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| 5
June (Wednesday)
5:00 PM Room
226 |
| Sašo
Živanović
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Department
of Comparative and General
Linguistics
University of Ljubljana
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Towards
a cognitively
plausible deductive
system
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We present a novel kind
of deductive system, which yields
formal deductions simple enough to
be part of a cognitively plausible
theory of human reasoning. The
main characteristics of our Deep
Deductive System (DDS) is that the
rules operate on constituents of
proof lines: in principle, any
constituent of a line can function
as a premise of a rule and the
conclusion of the rule can replace
any constituent of a line. (The
latter is also a property of the
Replacement rule of conventional
deductive systems.) In fact,
the "deep" nature of the system
makes it unnecessary to ever refer
to any line of proof but the
preceding line; proofs in DDS may
thus be viewed as a single,
dynamically changing formula.
The crux of the system is the
notion of premise scope (p-scope),
which determines which
constituents may function as a
premise for a given target.
P-scope is computed off the syntax
of the formula, crucially
employing the notion of polarity:
constituents within the scope of
an even/odd number of negations
have positive/negative polarity;
this requires that the formal
language uses only conjunction,
disjunction and negation as
sentential connectives.
The rules of DDS are very simple
(as usual, there are several
possibilities regarding the
specific choice of rules; only one
is presented here): Deep Axiom
Introduction, Deep Simplification
(a generalized conjunction
elimination), Deep Elimination and
Pruning rules (together, the
latter two function as a
generalized disjunctive
syllogism). The axioms are even
simpler: the system uses a single
axiom schema, Law of Excluded
Middle.
Several advantages of the system:
(i) Cognitive plausibility in
comparison to conventional
systems. (a) Deductions in DDS are
much simpler than in conventional
systems (this is particularily
true of Hilbert-style systems). In
particular, the statement and
proof of Deduction Theorem are so
trivial that it makes no sense to
ever employ it in practice; in
other words, conditional reasoning
is an integral part of the system.
(b) DDS introduces no machinery
specific to reasoning, like proof
trees of Gentzen-style systems, or
even proof lines of Hilbert-style
systems. The system works on the
structure of the formal language
alone.
(ii) While the system can be
easily accommodated to yield
standard logic, it seems even more
elegant as a deductive system for
inclusive logic, under the
additional assumption (very
plausible in application to
natural language) that only
restricted quantification is
allowed.
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