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The term logic
comes from the Greek word logos. The variety of senses that logos possesses
may suggest the difficulties to be encountered in characterizing the nature and
scope of logic. Among the partial translations of logos, there are
"sentence," "discourse," "reason,"
"rule," "ratio," "account" (especially the account
of the meaning of an expression), "rational principle," and
"definition." Not unlike this proliferation of meanings, the subject
matter of logic has been said to be the "laws of thought,"
"the rules of right reasoning," "the principles of valid
argumentation," "the use of certain words labelled 'logical
constants'," "truths (true propositions) based solely on the meanings
of the terms they contain," and so on. |
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7.1.1.1
Nature and varieties of logic.
It is relatively
easy to discern some order in the above embarrassment of explanations. Some of
the characterizations are in fact closely related to each other. When logic is
said, for instance, to be the study of the laws of thought, these laws cannot be
the empirical (or observable) regularities of actual human thinking as studied
in psychology; they must be laws of correct reasoning, which are independent of
the psychological idiosyncrasies of the thinker. Moreover, there is a
parallelism between correct thinking and valid argumentation: valid argumentation
may be thought of as an expression of correct thinking, and the latter as an
internalization of the former. In the sense of this parallelism, laws of correct
thought will match those of correct argumentation. The characteristic mark of
the latter is, in turn, that they do not depend on any particular matters of
fact. Whenever an argument that takes a reasoner from p to q is
valid, it must hold independently of what he happens to know or believe about
the subject matter of p and q. The only other source of the
certainty of the connection between p and q, however, is
presumably constituted by the meanings of the terms that the propositions p and
q contain. These very same meanings will then also make the sentence
"If p, then q" true irrespective of all contingent
matters of fact. More generally, one can validly argue from p to q if
and only if the implication "If p, then q" is logically
true--i.e., true in virtue of the meanings of words occurring in p and
q, independently of any matter of fact. |
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Logic may thus be
characterized as the study of truths based completely on the meanings
of the terms they contain. |
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In order to
accommodate certain traditional ideas with- in the scope of this formulation,
the meanings in question may have to be understood as embodying insights into
the essences of the entities denoted by the terms, not merely codifications of
customary linguistic usage. |
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The following
proposition (from Aristotle), for instance, is a simple truth of logic: "If
sight is perception, the objects of sight are objects of perception." Its
truth can be grasped without holding any opinions as to what, in fact, the
relationship of sight to perception is. What is needed is merely an
understanding of what is meant by such terms as "if-then,"
"is," and "are," and an understanding that "object
of" expresses some sort of relation. |
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The logical truth
of Aristotle's sample proposition is reflected by the fact that "The
objects of sight are objects of perception" can validly be inferred from
"Sight is perception." (see also Aristotelianism)
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Many questions
nevertheless remain unanswered by this characterization. The contrast between
matters of fact and relations between meanings that was relied on in the
characterization has been challenged, together with the very notion of meaning.
Even if both are accepted, there remains a considerable tension between a wider
and a narrower conception of logic. According to the wider interpretation, all
truths depending only on meanings belong to logic. It is in this sense that the
word logic is to be taken in such designations as "epistemic logic"
(logic of knowledge), "doxastic logic" (logic of belief),
"deontic logic" (logic of norms), "the logic of science,"
"inductive logic," and so on. According to the narrower conception,
logical truths obtain (or hold) in virtue of certain specific terms, often
called logical constants.
Whether they can be given an intrinsic characterization or whether they can be
specified only by enumeration is a moot point. It is generally agreed, however,
that they include (1) such propositional connectives as "not,"
"and," "or," and "if-then" and (2) the so-called
quantifiers "(
 x)"
(which may be read: "For at least one individual, call it x, it is
true that") and "(
 x)"
("For each individual, call it x, it is true that"). The dummy
letter x is here called a bound ( individual)
variable. Its values are supposed to be members of some fixed class
of entities, called individuals, a class that is variously
known as the universe of discourse, the universe presupposed in an
interpretation, or the domain of individuals. Its members are said to be
quantified over in "(
x)"
or "(
x)."
