Download Between Truth and Falsity

Between Truth and Falsehood
Besides developing the logic of
modality, the other serious development
in alternative logic is to challenge
bivalence. We saw a little bit of this in
Edginton’s discussion of conditionals,
but the argument against bivalence (the
assumption that ever sentence is either
true or false) is not restricted to
conditionals.
Presupposition
 Jones has not stopped beating his
wife.
 President Clinton does not regret his
affair with Margaret Thatcher.
 My mother never caught me having
sex with the farm animals.
Vagueness
 Dick Cheney is smart.
 Alan Iverson is tall.
 Logic is hard.
Sentences about fictional entities
 Harry Potter occasionally got
stomach cramps.
 Willy Wonka liked unsalted pretzels.
 Holden Caulfield had a crew cut
when he was 2.
Other sentences using words of
uncertain meaning or about entities
of uncertain metaphysical status
 God is ashamed of you.
 Some sets have an infinite number
of members.
 All statements are either true or
false.
Many Valued Truth Tables
Question: What is the meaning of the
logical connectives if we add an ‘I’ (for
indeterminate) to the list of possible
values in a truth table.
Answer: We preserve the classical
understanding whenever the relevant
values are only ‘T’ and ‘F’. But we must
assign truth values when ‘I’ comes into
play.
Bonevac presents a system named after
S.C. Kleene, which he calls K3. At
various points he compares it to G3 (a
system named after Gödel) and L3
(named after Lukasiewicz).
K3 Truth Table for Negation
A A
T F
I
I
F T
K3 Truth Table for Conjunction
&
A T
I
F
B
T I
T I
I I
F F
F
F
F
F
K3 Truth Table for Disjunction
v
A T
I
F
B
T I
T T
T I
T I
F
T
I
F
K3 Truth Table for the Conditional
→
A T
I
F
B
T I
T I
T I
T T
F
F
I
T
Implications: There are no tautologies in
K3.
E.g, If Holmes flossed his teeth, then
Holmes flossed his teeth.
L3 Truth Table for Conditional
→
A T
I
F
B
T I
T I
T T
T T
F
F
I
T
L3 at least preserves tautologies of the
form A → A. But it also assigns T to
statements like:
 If Holmes didn’t floss his teeth, then
he did floss his teeth.
This sentence has the form A → A. If
A is indeterminate, then A is also
indeterminate. It would be odd to assign
T to the conditional.
G3 Truth Table for Conditional
→
A T
I
F
B
T I
T I
T T
T T
F
F
F
T
The point of G3 is to correct another
apparent defect of K3. Besides
preserving (A →A) as a tautology, it
maintains the falsity of conditionals with
an indeterminate antecedent like:
 If Denzel Washington is keen, then
DW is immortal.
K3 assigns this statement the value I.
G3 assigns it false.
Defenders of K3 respond in Gricean
fashion that when we say (A →B),
knowing that B is false, this is really just
a way of saying that A is true.
 If you’re an A student, then I’m a
toss up.
Hence, (A →B), should always have the
truth value of A, which is what we have
with K3.
K3 Truth Table for Biconditional
↔
A T
I
F
B
T I
T I
I I
F I
F
F
I
T
Truth Table for Weak Negation
A -A
T F
I T
F T
Truth Table for Weak Affirmation
A +A
T T
I
F
T
F
Validity
In truth-functional logic we endorsed two
different ways of defining validity. One
of these is now wrong. Which one and
why?
1. An argument is deductively valid iff
its conclusion is true whenever it’s
premises are all true.
2. An argument is deductively valid iff
it’s premises can not all be true while
the conclusion is false.
Implication
Similar considerations hold for
implication.
1. A set of sentences implies a given
sentence just in case the truth of that
sentence is guaranteed by the truth of
all members of that set.
2. A set of sentences implies a given
sentence just in case it is impossible for
all the members of the set to be true and
the given sentence false.
