Truth and Reality
In order to judge the truth of statements,
i.e., their denotation,
one has to understand, at least in part, what they mean,
i.e., what their sense / linguistic content is.
-> see meaning
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[bdw] Rüdiger Vaas: Der kosmische Code; Bild der Wissenschaft 12/2003. My translation.
[Compl] Mario Bunge: Strife About Complementarity (two parts); The British Jounral for the Philopsophy of Science VI(21, 22) May, Aug 1955.
[DvR] Walter Robert Fuchs: Denkspiele vom Reißbrett - eine Einführung in die moderne Philosophie; Droemer Knaur 1972.
[KRCR] Nino B Cocchiarella: Knowledge representation in conceptual realism; 697-721: Int J Human-Computer Studies 43; 1995.
[OntRed] Reinhardt Grossmann: Ontological Reduction; Indiana Univ. Press 1973.
[PV] James H Fetzer: Program Verifaction: The Very Idea; 183-220 in: James H Fetzer: Computers and Cognition: Why Minds are not Machines; Kluwer 2001.
[PV2] James H Fetzer: Philosophy and Computer Science: Reflections on the Program Verifaction Debate
247-267 in: James H Fetzer: Computers and Cognition: Why Minds are not Machines; Kluwer 2001.
The referents of general terms like "red", "table", "tree", "familiar", "annual"
(that was is denoted by these terms)
are traditionally called universals (Plato: 'forms', Aristotle: 'properties'),
«understood as entities distinct from any of the particular things
describable by those terms» [x].
«The problem of universals boils down to the questions:
What are universals and how do they exist if at all?» [MB3 104].
Is There a Many-as-One, or Are There Only the Many Themselves?
Consider for example classes, groups, sets:
«A class is not something over and above its several members, and the
members are the class. Someone who admits that there is more than
one individual thereby admits that there is a class of more than
one individual. In particular, a class of several individuals is not a new,
higher-order, abstract individual. A class of several concrete individuals
is itself a concrete particular, though not a concrete individual. This
conception of classes, as `low-brow' collections rather than `high-brow'
abstract individuals, fits the linguistic phenomenon of plural reference
rather than the requirements of foundations of mathematics.» [Parts 145].
«The limiting case of a group is a class.
A group [eg. an orchestra] is several objects
fulfilling certain constitution conditions [emph. added].
The existence of the group members alone is insufficient to guarantee the
existence of the group. In the case of a class, it is sufficient, since a
class is simply the several objects; there are no further constitutive
conditions. The relation of member to class is tenseless, like identity, so
classes cannot fluctuate in membership even if their members exist at
widely different times» [Parts 146f].
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- Realism holds that individuals must have something identifiable
in common in order to be legitimately classified alike.
This something identical in all instances of a type T
is precisely the universal T.
That is, the presence of a universal (in, or related to the individuals)
is a prerequisit for individuals belong or not belonging to types
[x].
Consequently, an object can be analyzed into
(that which gives it) its type, its form,
and the typeless (formless) rest X, the individual's matter
(cf. categories).
How do these two relate?
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Platonism: Type = Cookie-Cutter Form out of matter, exists ante rem, "object instantiates form"
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Platonic types, called "forms", have existence
distinct and independent from (the existence of) individuals of those types;
they are eternal and unchanging (``ideal forms'') [AUOOP].
Hence one can talk of a particular universal (concrete universal)
and have them as genuine subjects of predication (universals are first-class citizens)
[x].
Moreover, types are responsible for objects being of the type they are,
they give them their `form' by imposing the `form' on the `matter' from which objects are made
(like a cookie-cutter cuts out forms from a dough)
[x].
«The Platonic doctrine of properties is, in a nutshell, the following:
(a) form is self-existent, ideal, and external to matter,
and (b) form precedes matter and can take on individuality,
or be realized (exemplified) in particulars.
Thus a white thing is said to "participate" in the universal Whiteness»
[MB3 103].
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Aristotelianism: Type is-a Accident of objects, exists in re, "object has form"
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Form and matter are inseparable constituents of all existing things
- matterless forms (`universals') (and formless matter)
are merely derived from them by a mental process of abstraction
and exist only in the mind [AUOOP].
Individuals (particulars?) are `primary substance' (plus properties)
and universals `secondary substance' (plus properties) [AUOOP],
existing only through/in/as-part-of their instances [x]
(but independent from any particular instance [x]).