Furthermore, (3) the concept of identity
(expressed by =) and (4) some notion of predication
(an individual's having a property or a relation's holding between several
individuals) belong to logic. The forms that the study of these logical
constants take are described in greater detail in the article LOGIC, in which
the different kinds of logical notation are also explained. Here, only a
delineation of the field of logic is given. (see also quantification,
bound variable)
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When the terms in
(1) alone are studied, the field is called propositional logic. When (1), (2),
and (4) are considered, the field is the central area of logic that is variously
known as first-order logic, quantification theory, lower predicate calculus,
lower functional calculus, or elementary logic. If the absence of (3) is
stressed, the epithet "without identity" is added, in contrast to
first-order logic with identity, in which (3) is also included. (see also propositional
calculus) |
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Borderline cases
between logical and nonlogical constants are the following (among others): (1)
Higher order quantification, which means quantification not over the individuals
belonging to a given universe of discourse, as in first-order logic, but also
over sets of individuals and sets of n-tuples of individuals.
(Alternatively, the properties and relations that specify these sets may be
quantified over.) This gives rise to second-order logic. The process can be
repeated. Quantification over sets of such sets (or of n-tuples of such
sets or over properties and relations of such sets) as are considered in
second-order logic gives rise to third-order logic; and all logics of finite
order form together the (simple) theory of (finite) types. (2) The membership
relation, expressed by
 , can
be grafted on to first-order logic; it gives rise to set
theory. (3) The concepts of (logical) necessity and (logical)
possibility can be added. |
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This narrower sense
of logic is related to the influential idea of logical form. In any given
sentence, all of the nonlogical terms may be replaced by variables of the
appropriate type, keeping only the logical constants intact. The result is a
formula exhibiting the logical form of the sentence. If the formula results in a
true sentence for any substitution of interpreted terms (of the appropriate
logical type) for the variables, the formula and the sentence are said to be
logically true (in the narrower sense of the expression). |
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Three areas of
general concern are the following. |
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7.1.1.2.1
Logical
semantics.
For the purpose of
clarifying logical truth and hence the concept of logic itself, a tool that has
turned out to be more important than the idea of logical form is logical
semantics, sometimes also known as model theory. By this is meant a
study of the relationships of linguistic
expressions to those structures in which they may be interpreted and of which
they can then convey information. The crucial idea in this theory is that of
truth (absolutely or with respect to an interpretation). It was first analyzed
in logical semantics around
1930 by the Polish-American logician Alfred
Tarski. In its different variants, logical semantics is the central
area in the philosophy of logic. It enables the logician to characterize the
notion of logical truth irrespective of the supply of nonlogical constants that
happen to be available to be substituted for variables, although this supply had
to be used in the characterization that turned on the idea of logical form. It
also enables him to identify logically true sentences with those that are true
in every interpretation (in "every possible world"). (see also metalogic)
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The ideas on which
logical semantics is based are not unproblematic, however. For one thing, a
semantical approach presupposes that the language in question can be viewed
"from the outside"; i.e., considered as a calculus that can be
variously interpreted and not as the all-encompassing medium in which all
communication takes place (logic as calculus versus logic as language). (see
also Analytic philosophy) |
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Furthermore, in
most of the usual logical semantics the very relations that connect language
with reality are left unanalyzed and static. Ludwig
Wittgenstein, an Austrian-born philosopher, discussed informally the
"language-games"--or rule-governed activities connecting a language
with the world--that are supposed to give the expressions of language their
meanings; but these games have scarcely been related to any systematic logical
theory. Only a few other attempts to study the dynamics of the representative
relationships between language and reality have been made. The simplest of these
suggestions is perhaps that the semantics of first-order logic should be
considered in terms of certain games (in the precise sense of game
theory) that are, roughly speaking, attempts to verify a given
first-order sentence. The truth of the sentence would then mean the existence of
a winning strategy in such a game. |
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Many philosophers
are distinctly uneasy about the wider sense of logic. Some of their
apprehensions, voiced with special eloquence by a contemporary Harvard
University logician, Willard Van Quine,
are based on the claim that relations of synonymy cannot be fully determined by
empirical means. Other apprehensions have to do with the fact that most
extensions of first-order logic do not admit of a complete axiomatization; i.e.,
their truths cannot all be derived from any finite--or recursive (see
below)--set of axioms. This
fact was shown by the important "incompleteness" theorems proved in
1931 by Kurt G?el, an
Austrian (later, American) logician, and their various consequences and
extensions. (G?el showed that any consistent axiomatic theory that comprises a
certain amount of elementary arithmetic is incapable of being completely
axiomatized.) Higher-order logics are in this sense incomplete and so are all
reasonably powerful systems of set theory. Although a semantical theory can be
built for them, they can scarcely be characterized any longer as giving actual
rules--in any case complete rules--for right reasoning or for valid
argumentation. Because of this shortcoming, several traditional definitions of
logic seem to be inapplicable to these parts of logical studies. (see also axiomatic
method ) |
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These apprehensions
do not arise in the case of modal logic,
which may be defined, in the narrow sense, as the study of logical necessity and
possibility; for even quantified modal logic admits of a complete
axiomatization. Other, related problems nevertheless arise in this area. It is
tempting to try to interpret such a notion as logical necessity as a syntactical
predicate; i.e., as a predicate the applicability of which depends only
on the form of the sentence claimed to be necessary--rather like the
applicability of formal rules of proof. It has been shown, however, by Richard Montague,
an American logician, that this cannot be done for the usual systems of modal
logic.