Sentence Validity
Again:
1. A sentence is valid iff it is true under
all possible interpretations.
2. A sentence is valid iff there is no
interpretation under which it is false.
Contradiction
Again:
1. A sentence is contradictory iff it is
impossible for it to be true.
2. A sentence is contradictory iff it is
false under every possible
interpretation.
Truth Tables
Since we now have three valued truth
tables for all of the connectives we can
do truth tables in the usual way. The
only difficulty is that they are longer.
Since there are three values now, a
truth table with n different atomic
formulas will have 3n lines. It’s worth
doing, maybe one.
Exercise: Use a truth table to show that
(p v q) ↔ (p →q) is not a tautology, but
that the formulas are both weakly
equivalent and strongly equivalent. (p.
308)
p
T
T
T
I
I
I
F
F
F
q
T
I
F
T
I
F
T
I
F
(p
T
T
T
I
I
I
F
F
F
v q) ↔ ( p → q)
T T T F T T T
T I T F T T I
T F T F T T F
T T T I I T T
I I I
I I I I
I F I
I I I F
T T T T F T T
I I I T F I I
F F T T F F F
→
A T
I
F
B
T I
T I
T I
T T
F
F
I
T
Terminology for Three-Valued Logic
Weak equivalence: Formulas imply each
other
Strong equivalence: Formulas always
agree in truth value.
Weak affirmation: +A, means either T or
I.
Weak negation: -A, means either F or I
Unexceptionable: Never False. (Hence,
always T or I.)
Contradictory: Never true (Hence,
always F or I).
Many Valued Truth Trees
In many valued logic we need a truth
tree method that identifies arguments as
invalid when they can have either
 true premises and a false
conclusion; or
 true premises and an indeterminate
conclusion.
Hence, it is not adequate to simply
assume the premises are true and the
conclusion is false, and check all
branches for contradictions, since this
method will not reveal arguments that
are invalid because it is possible to have
true premises and an indeterminate
conclusion.
Test for validity
Assume true premises and weakly
negate (-B) the conclusion (i.e., assume
the conclusion is either false or
indeterminate). If this produces all
closing branches then it is because it is
impossible to have true premises and a
false or indeterminate conclusion.
Hence, if the premises are true, the
conclusion must be true.
Test for formula validity
Assume –B. If all branches close, then
it is impossible for A to be false or
indeterminate. Hence it is valid. (But in
fact there are no valid formulas in K3)
Test for unexceptionability
Assume B. If this leads to a
contradiction, then the formula must be
always either true or indeterminate.
Test for contradictoriness
Assume formula is true. If the tree
closes it is never true (i.e., either false or
indeterminate) hence contradictory.
Test for satisfiability
Assume formula is true. If one branch
remains open, it is satisfiable.
Closure
All of the following represent closures of
branches.
A A +A
A - A
A
but not
+A
-A
Truth Tree Rules
Recall from above that to test whether a
formula is unexceptionable (never false)
or contradictory (never true) we do not
need to make use of weak affirmation
(+) or negation (-).
There are no tautologies in K3, so we
can not employ a truth table test of, say,
modus ponensn (→E), that requires
((A →B) & A) → B
to be a tautology. Modus ponens will be
a valid inference rule in K3 iff it is
unexceptionable, or never false. This,
of course, means that whenever ((A
→B) & A) is assigned true in the truth
table, B is also T. The reliability of MP
as an inference rule (put differently,
whether ((A →B) & A) implies B) does
not depend on interpretations when (A
→(B & A)) is false or indeterminate.
Since all of the classical inference rules
are valid by this test, we get to subsume
all of them in K3.
The only modification we need to make
in the truth tree method stems from the
fact that to generate a contradiction in
K3, we assume that the premises are
true and we weakly negate the
conclusion.
P1
P2
P3
.
.
-C
This is because premises imply a
conclusion in K3 iff the truth of the
premises guarantees the truth of the
conclusion, i.e., the assumption that the
premises are true and the conclusion is
false or indeterminate will produce a
contradiction.
