Between Truth and Falsity - Sacramento State



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

| |B |

| |& |T |I |F |

|A | | | | |

| |T |T |I |F |

| |I |I |I |F |

| |F |F |F |F |

K3 Truth Table for Disjunction

| |B |

| |v |T |I |F |

|A | | | | |

| |T |T |T |T |

| |I |T |I |I |

| |F |T |I |F |

K3 Truth Table for the Conditional

| |B |

| |→ |T |I |F |

|A | | | | |

| |T |T |I |F |

| |I |T |I |I |

| |F |T |T |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

| |B |

| |→ |T |I |F |

|A | | | | |

| |T |T |I |F |

| |I |T |T |I |

| |F |T |T |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

| |B |

| |→ |T |I |F |

|A | | | | |

| |T |T |I |F |

| |I |T |T |F |

| |F |T |T |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

| |B |

| |↔ |T |I |F |

|A | | | | |

| |T |T |I |F |

| |I |I |I |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 |T |

|F |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 |q |

| |→ |T |I |F |

|A | | | | |

| |T |T |I |F |

| |I |T |I |I |

| |F |T |T |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.

Problems

Test for validity. (p v (q → r), ((p & (r) ( ((q & (r)

Test for validity. q ( (p & r) v ((p & r)

Determine whether this is a tautology. (Devil’s Hint: There are no SL tautologies in K3).

(p → p) v –( p → 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 [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:

And the question mark is inserted for the negation in the standard rules for the conditional and biconditional.

Furthermore, the questionable conditional and questionable biconditional require shift lines, as follows.

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.

This is simply because if we have established (p, then it is true that p has not been established.

Also we have:

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

Preserves law of non-contradiction

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)

-----------------------

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)

–p +q

Weak Negated Conditional

–(p →q)

+p

–q

Weak Biconditional

(p ↔q)

+ p -p

+q -q

Weak Negated Biconditional

-( p ↔ q)

+p -p

-q +q

(p v (q → r)

√ ((p & (r)

√ – ((q & (r)

√+(q & (r)

+q

√+(r

(p

(r

–r

p q →r

(

(q r

( (

q

√ –((p & r) v ((p & r))

√–(p & r)

√–((p & r)

√+( p & r)

+p

+r

–p –r

Invalid

√ –((p → p) v –( p → p))

–(p → p)

– –(p → p)

(

p v – p

√?(p v (p)

?p

√? (p

p

Questionable Disjunction

√?(p v q)

?p

?q

Questionable Conjunction

√?(p & q)

?p ?q

Questionable Negation

√?( p

p

Negation

√( p

?p

Conditional

√(p →q)

?p q

Questionable Conditional

√?(p →q)

p

?q

Biconditional

√(p ↔q)

p ?p

q ?q

Questionable Biconditional

√?(p ↔ q)

p ?p

?q q

?((p & (p)

(p & (p)

p

(p

(

Props to Tim!!

Lanae says no props to Tim.

?((p ( q) ( ((p v q))

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