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These findings of
G?el and Montague are closely related to the general study of computability,
which is usually known as recursive
function theory (see MATHEMATICS,
FOUNDATIONS OF: The crisis in foundations following 1900: Logicism,
formalism, and the metamathematical method
) and which is one of the most important branches of contemporary
logic. In this part of logic, functions--or
laws governing numerical or other precise one-to-one or many-to-one
relationships--are studied with regard to the possibility of their being
computed; i.e., of being effectively--or mechanically--calculable.
Functions that can be so calculated are called recursive. Several different and
historically independent attempts have been made to define the class of all
recursive functions, and these have turned out to coincide with each other. The
claim that recursive functions exhaust the class of all functions that are
effectively calculable (in some intuitive informal sense) is known as Church's
thesis (named after the American logician Alonzo Church). |
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One of the
definitions of recursive functions is that they are computable by a kind of
idealized automaton known as a Turing
machine (named after Alan Mathison Turing, a British mathematician
and logician). Recursive function theory may therefore be considered a theory of
these idealized automata. The main idealization involved (as compared with
actually realizable computers) is the availability of a potentially infinite
tape. |
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The theory of
computability prompts many philosophical questions, most of which have not so
far been answered satisfactorily. It poses the question, for example, of the
extent to which all thinking can be carried out mechanically. Since it quickly
turns out that many functions employed in mathematics--including many in
elementary number theory--are nonrecursive, one may wonder whether it follows
that a mathematician's mind in thinking of such functions cannot be a mechanism
and whether the possibly nonmechanical character of mathematical thinking may
have consequences for the problems of determinism and free will. Further work is
needed before definitive answers can be given to these important questions. |
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In addition to the
problems and findings already discussed, the following topics may be mentioned. |
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7.1.2.1
Meaning and truth.
Since 1950, the
concept of analytical truth (logical truth in the wider sense) has been
subjected to sharp criticism, especially by Quine. The main objections turned
around the nonempirical character of analytical truth (arising from meanings
only) and of the concepts in terms of which it could be defined--such as
synonymy, meaning, and logical necessity. The critics usually do not contest the
claim that logicians can capture synonymies and meanings by starting from
first-order logic and adding suitable further assumptions, though definitory
identities do not always suffice for this purpose. The crucial criticism is that
the empirical meaning of such further "meaning postulates" is not
clear. (see also analytic proposition )
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In this respect,
logicians' prospects have been enhanced by the development of a semantical
theory of modal logic, both in the narrower sense of modal logic, which is
restricted to logical necessity and logical possibility, and in the wider sense,
in which all concepts that exhibit similar logical behaviour are included. This
development, initiated between 1957 and 1959 largely by Stig Kanger
of Sweden and Saul Kripke of
the U.S., has opened the door to applications in the logical analysis of many
philosophically central concepts, such as knowledge, belief, perception, and
obligation. Attempts have been made to analyze from the viewpoint of logical
semantics such philosophical topics as sense-datum theories, knowledge by
acquaintance, the paradox of saying and disbelieving propounded by the British
philosopher G.E. Moore, and the traditional distinction between statements de
dicto ("from saying") and statements de re ("from the
thing"). These developments also provide a framework in which many of those
meaning relations can be codified that go beyond first-order logic, and may
perhaps even afford suggestions as to what their empirical content might be.