So, to evaluate validity we need new
rules for weakly and strongly affirmed
sentences.
Weak Double Negation
– p
+p
Weak Negation
+p
-p
Weak Conjunction
+(p & q)
+p
+q
Weak Negated Conjunction
–(p & q)
-p
-q
Weak Disjunction
+(p v q)
+p
+q
Weak Negated Disjunction
–( p v q)
–p
–q
Weak Conditional
+(p →q)
–p
+q
Weak Negated Conditional
–(p →q)
+p
–q
Weak Biconditional
(p ↔q)
+p
+q
-p
-q
Weak Negated Biconditional
-( p ↔ q)
+p
-q
-p
+q
Problems
Test for validity. (p v (q → r), (p & r)
 (q & r)
(p v (q → r)
√ (p & r)
√ – (q & r)
√+(q & r)
+q
√+r
p
Test for validity. q  (p & r) v (p & r)
q
√ –((p & r) v (p & r))
√–(p & r)
√–(p & r)
√+( p & r)
+p
+r
–p
–r
Invalid
Determine whether this is a tautology.
(Devil’s Hint: There are no SL
tautologies in K3).
(p → p) v –( p
→ p)
√ –((p
→ p) v –( p → p))
–(p
→ p)
– –(p
→ p)

pv–p
Fuzzy Logic
Fuzzy logic, trumps the craziness of
trivalent logic by introducing the insane
idea that truth is a property that comes
in degrees.
Fuzzy logic may be understood as an
extension of multivalent logic to an
infinite number of values which are
represented by the real numbers.
Trivalent logic is subsumed under fuzzy
logic as follows
T = 1 = completely or absolutely true.
I = .5 = half true or half false.
F= 0 = completely or absolutely false.
Fuzzy logic may be understood as a
response to an old puzzle called the
paradox of the heap which is captured in
this argument:
One grain of sand is not a heap of sand.
If something is not a heap of sand,
adding a grain of sand to it will not make
it a heap of sand.
Therefore, there are no heaps of sand.
This argument appears to be sound, but
the conclusion is false. The problem
seems to be that “heap” is a fuzzy
concept. Some things are clearly heaps.
Other things are clearly not. But there is
no fine line one can draw between
heaps and non heaps.
Real philosophers have gradually come
to understand that most of our concepts
are fuzzy. Some are fuzzier than
others, but only mathematical and
logical concepts seem completely to
escape the fuzzy factor. Almost any
other concept C you think of will admit of
cases that cannot be unambiguously
resolved as either C or C.
Most properties, in other words, come in
degrees, including, btw, the concept of
fuzziness itself. (Heap is a very fuzzy
concept. Pregnant is not very fuzzy at
all.)
(This fact, btw, has significant
implications for philosophical method. It
means that, in general, the attempt to
identify necessary and sufficient
conditions for a concept will fail. Also,
there are many completely bogus
philosophical arguments that rest on a
failure to acknowledge that the relevant
concept comes in degrees.)
Numerical Truth Values
In fuzzy logic the truth value of A is
represented as [A]. Fuzzy logic is truth
functional, and the connectives can be
represented as mathematical functions.
Negation
For any given sentence A, [A] + [A] =
1. So, for example, if it is .8 true that
Virginia is swunk, then it is 1- .8 = .2
true that she is not.
[A] = 1 – [A]
Conjunction
Conjunction follows the weak link
principle. In bivalent logic the entire
conjunction is false if one of the
conjuncts is false. In fuzzy logic, the
conjunction takes the minimum value.
If it is .8 true that Virginia is swunk and
.3 true that Slim is kvatch, then it is .3
true that Virginia is swunk and Slim is
kvatch.
[A & B] = Min ([A],[B])
Disjunction
By similar reasoning, disjunction follows
the strong link principle.
