Copi’s rules are valid in K3
I
No valid formulas are theorems
I
No indirect proof or conditional proof
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Copi’s Introduction to Logic
I
Classical rules and inference patterns survive in K3.
I Double Negation: ¬¬A ⇔ A
I Conjunction Exploitation: A ∧ B |= A ; A ∧ B |= B
I Conjunction Introduction: A , B |= A ∧ B
I Disjunction Introduction: A |= A ∨ B; B |= A ∨ B
I Absorption: A ⊃ B |= A ⊃ (A ∧ B )
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Copi’s Introduction to Logic
I
Tautology: A ⇔ A ∧ A ; A ⇔ A ∨ A
I
DeMorgan’s Laws: ¬(A ∧ B ) ⇔ ¬A ∨ ¬B;
¬(A ∨ B ) ⇔ ¬A ∧ ¬B
I
Exportation: A ⊃ (B ⊃ C ) ⇔ (A ∧ B ) ⊃ C
I
Contraposition: A ⊃ B ⇔ ¬B ⊃ ¬A
I
Commutativity: A ∧ B ⇔ B ∧ A ; A ∨ B ⇔ B ∨ A
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Copi’s Introduction to Logic
I
Associativity: A ∧ (B ∧ C ) ⇔ (A ∧ B ) ∧ C;
A ∨ (B ∨ C ) ⇔ (A ∨ B ) ∨ C
I
Distribution: A ∧ (B ∨ C ) ⇔ (A ∧ B ) ∨ (A ∧ C );
A ∨ (B ∧ C ) ⇔ (A ∨ B ) ∧ (A ∨ C )
I
Disjunction Exploitation (Proof By Cases):
A ∨ B , A ⊃ C , B ⊃ C |= C
I
Constructive Dilemma: A ∨ B , A ⊃ C , B ⊃ D |= C ∨ D
I
Disjunctive Syllogism: A ∨ B , ¬A |= B; A ∨ B , ¬B |= A
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Copi’s Introduction to Logic
I
Modus Ponens: A ⊃ B , A |= B
I
Modus Tollens: A ⊃ B , ¬B |= ¬A
I
Hypothetical Syllogism: A ⊃ B , B ⊃ C |= A ⊃ C
I
Another Proof By Cases: A ⊃ B , ¬A ⊃ B |= B
I
Explosion: A , ¬A |= B
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Copi’s Introduction to Logic
I
This is interesting, for this is a complete set of inference rules
for classical propositional logic!
I
Or, at least, it almost is: the only thing missing is a tautology,
such as A ∨ ¬A or A ⊃ A .
I
This suggests that K3 and classical logic agree on every
argument that does not have a tautology as its conclusion.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Conditionals
I
This is uncontroversial for conjunction and disjunction,
relatively uncontroversial for negation (though it is also natural
to define a weak negation connective), but highly controversial
for the conditional.
I
K3 judges each of the following indeterminate:
I
If Holmes lived in London as an adult, he lived there as a child.
I
If Holmes lived in London as a child, he flossed his teeth.
I
If Holmes flossed his teeth, his associate was Dr. Moriarty.
I
All truth-functional systems of three-valued logic agree on (8).
If A is true and B is indeterminate, then A ⊃ B is
indeterminate.
I
K3’s judgments on the last two, however, are controversial.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Conditionals
I
If both A and B are indeterminate, what is A ⊃ B?
I
K3’s answer, that it is also indeterminate, seems plausible.
I
But it has a sweeping consequence: there are no tautologies
in K3!
I
All connectives in K3 treat a compound as indeterminate if all
its components are indeterminate.
I
That means that no formulas are true on every interpretation
of them. Thus, K3 takes as indeterminate:
I
If Holmes lived in London as a child, he lived in London as a
child.
I
But that seems true, not indeterminate.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Jan Lukasiewicz
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Lukasiewicz’s L3
I
Lukasiewicz’s L3 takes a conditional to be true if both its
components are indeterminate.
I
The idea is that a conditional is true unless the consequent is
”less true” than the antecedent.
I
That makes A ⊃ A a logical truth. Of course, it also makes
this true:
I
If Holmes lived in London as a child, he did not live in London
as a child.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
L3
I
L3, unlike K3, has validities, including A ⊃ A .
I
It is clear that the L3 conditional cannot be defined in terms of
the other connectives, for they always yield i when all inputs
are i.
I
One can define the other connectives, however, in terms of ¬
and ⊃; take A ∨ B ⇔ (A ⊃ B ) ⊃ B.
I
Lukasiewicz defined ^A and A in terms of the following
tables:
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
L3
Table: L3 Possibility
I
Daniel Bonevac
A
1
i
0
^A
1
1
0
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
L3
Table: L3 Necessity
I
Daniel Bonevac
A
1
i
0
A
1
0
0
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
L3
I
Lukasiewicz’s student Alfred Tarski showed that these are
definable in L3; ^A ⇔ ¬A ⊃ A , and A ⇔ ¬^¬A .
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
L3
I
L3’s validities all contain the conditional; formulas without the
conditional take i whenever all propositional parameters take i,
so none can be true on every interpretation.
I
This means that the law of noncontradiction, ¬(A ∧ ¬A ), is
not valid in L3.
I
The paradoxes of material implication, however, are.
I
Some classical tautologies can even turn out false in L3:
¬(p ⊃ ¬p ) ∨ ¬(¬p ⊃ p ), as A. R. Turquette showed (refuting a
conjecture of Arthur Prior), is an example.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
L3
I
Is L3 functionally complete?