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Especially in the
hands of Montague, the logical semantics of modal notions has blossomed into a
general theory of intensional logic; i.e., a theory of such notions as
proposition, individual concept, and in general of all entities usually thought
of as serving as the meanings of linguistic expressions. (Propositions are the
meanings of sentences, individual concepts are those of singular terms, and so
on.) A crucial role is here played by the notion of a possible world, which may
be thought of as a variant of the logicians' older notion of model, now
conceived of realistically as a serious alternative to the actual course of
events in the world. In this analysis, for instance, propositions are functions
that correlate possible worlds with truth-values.
This correlation may be thought of as spelling out the older idea that to know
the meaning of a sentence is to know under what circumstances (in which possible
worlds) it would be true. |
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Even though none of
the problems listed seems to affect the interest of logical semantics, its
applications are often handicapped by the nature of many of its basic concepts.
One may consider, for instance, the analysis of a proposition as a function that
correlates possible worlds with truth-values. An arbitrary function of this sort
can be thought of (as can functions in general) as an infinite class of pairs of
correlated values of an independent variable and of the function, like the
coordinate pairs (x, y) of points on a graph. Although propositions are
supposed to be meanings of sentences, no one can grasp such an infinite class
directly when understanding a sentence; he can do so only by means of some
particular algorithm, or recipe (as it were), for computing the function in
question. Such particular algorithms come closer in some respects to what is
actually needed in the theory of meaning than the meaning entities of the usual
intensional logic. |
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This observation is
connected with the fact that, in the usual logical semantics, no finer
distinctions are utilized in semantical discussions than logical
equivalence. Hence the transition from one sentence to another
logically equivalent one is disregarded for the purposes of meaning concepts.
This disregard would be justifiable if one of the most famous theses of Logical
Positivists were true in a sufficiently strong sense, viz., that
logical truths are really tautologies (such as "It is either raining or not
raining") in every interesting objective sense of the word. Many
philosophers have been dissatisfied with the stronger forms of this thesis, but
only recently have attempts been made to spell out the precise sense in which
logical and mathematical truths are informative and not tautologous. (see also foundations
of mathematics ) |
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Among the
ontological problems--problems concerning existence and existential
assumptions--arising in logic are those of individuation and existence. |
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7.1.2.2.1
Individuation.
Not all interesting
interpretational problems are solved by possible-world semantics, as the
developments earlier registered are sometimes called. The systematic use of the
idea of possible worlds has raised, however, the subject of cross
identification; i.e., of the principles according to which a member of
one possible world is to be found identical or nonidentical with one of another.
Since one can scarcely be said to have a concept of an individual if he cannot
locate it in several possible situations, the problem of cross-identification is
also one of the most important ingredients of the logical and philosophical
problem of individuation. The criticisms that Quine has put forward concerning
modal logic and analyticity (see above Limitations
of logic ) can be deepened
into questions concerning methods of cross identification. Although some such
methods undoubtedly belong to everyone's normal unarticulated conceptual
repertoire, it is not clear that they are defined or even definable widely
enough to enable philosophers to make satisfactory sense of a quantified logic
of logical necessity and logical possibility. The precise principles used in
ordinary discourse--or even in the language of science--pose a subtle
philosophical problem. The extent to which special "essential
properties" are relied on in individuation and the role of spatio-temporal
frameworks are moot points here. It has also been suggested that essentially
different methods of cross identification are actually used together, some of
them depending on impersonal descriptive principles and others on the
perspective of a person. |
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Because one of the
basic concepts of first-order logic is that of existence, as codified by the
existential quantifier "(
x),"
one might suppose that there is little room left for any separate philosophical
problem of existence. Yet existence, in fact, does seem to pose a problem, as
witnessed by the bulk of the relevant literature. Some issues are relatively
easy to clarify. In the usual formulations of first-order logic, for instance,
there are "existential presuppositions" present to the effect that
none of the singular terms employed is without a bearer (as "Pegasus"
is). It is a straightforward matter, however, to dispense with these
presuppositions. Though this seems to involve the procedure, branded as
inadmissible by many philosophers, of treating existence as a predicate, this
can nonetheless be easily done on the formal level. Given certain assumptions,
it may even be shown that this "predicate" will have to be "(
x) (x = a)" (for "a exists"--literally,
"There exists an x such that x is a") or
something equivalent. Furthermore, the logical peculiarities of this predicate
seem to explain amply philosophers' apparent denial of its reality. |
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The interest in the
notion of existence is connected with the question of what entities a theory
commits its holder to or what its "ontology" is. The "predicate
of existence" just mentioned recalls Quine's criterion of ontological
commitment: "To be is to be a value of a bound variable"--i.e., of
the x in (
x)
or in (
x).