[A v B] = Max ([A],[B])
Conditional
As usual, the conditional is the trickiest,
but it is not too difficult to grasp.
Basically, the truth value of (A → B) is a
function of the degree to which A and B
correspond in truth value.
b
If A and B have the same truth value,
then [(A →B)] = 1. This reflect the
bivalent truth table, where T,T and F,F
both are assigned true.
When [A] is less then [B], this
corresponds to the bivalent assignment
A= F and B =T, with the result that (A
→B) is T. Fuzzy logic tracks this
assignment, reasoning that B being
more true than A can not weaken the
truth value of (A →B), so in this case [(A
→B)] also = 1.
T, F is the only assignment in bivalent
logic that results in (A →B) being F.
Fuzzy logic distinguishes degrees of
falsity here, depending on how much
greater [A] is than [B].
[(A →B)] = 1 – ([A] – [B]).
Biconditional
Recalling that the biconditional is a
conjunction of conditionals, it will follow
from the definition of conjunction that [(A
↔ B)] = 1 whenever A and B agree in
truth value.
When A and B have different truth
values, then [(A ↔ B)] will be equal to
the lesser of the truth values of the
corresponding conditionals. This is
captured by an equation using the
absolute value of the difference between
[A] and [B]:
[(A ↔ B)] = 1 – │([A] – [B]) │
Fuzzy Logic Implication
In bivalent (and trivalent) logic
implication is defined as truth
preservation. If the premises are true,
the conclusion must be true. When we
incorporate the mathematical thinking of
fuzzy logic, according to which F<T,
then the idea of truth preservation
becomes, roughly:
If the premises have a certain degree of
truth, then the conclusion must not have
less than that degree of truth.
More precisely, for any conclusion
supported by a finite set of premises,
the conclusion must not have less truth
than the premise with the least amount
of truth. For example:
Simon’s got hench (.6)
Billiam’s got hench (.3)
Therefore, both Simon and Billiam have
got hench. (.3)
This argument form is valid, because
the conclusion has the same amount of
truth as the least true conclusion.
Fuzzy Implication Defined
Some sets are infinitely large, and it is
possible to countenance a set of
sentences with fuzzy truth values that
get infinitely close to some finite number
without there being an actual lowest
value.
If we call the greatest number that is
less than every member of a set the
greatest lower bound, or GLB, we can
define validity as follows.
 A set of sentences  implies
sentence A iff, in every possible
circumstance, GLB [} ≤ [A].
For finite sets, since the conjunction of
all the sentences in  (which we can
abbreviate as &) will have the smallest
truth value of any sentence in , we
preserve the requirement that
 When  implies A, [(& → A)] = 1
Fuzzy tautologies
In fuzzy logic, a tautology is a sentence
that has the truth value 1 in ever
possible circumstance. Are there
tautologies in fuzzy logic?
Consider:
 (p → p)
 (p → (q →p))
 (p v p)
Modus Ponens
It’s not too hard to see why fuzzy logic
must reject modus ponens. Return to
the paradox of the heap. Let’s suppose
that we take 100 grains of sand and
assign truth values as follows.
[1 grain of sand is not a heap] = 1
[2 grains of sand is not a heap] = .99
[3 grains of sand is not a heap] = .98
.
.
[101] grains of sand is not a heap] = 0
Because there is always a difference of
.01 in the truth values of each
succeeding sentence, it follows from the
fuzzy truth table for the conditional that:
[If n isn’t a heap, then n +1 isn’t a heap]
= .99
So we have independently assigned the
following truth values to the sentences
of this argument:
[Pile 2 is not a heap]
.99
[If 2 isn’t a heap, then pile 3 isn’t a heap]
.99
[Pile 3 is not a heap]
.98
In other words, accepting the validity of
modus ponens, contradicts the
requirement of validity, that the
conclusion have at least as much truth
as the least true premise.