Hence, in a predication, only individuals are meaningful subjects,
and only universals can be predicated of them.
Statements which grammatically appear to talk about a universal have to be reinterpreted:
«'Socrates is wise' would predicate the universal, wisdom, of
the particular, Socrates, and 'Wisdom is a characteristic of Socrates' would
be a grammatically misleading way of predicating that same universal of that
same particular, while 'Wisdom is a primary virtue' would be a grammatically
misleading way of saying that any person having wisdom has a primary virtue»
[x].
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Object-independent properties/forms as well as substance/matter
are fictions
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«Every real thing has a number of properties: there is no formless
substance except as a useful fiction. Nor are there pure forms hovering above
matter.» [MB3 57].
«Neither properties nor individuals are independently real.
Some individuals and some properties constitute things, their states and their changes of state
- the only realities. In the fact that thing b pulls thing c
what is real is neither of the three items separately but the fact as a whole.
Only in pure mathematics does one find (or rather create) individuals
bereft of all relations, namely the members of structureless sets.
And again only in pure mathematics can one maintain that either the
individuals or the relations (in particular the functions) constitute
the basic objects out of which everything else can be built.
Thus set theory takes the view that individuals and collections of such are base,
everything else being reducible to them;
and category theory starts on the other hand from the functions
which the individuals happen to satisfy.
Neither point of view carries over to ontology:
here we must regard individuals as well as their properties
as so many abstractions. The real thing is the substantial individual
with all its intrinsic and mutual properties.
Everything else is fiction» [MB3 100f].
«Surely every real thing consists of
some definite stuff endowed with definite properties:
there are no bare individuals except in our imagination.
This is a tacit assumption of factual science.
In other words the Platonic concept of formless matter or the Aristotelian
notion of unchanging substratum [=substance?] have been shelved by science.
Still, although we no longer believe in the actual existence of nondescript things,
we find it methodologically convenient to feign them - only to endow them
with further properties later on ...» [MB3 26]
«The expressions `pure form' ...
make no sense except as abbreviations or abstractions.
In fact, a mathematical and semantical analysis of the concept of a property
will show us shortly that every property - except of course the null property,
which is a fiction - is possessed by some individual or other ...
In other words, an attribute can only be attributed to, or predicated of,
some subject or other. ... whiteness can be predicated of snowballs
... but it does not exist separately ... In other words there are no
universals in themselves but only properties that are universal in a given
set of individuals, i.e. that are shared by all the members of the set.
All this is rather obvious from the way properties are handled in science
and predicates are analyzed in semantics. In both cases every property
is represented by a function mapping individuals ... into statements
of the form `Individual (or n-tuple) x is assigned attribute A'.
Such propositional functions, or proposition-valued functions,
will be called attributes or predicates.» [MB3 62]
- Conceptualism «holds that our classification of particulars under general terms
is a product of our selective human interests
rather than a reflection of metaphysical truth»
[x].
Cf. Jevons's (1835-1882) and Whewell's (1794-1866)
critique on the universality of classification
[history].
- Nominalism.
The [realist] hypothesis that things have properties is challenged
by nominalists. They «wish to dispense with properties, which they
regard as Platonic fictions, and attempt to reduce everything to things,
their names, and collections of such (...)» [MB3 57].
«All properties, whether intrinsic or mutual, are unreal: only individuals are real.
This is ... the nominalist thesis ... According to it a property,
if intrinsic, is identical with a collection of individuals and,
if mutual, is nothing but a collection of ordered n-tuples.
... [But:] The relata constitute the extension or graph of the relation,
not the relation itself» [MB3 100].
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Extensionalism: Type = extension, exists post rem
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«At the very least the nominalist
will want to construe every property as a class of individuals, or of
n-tuples of individuals - i.e. he will adopt an ontological interpretation
of the semantic doctrine called extensionalism, which equates
predicates with their extensions.» [MB3 57]
«But ... extensionalism is undefensible even
in mathematics, if for no other reasons that (a) some of the basic mathematical
predicates, notably the membership relation, are not defined as sets, and
(b) coextensive predicates need no be cointensive or have the same meaning.
...
If extensionalism fails so does its ontological interpretation, namely
the reduction of the world to a collection of individuals devoid of properties.»
[MB3 57f]
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Type = resemblance (relation), exists post rem
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Nominalism «holds that resemblances between particulars are sufficient to
justify our application of the same general term to them without appeal to
any additional entity» [x].
