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#Post#: 117--------------------------------------------------
Sample Answer: Deductive Logic
DIR By: CaseyEnos
Date: August 18, 2012, 12:53 am
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Deductive Logic is another subject that comes up on every exam.
This is a rehearsal draft for an answer I used on last years
exams.
"Any justification of deductive reasoning will itself rely on
deductive reasoning".
A deductive argument is one formulated so that, if validly
constructed and the premises are true, the conclusion must
likewise be true. Logical laws, such as Modus Ponens and the
functioning of syllogisms, are used to bring the argument
forward to its conclusions. The question immediately arises:
what garuntees the truth of the conclusion? If it is the logical
laws, the rules of deduction, what is that garuntees their
validity? It seems to most that the rules of deduction must be
justified, if they can be justified at all, by themselves, thus
generating a circular system.
One entertaining challenge to the system of deductive logic was
presented by Carroll in the story of the Tortoise and Achilles.
The former, presented with a straight forward case of Modus
Ponens (if P then Q, therefore Q) simply refuses to grant the
conclusion. Every time Achilles attempt to introduce a new
premise to the argument showing that if one accepts P one must
also accept Q, the stubborn tortoise simply continues to call
the process into question. As Achilles adds more and more
premises, an infinite regress beckons and finally he is forced
to admit the impossibility of proving the validity of the
process he is using other than in its own terms. Finding an
adequate response to the tortoise's disbelief is the task of
modern logicians who wish to provide some element of
justification to the use of deduction.
Reactions to Carroll's story have varied considerably. Some have
concluded that deductive reasoning is by nature circular, like
other human activities it can operate only with the introduction
of rules, and function only within those rules, thus for better
or worse a circles is inescapable. Searle, in his lectures,
countered what he considered to be the key mistake, that of
treating the deductive basis of the argument as if it was simply
another premise; in fact it is the fundamental basis for thought
and as such cannot be intelligibly challenged. Russell
addressed the specific challenge to Modus Ponens by drawing a
distinction between "implies" in the traditional logical sense;
and between P and Q, either or both of which or both of which
may or not obtain; and "entails", which links together a P and Q
asserted by the argument. Clearly the two are very different
propositions, with the later allowing the unchallenged move from
P to Q, but with the introduction of A Prosteri elements,
which, as we will see, are disallowed by most accounts of
deduction.
Another, more serious challenge, was posed by Hume. Hume began
by challenging the concept of deductive reasoning, attacking it
on the notion that any sort of reasoning which has worked so far
will only hold valid on the untestable hypothesis that the
future will resemble the past, an objection presented more
formally by Goodman. Therefore, any attempt to support deduction
through the use of inductive reasoning will inevitably fall
short of the absolute certainty required for deduction. For
Hume, the uncertain validity of inductive seasoning leaves
deduction without any sort of support, a position which he
viewed as untenable. At best, deductive reasoning, which is true
only by the relationship of the ideas it contains, is separated
from any sort of contact with the outside world
Modern attempts to provide justification for deduction in the
face of such challenges have focused on the notion of "logical
consequences", that is on the provisions of logic which allow
sentence to be deduced one another, thus allowing the movement
from premise to conclusion. These "deductive theoretic" systems
are based on the principle that 'X is a logical consequence of K
if and only if X can be deduced from K in a deductive system'.
Such a system, relying entirely on deduction, must be carefully
separated from a "modal-theoretic" system in which X is a
consequence of K in all available models, a type of system which
operates under different, not forcibly deductive premises, in
attempting to establish that X is a necessary truth given K.
Tarski, attempting to define "logical consequences" in terms of
its normal usages, gives three rules for the deduction of one
sentence from another:
1. The logical relation depends entirely on the formal aspects
of the systems involved
2. The relation is a A Proiri , that is, independent of any
outside evidence.
3. The deduction is necessary, given the system of rules in
which one is operating.
While this is one of the most developed of criteria for
deductions, possible questions criticisms are immediately
evident. Most importantly for the question we are considering is
the requirement that justification be conducted in an A
Priori[i][/i]manner and based entirely on formal elements, these
together ruling out any attempt to base a system of deduction on
anything other than its own rules and axioms. Ayer agreed,
noting that deductive statements are true only by the
definitions of the operators involved, however even this view is
a problematic one since the definition of logical operators is
itself a subject of continuous controversy.
Based on these criteria for deduction, we are still left with an
entirely circular system, one that we can only be extracted from
by assertions like Searles that deductive logic forms an
unchallengeable foundation for thought. However, it is very
difficult so see on what grounds deduction could then be
justified, other than by observation of how it holds across
cases, so that by definition it would no longer deductive, or by
accepting it as not in need of justification. The later is also
a highly questionable option, especially given recent challenges
posed by modern physics to basic logical laws used in deduction.
Deduction has been defined by the metaphor of someone in a dark
room who has complete knowledge of their language, but none of
the outside world; whatever conclusions they can piece together
must be deductively valid. Even a system with such a limited
scope may be open to challenge. Quine stressed that no portion
of human knowledge can be held to be immune from revision, even
the logical laws in which deduction are based on are open to
revision, notably since quantum mechanics has recently
provided a strong challenge to the law of the excluded middle
and other previously basic rules. Therefore, it seems that as
human knowledge expand, even what was at one point considered
true by definition is open to revision, and deductive logic is
left without any tenable basis. Because logical truths are by
definition to hold across an infinite number of possible cases,
yet they are based on definitions which are subject to change
and revision, they cannot be said to have any justification at
all.
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