The Role of Logic
R.E. Jennings
START PULLQUOTE
Logic has a fundamental role to play in data exploitation, not
as a tribunal, and certainly not as a judge issuing forth
necessities, but as a laboratory of imaginative
experimentation and as a source of insight into possibilities.
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Reasoning about even the most mundane things can be tricky. For example, most of us
believe that we’re not infallible. Being people of reasonable humility, we accept that at
least some of our beliefs must be false. But which ones? The fact that we have accepted the
beliefs we have means that we think they are all true. But if so, how can we also accept the
belief that some of them are false? One might hope that it is the job of logic to help resolve
this kind of puzzle. But if so, how?
Logicians have traditionally distinguished between factual claims and logical
claims. The claim that Russell has a brother is a factual claim. In contrast, the claim that
“Russell has a sibling” follows from “Russell has a brother” is a logical claim.
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Traditionally it has been thought that factual claims are the kinds of things capable of being
learned by experience. Logical claims, in contrast, traditionally have been thought to be
learned by reason alone. False beliefs may turn out to be either factual or logical in nature.
If you mistakenly think that Russell has no brothers, this is a mistaken belief about how the
world is organized. If you think that Russell has no siblings even though he has a brother,
this is not so much a mistake about the world, but about which sentences follow from others.
In short, the former is a factual pickle; the second is a logical one.
To put the matter a little differently, in some cases we are mistaken about the truth
of a factual claim. In other cases, we are mistaken about the preservation of truth within an
inference. Or to use yet other words, sometimes confidence in a belief comes directly
through observation; sometimes it comes indirectly because it is inherited from other
sentences. Confidence that comes directly from experience begets confidence that does not.
The same is true of other properties such as reasonableness and believableness. On the
other hand, misgivings need not be inherited. Just because I might have misgivings that
Russell has a brother, it need not follow that I also have misgivings that he has a sibling.
Direct experience bestows a status upon some sentences that it does not bestow
upon others. It bestows confidence, say, that Russell has a brother. That confidence is then
inherited by other perhaps originally unforeseen sentences, just as citizenship that is
bestowed with ceremony upon members of one generation will be inherited without
ceremony by offspring as yet unconceived. The revoking of citizenship, by contrast, need
not affect the status of children.
This simile provides a key to understanding the science of logic. Just as in a
political community where some goods will be inherited between generations and some
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will not, some “goods” will also be inherited between “generations of sentences” while
others will not. Logic thus becomes an experimental branch of “property law” for
“communities of sentences.” Discovering which logical claims are true will not be
something to be discovered by reason alone since many of logic’s most fundamental
questions will only be capable of being answered within context-specific circumstances.
For example, within a given system of sentences and a given context, what constitutes a
good? And under what conditions will such goods be inherited?
Traditional property law originally evolved from tribal practices in which perhaps
only land, livestock and their derivative products originally were considered to be goods.
Today, logic is evolving in a similar way. Just as property law evolved as cooperative
societies created new goods, logic is extricating itself from an archaic custom in which
truth was considered the only sentential good. Nowadays governments charge
work-groups with anticipating legislative challenges that would arise from the many ways
that technology might force a broadening of the conception of social goods. But
technology and theoretical science also enrich human language as new philosophical
methods better enable us to understand language in general. With these changes comes a
new flexibility in our conception of what is of value in sentences; and this conception
evolves as our “communities of sentences” evolve. We can find out whether “The golf
clubs are in the closet” is true or false by looking in the closet, but there is no guarantee that
every sentence that a new theory throws up will have these, rather than some other,
properties. For example, “The neuroptron has inverse angularity” may turn out to be
neither true nor false in the physics of quasi-bleptonal curvatures, but it will be either
1/3-closed or 1/3-open or 1/3-clopen. Such new sentences of new languages may defy
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classification within the hoary old categories of truth and falsity. In bleptoplanar logic for
example, it is the property of 1/3-clopenness, not the property of truth that is inherited
under blepto-implication. (Sadly, for further information about the fascinating world of
bleptonal surface physics, and prolapsive physics more generally, the reader must wait
until the subject is invented.) Or here is a second example: if we are convinced that it is
both blowing and either snowing or raining we can readily convince ourselves that it is
either blowing and snowing or blowing and raining. But what holds universally for
meteorological phenomena need not hold universally for quantum phenomena. The
quantum physicist, confident in some degree that state A holds and state B or C holds can
remain uncommitted as to whether either state A and B holds or state A and C holds.