According to Quine, a theory is committed to those and only those entities that
in the last analysis serve as the values of its bound variables. Thus ordinary
first-order theory commits one to an ontology only of individuals (particulars),
whereas higher order logic commits one to the existence of sets--i.e., of
collections of definite and distinct entities (or, alternatively, of properties
and relations). Likewise, if bound first-order variables are assumed to range
over sets (as they do in set theory), a commitment to the existence of these
sets is incurred. |
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The doctrine that
an ontology of individuals is all that is needed is known as (the modern version
of) nominalism. The opposite
view is known as (logical) realism.
Even those philosophers who profess sympathy with nominalism find it hard,
however, to maintain that mathematics could be built on a consistently
nominalistic foundation. |
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The precise import
of Quine's criterion of ontological commitment, however, is not completely
clear. Nor is it clear in what other sense one is perhaps committed by a theory
to those entities that are named or otherwise referred to in it but not
quantified over in it. Questions can also be raised concerning the very
distinction between what in modern logic are usually called individuals
("particulars" would be a more traditional designation) and such
universals as their properties and relations; and these questions can be
combined with others concerning the "tie" that binds particulars and
universals together in predication. |
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An interesting
approach to these problems is the distinction made by Gottlob
Frege, a pioneer of mathematical logic in the late 19th century,
between individuals--he called them objects--and what he called functions (which
in his view include concepts) and his doctrine of the unsaturated character of
the latter, according to which a function (as it were) contains a gap, which can
be filled by an object. Another approach is the "picture theory of
language" of Wittgenstein's Tractatus Logico-Philosophicus, according
to which a simple sentence presents a person with an isomorphic representation
(a "picture") of reality as it would be if the sentence were true.
According to this view (which was later given up by Wittgenstein), "a
sentence [or proposition, Satz] is a model of reality such as we think of
it as being." |
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The natures of most
of the so-called nonclassical logics can be understood against the background of
what has here been said. Some of them are simply extensions of the
"classical" first-order logic--e.g., modal logics and many
versions of intensional logic. The so-called free logics are simply first-order
(or modal) logics without existential presuppositions. |
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One of the most
important nonclassical logics is intuitionistic logic, first formalized by the
Dutch mathematician Arend Heyting
in 1930. It has been shown that this logic can be interpreted in terms of the
same kind of modal logic serving as a system of epistemic logic. In the light of
its purpose to consider only the known, this isomorphism is suggestive. The
avowed purpose of the intuitionist
is to consider only what can actually be established constructively in logic and
in mathematics--i.e., what can actually be known. Thus, he refuses
to consider, for example, "Either A or not-A" as a logical truth, for
it does not actually help one in knowing whether A or not-A is the case. This
does not close, however, the philosophical problem about intuitionism. Special
problems arise from intuitionists' rejection (in effect) of the nonepistemic
aspects of logic, as illustrated by the fact that only a part of epistemic logic
is needed in this translation of intuitionistic logic into epistemic logic. (see
also intuitionistic calculus)
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Other new logics
are obtained by modifying the rules of those games that are involved in the
game-theoretical interpretation of first-order logic mentioned above. The
logician may reject, for instance, the assumption that he possesses perfect
information, an assumption that characterizes classical first-order logic. One
may also try to restrict the strategy sets of the players--to recursive
strategies, for example. |
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Among the oldest
kinds of alternative logics are many-valued
logics. In them, more truth values than the usual true and false are
assumed. The idea seems very natural when considered in abstraction from the
actual use of logic. But a philosophically satisfactory interpretation of
many-valued logics is not equally straightforward. The interest in finite-valued
logics and the applicability of them are sometimes exaggerated. The idea,
however, of using the elements of an arbitrary Boolean algebra--a generalized
calculus of classes--as abstract truth-values has provided a powerful tool for
systematic logical theory. |
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7.1.3.1
Technical disciplines.