Cf. Wittgenstein's `game' example and the notion of family resemblance
[history].
Natural laws are a kind of universals,
since they transcend particular occurences of phenomena.
An old controversy among philosphers and among scientists is whether natural laws are primarily
discovered (which means that they are a-posteriori[?]),
or rather invented (meaning they are synthetic[?]).
In other words, how real are the natural laws?
When we write down a natural law (informally or as a mathematical formula),
do we describe something that exists in the real world?
Are we "Describing the Real World"?
- (Critical) Realists - say that natural laws are discovered.
E.g. Russell.
- (Critical) Idealists - say that natural laws are
invented/created/constructed/contrieved
to match the collection of empirical data.
E.g. Kant (founder of critical idealism), Wittgenstein.
«In order to be classified as an idealist one does not need to speak
the whole day long about the spirit, or to maintain that life is a dream;
it is enough to maintain that nothing exists or appears by itself,
autonomously, independently from some mind.
[Berkeley's esse est percipi]
Nowadays it is hard to maintain [Berkeley's] subjective idealism in ordinary life;
it is easier to maintain it for a domain accessible only to the
specialist--for instance, atomic physics. Thus, we often find the amusing
spectacle that subjective idealism is asserted with regard to microscopic events,
whereas some sort of materialism is retained for the macroscopic level.» [Compl 5].
Quantum mechanics (QM), in particular uncertainty
/ complementarity (cf. the and in de Broglie's 'wave and particle')
/ collapse of the wave function by obervation,
implies that there are limits to what we can know
(an epistemological question).
This lead scientists to debate what this means for their object of study, reality
(the ontological question).
The two main camps in the debate were,
and still are so-called materialists and positivists:
«[M]ost scientists, at least when they are doing research,
share the materialistic principle of the objective existence
of a gradually knowable thing-in-itself,
whereas positivism maintains that there is no such `hidden' reality
behind appearance, since the object is exhausted by its perception
(nowadays by its measurement). This positivistic axiom is very old,
but in modern times it was first clearly stated by Berkeley ...» [Compl 7]
Materialism: Ontological position: there is more lawfullness than we yet know
+ epistemological position: we can always learn more (but may/will never find out all)
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Anti-Atomism/Holism (epistemological):
decomposability / analysability => further knowledge
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«Scientific materialism ... rejects ultimate atomism,
which is a feature of mechanism, and the irrationalist dogma of the unanalysability
of wholes, asserting instead that nature and the knowledge of nature
are qualitatively infinite and inexhaustible. They maintain that at every level
there are wholes which are so tightly knit, so predominantely determined
by their inner motions and the interconnections of their parts,
that they actually behave as atomic at their level;
but that there is nothing that guarantees us that these atoms are undecomposable
at other levels, so that they may sometimes be regarded as undivided,
but not as indivisible. This holds in particular for the unit formed
by subject and object. In sum, scientific materialists reject atomism as an
ultima ratio and the correlate assertion of the exhaustability of knowledge.
» [Compl 143]
Positivism: Ontological position: no hidden lawfulness beyond QM
+ epistemological position: all laws of nature can be found
(in case of quantum phenomena, all laws have been found - see QM).
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For example Schlick: «`Quantum physics teaches inexorably that the detailed prediction
of future events is in principle impossible. Hence, it sets upon the knowability
of nature an unsurpassable limit. ...' ... such processes, elucidates Schlick,
are not hidden, they simply are not, there is nothing beyond
the unsurpassable limits set up by quantum mechanics.
According to positivism, quantum mechanics imposes a limitation
upon our knowledge and at the same time it gives us a complete
description of all there is to know. On a materialist theory of knowledge
this would be contradictory; on the positivist epistemology it is not.
... [Schlick:]
`The quantum laws do honour the pretension of a complete, exhaustive description
of nature in the sense that in principle they say everything
that there is to say ... about any natural process.
... [W]hen we say that ... the knowability of nature is anyhow limited
... we have not to do here with a limit between known and forever unknown
natural laws; the limit of knowability is at the same time the limit
of the lawfulness (Gesetzmässigkeit) of nature.'»
[Compl 141-143 <>]
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Phenomenalism/Idealism (ontological position):
only phenomena really exist, things exists only in the observer
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The methodological prescription in positivism is to give account of phenomena.