As governments charge committees of legal theorists with thinking about various
goods and their transfer, so universities and industry charge logical theorists with
discovering and thinking about preservation-worthy properties of sentences specific to
particular domains, whether it be the domain of quantum objects, or pop-up toasters or the
surface geometry of an air foil. The language in which such properties are expressed is by
no means fixed, and therefore neither is the range of values to which its sentences are
susceptible.
This relegation of truth to informal conversational uses should not be surprising. In
moments when we are as frank as Pontius Pilate, we realize that we haven’t much of an
idea what truth is. In introductory philosophy classes, instructors canvass a few classical
theories of truth, but soon let the subject drop, and no one worries when the vocabulary
recurs, as it does throughout their professional life and writings, without the matter ever
having been settled. For the logician, by contrast, the language of truth and falsity provides
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a reading for two values that are often labelled 1 and 0; but all the logician needs to know
for his purposes is that the two values are distinct. Which of the two is designated for
preservation is a matter of convenience and tradition. Moreover, the use of numerals
signals a readiness to add more values (1/2 or 2, say, or all the real numbers in the unit
interval) for greater diversity, accordingly as an application requires it or as an
experimental fancy takes him; and for a logician nothing dictates an interpretation of these
values except convenience or application. If you have an appliance that boasts fuzzy logic,
you can be sure that whatever logic is genuinely embodied in its design, it has more than
just two values.
And one last point about truth, before we move along. Since we don’t know what it
is, we don’t know whether it is a completely simple property with no properties of its own
or a relatively complex one. If it’s complex, then we might expect there to be closely
related subspecies of truth, which we are perhaps as yet unable to distinguish. In this case,
we might want not just to preserve truth, but to preserve particular types of truth. In this
case, we will need some way, perhaps numerical, of marking the distinctions. So here again
is opportunity for experimental logic, as we ask what combinations of logical principles
reliably preserve which kinds of truth.
This point also invites us to think about logic as experimental property law. In more
primitive times, goods were items within the disposition of an individual. We have said
already that the variety of such items has been vastly increased. But we must also
recognize that the class of items regarded as capable of having goods has also become a
good deal more complex, so that it now includes political entities such as cities, districts,
provinces, states and nations, as well as civil entities such as partnerships, corporations,
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foundations, universities and so on. Such entities can possess many of the goods that
individuals can, but the creation of such political and civil fictions has also created new
kinds of goods disposable only derivatively by component individuals.
A similar kind of development has occurred within the study of logic. Logicians
have long realized that even on no more than their non-committal understanding of truth,
questions can arise about its preservation from one body of data to another. Two avenues of
investigation have thus traditionally corresponded to two obvious generalizations of
inheritance. First, what system of logical principles would guarantee that if all of the
sentences in one body of data are true, then all of the sentences in the other one are too?
And second, what system would guarantee that if all of one body are true, then at least one
sentence of the other body is true?
More recently, a third question has led to an intriguing variety of theories with
much more general applications in data management and automated reasoning. This is the
question of which properties are preserved when a body of data is augmented one sentence
at a time. This of course takes us back to the beginning. Suppose that you have a body of
beliefs all gained directly from experience, and all individually true. This gives a derivative
but specifically “corporate” property to the set of beliefs, namely that all of its members are
true. Now we can ask which additional, unobserved sentences inherit truth from this set of
beliefs? If we can distinguish those sentences from those that do not, we can then add
sentences of the former sort one by one to the original body of beliefs such that each
resulting body of beliefs will inherit the corporate property of its predecessor set, namely
that all of its members are true. Standard logic performs this role adequately, even superbly
in some circumstances. Even if there are many sentences that inherit truth from the set but
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that standard logic does not let us infer, we know that any sentences that it does pick out
can be added to the set without loss of that corporate property.