This is pretty shocking, since our
bivalent intuitions tell us that if anything
is valid, modus ponens is, but according
to fuzzy logic, our bivalent intuitions are
just wrong.
Mathematical evaluation of
arguments in fuzzy logic. (p.338)
Recall that to evaluate an argument
form in bivalent logic we can proceed
indirectly. Assume the premises are
true and the conclusion is false; then try
to derive a contradiction.
The same approach holds up in fuzzy
logic. If we want to know whether an
argument form is valid, we simply
assume that the GLB, or in the case of
finite sets, the minimum truth value
represented in the premises is greater
than the truth value of the conclusion.
Bonevac does this on modus ponens.
Assume for indirect proof that:
Min ([p], [p →q]) > [q]
This means that
[p] > [q]
and
[p →q] > [q]
Now, for the conditional, when [p] > [q]
we know that
[p →q] = 1 – ([p] – [q]).
So, we know that
1 – ([p] – [q]) > [q]
This simplifies to
1 – [p] + [q] > [q]
and ultimately
1 > [p].
This is not a contradiction. In other
words, with modus ponens, there is no
contradiction in assuming that the
minimum value of the premises exceeds
that of the conclusion. For example:
Suppose
[p] is .9
[q] is .5
Then
[p] → [q] = 1 - .4 = .6
So, Min ([p], [p →q]) = Min ( .9, .6) = .6
which is > [q].
The idea here, then, is simply every
single truth inference rule can be
evaluated mathematically. How hench
is that?
Intuitionistic Logic
Intuitionistic logic rests on a rather
swunk way of thinking about logical and
mathematical truth. Realism provides
the simplest and seemingly most
intuitive way of thinking about
mathematical truth. Realism implies, for
example, that Goldbach’s conjecture:
 Every even number is the sum of two
primes.
is either true or false, even though we
are currently unable to demonstrate
which.
Intuitionists don’t buy this. They claim to
assert that a statement is true is
equivalent to saying that we have a
proof of it. This is a generalization of
what we call verificationism, which can
be defined as
 The meaning of a statement is it’s
method of verification.
According to verificationism, in the
absence of a clear method for
determining whether a statement is true
or false, the statement actually has no
meaning, and hence its truth value is
indeterminate.
Intuitionism is famously associated with
the denial of the law of the excluded
middle.
 p v p
This is simply because not every
statement has been verified or refuted.
Goldbach’s conjecture, from an
intuitionistic logic point of view is neither
true nor false, but indeterminate.
Philosophical intermezzo
There are plenty of philosophers who
hate all this stuff we are taking seriously.
Their basic complaint against
multivalent logic, fuzzy logic, and
intuitionism is that indeterminacy,
vagueness, and fuzziness are just
properties of language, and/or our
epistemic relation to reality. It is not a
property of reality itself.
Realism: Reality itself is fully
determinate.
Hence, if a sentence is too vague to be
given a determinate truth value, this is
because it does not express a clear
proposition (propositions, on this view),
being the primary bearers of truth
values.
Similarly, if we do not know, say,
whether the decimal expansion of π ever
repeats, or exactly how many neutrinos
have been expelled from the sun as a
result of nuclear fusion, that doesn’t
mean the truth values of the associated
sentences are indeterminate.
This position is clear, intuitive, and
certainly should give anyone who takes
indeterminacy seriously some pause.
But in the end, it is not just obviously
true that reality is determinate. It’s nice
to think that, but in fact it is not the
mainstream view in physics. According
to mainstream quantum physics, reality
is indeterminate until it has been verified
by experiment.
Now, back to work.
Proof in Intuitionisitic Logic
According to intuitionism, there is a big
difference between showing that A is
not the case and showing that A is the
case. To focus on the conditional.
A→B
can be refuted simply by showing
A & B.
However, to show that A → B is true,
we must actually derive B from A.
Specifically, it will not be enough to
simply show that A, and derive A →
B from there.