I
No. Being able to define ^A as ¬A ⊃ A enables us to define
any function taking all its values among 1 and 0, but there is
still no way to define any arbitrary function.
I
In particular, Jerzy SLupecki’s T-function (1936), which always
outputs i, cannot be defined in L3.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Conditionals
I
Conditionals with indeterminate antecedents and false
consequents seem strange.
I
If the present King of France is bald, France is part of Italy.
I
If Austin is large, so is Salt Flat.
I
If there will be a sea battle tomorrow, I’ll eat my hat.
I
If for every n there are n consecutive ns in the decimal
expansion of π, 7 + 7 = 41.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
G3
I
K3 treats these as indeterminate.
I
From one perspective, however, these sentences seem false.
I
One, for instance, seems to assert that Austin and Salt Flat
are similar in size, which is silly.
I
If we alter the L3 table and take a conditional with an
indeterminate antecedent and a false consequent to be false,
we obtain Gödel’s system G3.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
K3
I
There is a point behind both these objections.
I
Nevertheless, there are good reasons to stay with K3.
Someone who says “If there will be a sea battle tomorrow, I’ll
eat my hat,” seems to be saying, in a flamboyant way, that
there won’t be a sea battle tomorrow. The other sentences
could be viewed as ways of asserting that the present King of
France is not bald, that Austin is not large, etc.
I
And those, according to our table for negation, are
indeterminate.
I
It seems reasonable, in other words, to treat A ⊃ B, where B
is false, as having the same truth value as ¬A .
I
K3, like L3, does that; G3 does not.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Functional Completeness
I
K3, like L3, is not functionally complete.
I
It is easy to show that any formula of K3 whose propositional
parameters are all indeterminate is itself indeterminate. That
means that any truth function taking indeterminacy into truth
or falsehood cannot be defined in K3.
I
Moreover, K3, being a normal system, takes all classical truth
values into classical truth values, and so can never define a
function taking a combination of classical inputs into i.
I
Given these connectives, therefore, it makes sense to expand
our language to make it functionally complete.
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Post’s Cyclic Negation
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Post’s Cyclic Negation
I
One way to do this is to add Post’s cyclic negation function:
Table: Post’s Cyclic Negation
A
1
i
0
Daniel Bonevac
yA
i
0
1
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
K3+
I
Call this logic K3+.
I
K3+ is functionally complete.
I
Unlike K3, it has valid formulas, such as A ≡ yyy A ,
A ∨ y A ∨ yy A , and y (A ∧ y A ∧ yy A ).
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Weak Negation
I
We will add two more intelligible connectives.
I
The table for negation reflects a common use of negation.
I
But there are other uses. An indeterminate sentence, for
example, is not true. But it is not false either.
I
What does not mean in these two sentences?
I
It cannot have the truth table above, according to which ¬A is
true if and only if A is false. It must instead be a weak
negation, having the table
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Weak Negation
Table: Weak Negation
I
Daniel Bonevac
A
1
i
0
−A
0
1
1
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Weak Affirmation
I
Similarly, it is useful to be able to say that a formula is not
false—that is, is either true or indeterminate.
Table: Weak Affirmation
A
1
i
0
Daniel Bonevac
+A
1
1
0
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Interdefinability
I
These are interdefinable: +A is equivalent to −¬A , and −A is
equivalent to +¬A .
I
(Notice that weak affirmation is just Lukasiewicz’s possibility
function, and weak negation is just the negation of his
necessity function.)
I
Both can be defined with Post’s cyclic negation: +A is
equivalent to ¬(¬A ∧ y A ), and −A to ¬(A ∧ y ¬A ).
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Tableau Rules
I
Entries have the forms A , + or A , −.
I
Tableaux begin with an initial list X , +, A , −.
I
Rules for disjunctions and conjunctions are as expected.
I
Those for negated conjunctions and disjunctions are
DeMorgan’s laws; there are also rules for double negation.
I
Define the rules as in Priest (except that I have played out the
DeMorgan rules, which actually stresses the parallel between
them and the corresponding classical rules):
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Tableau Rules
¬¬A , +
¬¬A , −
A, +
A, −
Daniel Bonevac
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Tableau Rules
A ∧ B, +
A ∧ B, −
A, −
A, +
B, +
Daniel Bonevac
B, −
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Tableau Rules
A ∨ B, +
A, +
A ∨ B, −
B, +
Daniel Bonevac
A, −
B, −
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Tableau Rules
A ⊃ B, +
¬A , +
B, +
Daniel Bonevac
A ⊃ B, −
¬A , −
B, −
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Tableau Rules
¬(A ∧ B ), +
¬A , +
¬B , +
Daniel Bonevac
¬(A ∧ B ), −
¬A , −
¬B , −
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Tableau Rules
¬(A ∨ B ), +
¬(A ∨ B ), −
¬A , −
¬A , +
¬B , +
Daniel Bonevac
¬B , −
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Tableau Rules
¬(A ⊃ B ), +
¬(A ⊃ B ), −
A, −
A, +
¬B , +
Daniel Bonevac
¬B , −
Many-Valued Logic
Rationales for Many Valued Logic
The Strong Kleene System K3
Lukasiewicz’s L3
Many-Valued Tableaux
Tableau Rules
I
Branches close if they have nodes of the form A , + and A , −
or A , +, ¬A , +for some A .
I
Open branches determine countermodels.
I
These rules are sound and complete for K3.
Daniel Bonevac
Many-Valued Logic