Understanding logic in this way gives us a partial answer to our earlier question of
what constitutes a good. For a body of data, a good must at least be a property that merely
random enlargements cannot be relied upon to preserve. The application of logic must
then allow us to distinguish enlargements that inherit the good from enlargements that do
not. A good logic cannot be distinguished from a bad one independently from the
specification of a good to be preserved. As we have remarked, if the good to be preserved is
that all data in the body are true, then standard logic will normally be adequate if not
sufficient. But if not all of the data are true, then standard logic becomes inadequate
because the only property it preserves is not present to be preserved. Moreover, if we
distinguish only the conditions all true and not all true, the remaining property is not a
good, since it is inherited by any random enlargement of the body of data. Since in real life
most bodies of data are not, and in some cases such as the totality of one’s beliefs, cannot
be all true, practical applications have demanded that a wider range of corporate goods be
identified, and enlargements regulated in such a way as to preserve those goods that the
applications require to be inherited.
Recent logical theorists have occupied themselves with identifying corporate
goods that even inconsistent bodies of data can possess, and articulating the principles that
ensure their inheritance. Take, for example, the set of beliefs of an intellectually humble
person. Surely this will be an inconsistent set, but if the sole inconsistency is a
consequence of humility (believing that some of one’s beliefs must be false when in fact
this is one’s only false belief), then the inconsistency is, as we might say, maximally dilute,
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since it requires the whole body of beliefs to reveal itself. At the other extreme, imagine
believing a genuine contradiction such as “3 + 3 = 6 and 3 + 3
6.” This would be a
maximally concentrated contradiction, since it reveals itself in a single belief. Between the
two extremes, it is easy to see that all of the intermediate dilutions are possible, and that the
more dilute the better, since the less likely we are to reason from an inconsistent subset. In
fact any dilution sufficient to defeat our capacity to consider beliefs jointly should keep the
inconsistency below the radar of conscious thought. So a system of inference that
permitted no augmentation that increased concentration of inconsistency would ensure that
its dilution was preserved.
One other illustration will bring our discussion to a close. We should observe that if
a body of data has no single inconsistent sentence, that is, it has less than maximally
concentrated inconsistency, then other identifiable properties may assume importance as
corporate goods. Imagine that you are collecting data from, say, five independent sources,
or that a robot is collecting data from five difference senses, each of which is able to ensure
the consistency of its submissions. You, on the receiving end of the data might nevertheless
be faced with inconsistencies between submissions. In the worst case, you might find that
no pair of submissions could consistently be merged. Even in this dire circumstance, the
data you have received might have features worth preserving. For example, you might not
want to augment the data by adding inferences that required the total data now require
parcelling into six consistent batches, where before it required only five. You might add no
more than the data common to all five sources. This ultra-conservative strategy would take
unanimity as a good, and preserve that. A less conservative strategy would aim to preserve
what could be called the level of inconsistency of the data, that is, the least number of
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consistent bundles into which the data could be separated. One still conservative strategy
that achieves this aim would be to regard yourself as forced to add only those data that
could be standardly inferred from at least one bundle on every such separation. The task of
the applied logician is to devise a system of inferential rules that permit us to infer all and
only those sentences that can be added without increasing the level of incoherence.
Now, however difficult it may have been to accept that inconsistent data can have
any assets worth preserving, we have to admit that there are different kinds of
inconsistency, and that we can see that the nature of the inconsistency of a particular body
of data may be (economically, strategically, politically) important information.
Understood by analogy with the law of properties, logic has a fundamental role to play in
data exploitation, not as a tribunal, and certainly not as a judge issuing forth necessities, but
as a laboratory of imaginative experimentation and as a source of insight into possibilities.