Semantics for Intuitionism
Recall that we developed the semantics
of modal logic by introducing the idea of
truth in a world. Intuitionism introduces
the idea of truth at a stage. This
corresponds to the idea that sentences
become true and false as they are
proved or disproved.
A sentence letter is true/false at a stage,
iff it is assigned truth directly at that
stage. If it is assigned a truth value at
one stage, then it must have that value
at all future stages.
The entire semantics is on page 345 of
Bonevac.
The idea here, roughly speaking, is that
intuitionism preserves the idea that the
truth values of sentences can’t switch
back and forth between truth and falsity.
The main interesting features of
intuitionistic semantics occur in
connection with the truth of formulas
whose main connectives are , and →.
 A is true at a stage iff A is not true
at that, or any later stage.
 A → B is true at a stage iff, if A is
true at that or any later stage, then B
is also true.
This means that intuitionistic logic is
ultimately non truth-functional, for the
same reasons that possible worlds
semantics is non truth-functional. Like
“true at all possible worlds,” “true at all
future stages” can not be determined
simply by determining the value at a
particular stage.
Truth trees in intuitionistic logic
The key move for the truth tree method
is to realize that in a valid intuitionistic
argument if the premises have been
established, then the conclusion has
been established.
Hence, to establish validity, we assume
that the premises have been established
and the conclusion has not been
established.
But, of course,
“has not been established that A” 
“has been established that not A”
So we introduce “?” as a new symbol
meaning “has not been established’.
For the tree method, then we assume
the premises (A1...An) and ? B
and try to derive a contradiction.
Contradictions occur on a branch when
both A & A, or A & ?A occur live.
The ? works exactly like  in the
standard truth tree rules. For example:
Questionable Disjunction
√?(p v q)
?p
Questionable Conjunction
√?(p & q)
?p
?q
And the question mark is inserted for the
negation in the standard rules for the
conditional and biconditional.
Conditional
√(p →q)
?p
q
Biconditional
√(p ↔q)
p
q
?p
?q
Furthermore, the questionable
conditional and questionable
biconditional require shift lines, as
follows.
Questionable Conditional
√?(p →q)
p
?q
Questionable Biconditional
√?(p ↔ q)
p
?q
?p
q
These are not world shift lines, but shifts
to a future stage of knowledge.
Essentially, to say that it is questionable
that (p q) is to say that if at some
future stage of knowledge we knew that
p, q would still be questionable. (Note,
that these future stages are not stages
that will develop, but that might develop.
So there is kind of a hidden reference to
possibility here.)
The introduction of shift lines helps us to
take account of the fact that as we move
through stages of knowledge, it’s
permissible to go from ?A to A, but not
from A to ?A.
We represent this fact by stipulating that
shift lines kill off all previous ?’d
formulas.
So, while a branch like this closes:
?p
p

A branch like this does not close:
?p
p
.
But a branch like this does close:
p
?p

.
Finally, the rules for negation are kind of
weird. First of all, there is no double
negation rule because double negation
doesn’t actually hold in intuitionistic
logic. To say that P has been
established is not to say that P has
been established to be false. These are
distinct procedures.
However we do have this strange rule.
Negation
√ p
?p
This is simply because if we have
established p, then it is true that p has
not been established.
Also we have:
Questionable Negation
√? p
p
This is because ?p means that p may
be established at some future stage.
(This is frankly pretty bizarre, intuitively,
which is funny when you think about it.)
Standard results in intuitionistic logic
Denial of law of excluded middle
√?(p v p)
?p
√? p
p
Preserves law of non-contradiction
?(p & p)
(p & p)
p
p

Props to Tim!!
Lanae says no props to Tim.
Rejects equivalence of (p q) and (p v
q).
To say that (p q) is to say only that
either p has not been established or q.
On the other hand, to say that (p v q)
is in fact to imply that if p, then q. So
the inference goes one way, but not the
other. (p. 350)
?((p  q)  (p v q))