The relations of
logic to mathematics, to computer
technology, and to the empirical sciences are here considered. |
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7.1.3.1.1
Mathematics.
It is usually said
that all of mathematics can, in principle, be formulated in a sufficiently
theorem-rich system of axiomatic set theory. What the axioms of a set theory
that could accomplish this might be, however, and whether they are at all
natural is not obvious in every case. (The recent development in abstract
algebra known as category theory offers the most conspicuous examples of these
problems.) The axioms of set theory may be presumed to hold in virtue of the
meanings of the terms set, member of, and so on. Thus, in some loose sense all
of pure mathematics falls within the scope of logic in the wider sense. This
assertion is not very informative, however, as long as the logician has no ways
of analyzing these meanings so as to be able to tell what assumptions (axioms of
set theory) should be adopted. The definitions of basic mathematical concepts
(such as "number") in logical terms proposed by Gottlob Frege (in
1884), by Bertrand Russell (in 1903), and by their successors do not help in
this enterprise. It is not clear that more recent insights in logic help very
much, either, in the search for strong set-theoretical assumptions. The
relationship of mathematics to logic on this level therefore remains ambiguous. |
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Notwithstanding
these deep problems, virtually all normal mathematical argumentation is carried
out in logical terms--mostly in first-order terms, but with a generous
sprinkling of second-order reasoning and various principles of set theory.
Historically speaking, most specific early examples of nontrivial logical
reasoning were taken from mathematics. |
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Often these
examples were set in contrast to logical arguments understood in a narrow
traditional sense--in a sense narrower still than the idea of logic as being
exhausted by quantification theory. According to this traditional view, logic is
equated with syllogistic; i.e.,
with a part of that part of first-order logic that deals with properties and
not with relations. Much of what earlier philosophers said of mathematical
reasoning must, thus, be understood as applying to relational (first-order)
reasoning. The present-day philosophy of logic is therefore as much an heir to
traditional philosophy of mathematics as to traditional philosophy of logic. |
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Specific logical
results are applicable in several parts of mathematics, especially in algebra,
and various concepts and techniques used by logicians have often been borrowed
from mathematics. (Thus one can even speak of "the mathematics of
metamathematics.") |
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It has already been
indicated that recursive function theory is, in effect, the study of certain
idealized automata (computers). It is, in fact, a matter of indifference whether
this theory belongs to logic or to computer science. The idealized assumption of
a potentially infinite computer tape, however, is not a trivial one: Turing
machines typically need plenty of tape in their calculations. Hence the step
from Turing machines to finite automata (which are not assumed to have access to
an infinite tape) is an important one. |
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This limitation
does not dissociate computer science from logic, however, for other parts of
logic are also relevant to computer science and are constantly employed there.
Propositional logic may be thought of as the "logic" of certain simple
types of switching circuits. There are also close connections between automata
theory and the logical and algebraic study of formal languages. An interesting
topic on the borderline of logic and computer science is mechanical theorem
proving, which derives some of its interest from being a clear-cut instance of
the problems of artificial intelligence, especially of the problems of realizing
various heuristic modes of thinking on computers. In theoretical discussions in
this area, it is nevertheless not always understood how much textbook logic is
basically trivial and where the distinctively nontrivial truths of logic
(including first-order logic) lie. |
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The quest for
theoretical self-awareness in the empirical sciences has led to interest in methodological
and foundational problems as well as to attempts to axiomatize different
empirical theories. Moreover, general methodological problems, such as the
nature of scientific explanations, have been discussed intensively among
philosophers of science. In all of these endeavours, logic plays an important
role. (see also social science)
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By and large, there
are here three different lines of thought. (1) Often, only the simplest parts of
logic--e.g., propositional logic--are appealed to (over and above the
mere use of logical notation). Sometimes, claims regarding the usefulness of
logic in the methodology of the empirical sciences are, in effect, restricted to
such rudimentary applications. This restriction is misleading, however, for most
of the interesting and promising connections between methodology and logic lie
on a higher level, especially in the area of model theory. In econometrics,
for instance, a special case of the logicians' problems of definability plays an
important role under the title "identification problem." On a more
general level, logicians have been able to clarify the concept of a model as it
is used in the empirical sciences. |
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In addition to
those employing simple logic, two other contrasting types of theorists can be
distinguished: (2) philosophers of science, who rely mostly on first-order
formulations, and (3) methodologists (e.g., Patrick Suppes, a U.S.
philosopher and behavioral scientist), who want to use the full power of set
theory and of the mathematics based on it. Both approaches have advantages.