«This phenomenalist attitude, typical of positivism since Comte,
has been adopted by the upholders of the official philosophy of
quantum mechanics, one of whose best representatives, Heisenberg,
has explicitly stated that the quantum theory does not assume the existence
of a Ding an sich behind the phenomena (or appearances).» [Compl 7]
«[W]e cannot attribute an autonomous physical reality (i.e. a reality
independent of the experimenter) to objects at the atomic scale ...
[W]hat we call electron is not a bit of matter but a set of symbols ...»
[Compl 3]
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Holism (epistemological): Reason for the uncertaintly-limitation-on-knowledge
is non-decomposability/analyzability of system + observer
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«[F]or Bohr and Rosenfeld, the observed system and the
instrument of observation form a whole, a `sealed unit' defined [by?] the
phenomenon ... a totality that cannot be rationalised and hence represents
a limit, a non plus ultra, to human knowledge. ...
This modern version of atomism is as mechanistic as ancient atomism
and as irrationalistic as any obscurantist world outlook.
The irrationalist feature lies in the claim that wholes are unanalysable,
that their analysis and understanding is forever beyond all human possibility.»
[Compl 141f]
C. A More Current Classification on Scientific Models vs. Laws
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Realism: Laws in nature cause oberveable regularities <-- science discovers them, "laws" in science are true or false
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«Realists or Platonists believe as followers of the necessity thesis
that natural laws exist independently of their formulations.
They are as real as chairs» [bdw].
Gottlieb Frege wrote «A natural law is not thought up by us, but discovered» [bdw].
Henning Genz:
«The way of the researcher to his theories is his private business.
But if this theory is important enough,
it is tested independently of the motivation
by the researcher community
and, if necessary,
added to the canon of natural laws.
Natural laws are, as Albert Einstein said, "free inventions of the human mind"
- but which have to stand the test
in order to become, after this and by this, inventions.
If no contradictions show up during tests despite numerous possibilities,
the theory can be confered the honary title "natural law".
Provisionally, of course, since it can still turn out
that the theory has to be modified by restricting its scope,
that another, better theory takes its place,
or that it is even wholly disproved.»
A theory must make the right predictions, but
«[m]ain purpose of a theory is to explain observations.
A theory which explains nothing is at most a duplicate of nature.
We can ask it like an oracle.»
This would help to predict the consequences of our decisions
but does not give us insight into the possible choices.
[Henning Genz: "So real wie Stühle und Steine"; Bild der Wissenschaft 12/2003, my translation, my underlining]
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Nominalism, Positivism: no Laws in nature <-- science develops models ("laws") to describe oberved regularity
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Stephen Hawking: «Physical theories are only mathematical models,
which we construct. We cannot ask what the reality is, since we have no
model-independent verifications of that what is real» [bdw].
«Really existing in nature are only the single effects,
the laws exist merely in the ideas of the natural scientist
or in the system of the natural sciences»
- so wrote Johannes Samuel Traugott Gehler in his "Physical Dictionary" of 1798.
Others: Mach, Wittgenstein.
- Doesn't this include: Instrumentism and Constructivism
sees natural laws as a means for description.
«As opposed to realists ...for instrumentists [natural laws] are only tools
and insofar practical or unpractical, but not true or false.
The constructivists take the tool character nearly literally
- as specification for the construction and operation
of devices and measuring instruments.
Natural science is then applied technology, not vice versa.
"It is a fact which cannot be denied
that our experimental equipment does not lie around in nature,
but that we have to construct it first. We are gided in that by the aim
of reproducing without disturbances
processes of a particular type" argues Holm Tetens,
now professor of philosophy at the Free University Berlin ...» [bwd].
He denies vehemently «"that one first has to know out of the natural sciences
how the world is before one can successfully act technically in it;
on the contrary
one must be able to act technically successfully in it
in order to know how the world shows itself in the framework of scientific methods»
[bwd].
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Pragmatism: science choses the most useful models ("laws")
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«Pragmatists try to keep out of the debate and
see natural laws as a useful tool for the description of the phenoma
and observed regularities, whose reality they do not want to negate.
But by not insisting on necessity, they are also followers of the regularity thesis.
"I am interested in the model which explains the observations most efficiently",
says the physicist and cosmologist Paul Steinhardt ...
"Whether it matches the reality is an abstract questions. Models are
always simplifications. We are not necessarily concerned with the reality.
What we want is a model which can capture the largest possible diversity
of complex phenomena with the simples set of concepts
which are understandable for the human brain and allow predictions.