Usually realistic axiomatizations and other reconstructions of actual scientific
theories are possible only in terms of set theoretical and other strong
mathematical conceptualizations (theories conceived of as "set-theoretical
predicates"). In spite of the oversimplification that first-order
formulations often entail, however, they can yield theoretical insights because
first-order logic (including its model theory) is mastered by logicians much
more thoroughly than is set theory. |
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Many empirical
sciences, especially the social sciences, use mathematical tools borrowed from
probability theory and statistics, together with such outgrowths of these as
decision theory, game theory, utility theory, and operations research. A modest
but not uninteresting beginning in the study of their foundations has been made
in modern inductive logic. |
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The relations of
logic to linguistics, psychology, law, and education are here considered. |
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7.1.3.2.1
Linguistics.
The revival of
interest in semantics among theoretical linguists in the late 1960s awakened
their interest in the interrelations of logic and linguistic theory as well. It
was also discovered that certain grammatical problems are closely related to
logicians' concepts and theories. A near-identity of linguistics and
"natural logic" has been claimed by the U.S. linguist George Lakoff.
Among the many conflicting and controversial developments in this area, special
mention may perhaps be made of attempts by Jerrold
J. Katz, a U.S. grammarian-philosopher, and others to give a
linguistic characterization of such fundamental logical notions as analyticity;
the sketch by Montague of a "universal grammar" based on his
intensional logic; and the suggestion (by several logicians and linguists) that
what linguists call "deep structure" is to be identified with logical
form. Of a much less controversial nature is the extensive and fruitful use of
recursive function theory and related areas of logic in formal grammars and in
the formal models of language users.
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Although the
"laws of thought" studied in logic are not the empirical
generalizations of a psychologist, they can serve as a conceptual framework for
psychological theorizing. Probably the best known recent example of such
theorizing is the large-scale attempt made in the mid-20th century by Jean
Piaget, a Swiss psychologist, to characterize the developmental
stages of a child's thought by reference to the logical structures that he can
master. |
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Elsewhere in
psychology, logic is employed mostly as an ingredient of various models using
mathematical ideas or ideas drawn from such areas as automata or information
theory. Large-scale direct uses are rare, however, partly because of the
problems mentioned above in the section on logic and information. |
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Philosophies
of the Branches of Knowledge
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Of the great
variety of kinds of argumentation used in the law, some are persuasive rather
than strictly logical, and others exemplify different procedures in applied
logic rather than the formulas of pure logic. Examinations of "Lawiers
Logike"--as the subject was called in 1588--have also uncovered a variety
of arguments belonging to the various departments of logic mentioned above. Such
inquiries do not seem to catch the most characteristic kinds of legal
conceptualization, however--with one exception, viz., a theory developed
by Wesley Newcomb Hohfeld, a pre-World War I U.S. legal
scholar, of what he called the fundamental legal conceptions. Although
originally presented in informal terms, this theory is closely related to recent
deontic logic (in some cases
in combination with suitable causal notions). Even some of the apparent
difficulties are shared by the two approaches: the deontic logician's notion of
permission, for example, which is often thought of as being unduly weak, is to
all practical purposes a generalization of Hohfeld's concept of privilege.
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After having been
one of the main ingredients of the general school curriculum for centuries,
logic virtually disappeared from liberal education during the first half of the
20th century. It has made major inroads back into school curricula, however, as
a part of the new mathematical curriculum that came into fairly general use in
the 1960s, which normally includes the elements of propositional logic and of
set theory. Logic is also easily adapted to being taught by computers and has
been used in experiments with computer-based education. (see also education,
history of ) |
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(K.J.Hi.) |
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