... Reality is not always what one wants. One rather wants simple understanding"»
[bdw]
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Conventionalism: science choses models ("laws") depending on influence/majority
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Constructivism is more radical than pragmatism.
It emphasize beyond usefulness the social aspect:
«In the extreme case natural laws are then purely social constructs
of an influencial opinion leadership or majority of researchers.
... Whether the sun revolves around the earch or vice versa
is then merely a mood of history.» [bwd]
Einstein showed that this old question is unscientific: everything is relative
- no observation of any physical property can be made
that would allow us to decide between these two ontologically different scenarios.
LOST: Ontological Status of Negated Properties: Perception or Conclusion?
- Representational view (realism?).
There is a difference in the substantial properties
a concrete object (substantial individual) has, and
the attributes we attribute to it in a cognitive act
by means of a conceptual model of it.
«The strong distinction between properties and their representations
helps clarifying a common misinterpretation of what a mutual property
means. When we say, for example, that "John and Mary are married to
each other", we acknowledge the existence of "being married to each
other" as a mutual property of John and Mary. At first glance, one
could find it counterintuitive to classify this as a mutual property
since "being married to Mary" would be the property of John while
"being married to John" would be the property of Mary. However, one
should notice that we can only get in contact with properties of
things via their attributes. Thus, "being married to Mary" is actually
an attribute function that represents this mutual property from John
s point of view, and not a property.» [GOF/CM]
[MB3 58f, continued from,
underlining added] «Because every model of a substantial
individual is built with concepts, it contains attributes or predicates;
and insofar as the model represents a substantial individual, some of those
attributes or predicates represent substantial properties.
In the case of a conceptual object, such as a set or a theory, the words
`attribute' and `property' are exchangeable because a conceptual object has
all the properties we consistently attribute to it. But in the case of a
substantial individual we must distinguish a substantial property or
objective trait from the corresponding attribute(s) if any. A substantial
property is a feature that some substantial individual possesses even if we
are ignorant of this fact. On the other hand an attribute or predicate is
a feature we assign or attribute to some object: it is a concept. A predicate
may be conceptualized or represent a substantial property; but then it may not
... On the other hand the possessing of a propery is not a matter of truth or falsity;
only our knowledge of properties can be more or less true or adequate. ...
... Again: whereas the possessing (or the acquiring or the losing) of
a property by a substantial individual is a fact often beyond our reach,
our attribution of a propery (via some predicate) is a cognitive act.
In other words, whereas we control all predicates because we make them,
we control only some properties.»
Predicates not denoting a substantial property
- Membership. «[B]eing an element of an arbitrary set does not count as a substantial
property. For example, although I may decide to group my shoes together with
President X's latest statement, such a common membership does not constitute
an objective property shared by the two items. In short, not all sets are kinds.»
- Negation not P. «`Neutrons are not electrically charged' denies the being charged
instead of attributing a property of not-being-charged. «We certainly need
negation to understand reality and argue about it or anthing else, but
external reality wears only positive traits. ...
In short, negation is de dicto not de re: it is defined
on the set A of attributes not on the set P of properties.»
- Disjunction P \/ Q. «Likewise ... there is no substantial individual possessing the property
of being heavy or transludid ... (In other words, we employ the
disjunctive attribute "heavy or translucid" but do not have it represent
a property possessed by any substantial individual.)»
OTOH -> conjuction P /\ Q denotes a complex property
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[MB3 59f, underlining added]
«Surely we know properties only as attributes or predicates,
i.e. as components of our view of things. Still, we must distinguish the
object represented from its representations if we wish to account for
discovery (or invention) and ignorance, truth and error. ...
... The attribute-property correspondence is a particular case of the
knowledge-reality, or mind-world relation. ...
The representation function is a correspondence between a proper
subset of the conceivable attributes (predicates) and the ill-defined
set of all (known and unknown) substantial properties. That is, there are
attributes with no ontic correlate. Among them we find membership in a set,
the negative attributes, and the disjunctive ones.»
[MB3 62]
«The attribute-property gap bars any attempt to read the predicate
calculus in ontological terms, i.e. to get a theory of properties without
tears. Therefore we must try and build a property calculus as distinct from
the predicate calculus.»
- Idealism.
[MB3 59] «Needless to say, no such difference between attributes and substantial
properties is admitted by idealism: in such a philosophy the acquiring of a
property coincides with its attribution. (Which is why the inadequacies of this
philosophy are hardly realized in the realm of formal science.)»
Different levels of expressive power of logic can be distinguished
w.r.t. the use of universals (see above),
i.e., object types, relations, predicates, etc.
Each such logic may allow more or less different interpretations
with different ontological committments.
According to the classical view, second-order quantification (in arithmetics)
«ranges over all predicates on the integrers,
ie, extensionally, all subsets of the integers» [APKR]
(see realism below). But other views are possible.
- Nominalism and predicative logic:
«On a slightly more modern view due to Henkin, [second-order quantification]
ranges rather over all nameable predicates,
ie over all predicates whjich can be named by a lambda-expression.
These are not the same set: the second is a countable subset of the
uncountable classical predicate universe» [APKR].
Hence the logic is weaker, it is predicative second-order logic.
The Henkin view / predicative logic as two advantages:
- The logic is axiomatizable, has a complete formalization.
- It allows to avoid committing
to the individual existence of relations and sets,
and still do some second order logic.
[<] «In nominalism,
as a theory of predication (or formal ontology),
only first-order, objectual quantifiers have ontological significance,
which means that predicate quantifiers (and variables) are either omitted
altogether or given only a substitutional interpretation.
The latter option brings with it constraints on the logic of predicate
quantifiers that in effect exclude impredicative formulas,
i.e. formulas containing bound predicate variables,
from being substituends of predicate variables
(i.e. substituends that preserve validity according to nominalism).
The system that results by imposing these constraints is known as
(standard) predicative second-order logic.
(Nominalism can be extended into standard ramified predicative second-
order logic in which formulas with bound predicate variables of a given
level of ramification can be substituends of predicates of any higher
level; but it still excludes a comprehension principle in which formulas
with bound predicate variables of a given level can be substituends of the
predicate variables of that or any lower level.)» [KRCR]
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Nominalism and predicative polymorphism:
Compare logic to parametric polymorphism (second-order type systems)
in programming languages:
Let M be a program term (e.g., function or class definition)
containing formal type parameters.
If limited to predicative use (as in ML, C++, Eiffel, ...),
the nominalist can do without committing
to there being any kind of value that M denotes---
M can be understood as
an uninterpreted syntactic schema, or template,
from which a value (function, class) can be obtained whenever needed
by instantiating M with actual types.
From this perspective,
talk of (predicative) polymorphic functions or classes
is only a façon de parler,
and whether M can be compiled to code, or only its instances,
is a matter of the compiler, not the programming language concept.
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- Conceptualism «goes beyond nominalism in its recognition of
predicable concepts as values of predicate variables (with respect
to which predicate quantifiers have referential significance.» [KRCR]
«Concepts ... are intersubjectively realizable ...
cognitive structures that underlie our
ability to think and communicate. They are not objects (and therefore
cannot be values of individual variables) but are rather unsaturated
cognitive structures that are realized (saturated) in particular mental
acts--including speech acts ...
Predicable concepts in particular are based on cognitive capabilities to
identify, characterize and relate objects in various ways, and they
underlie our ability to follow the rules of language regarding the use of
predicate expressions.» [KRCR]
- Constructive conceptualism «imposes a constraint
that precludes impredicative concept-formation, i.e.
the formation of concepts on the basis of a totality to which those
concepts belong or form part. The logic resulting from these
constraints is a nonstandard form of predicative second-order logic
... which is similar and yet different from the standard second-
order logic of nominalism» [KRCR].
- Holistic conceptualism allows impredicative concept formation
«but without nullifying the (ramified) predicate quantifiers that
range over the so-called predicative concepts of
constructive conceptualism» [KRCR] ???
Predicable concepts: impredicative + predicative-constructive ???
Referential concepts: common-name concepts (non-sortal and sortal).
- [Realism and] Higher-order predicate logic:
«Higher-order predicate logic goes beyond the impredicative second-order
logic of holistic conceptualism by introducing a syntactic operation that
transforms predicates (and formulas) into abstract singular terms, i.e.
into nominalized predicates (and formulas). For example, in addition to the role of F in F(x)
as a predicate expression, we also have the role of the nominalization of
F as an abstract singular term in G(F) and R(y,F)--
as well as in F(F), where F occurs first with a pair of
parentheses as a predicate expression, and then whithout the parentheses as
a singular term.» [KRCR]
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Realism and nominalized polymorphic terms (impredicative polymorphism):
Let the type-parameterized term M
be passed as argument of a function: f(y,M).
For instance, M could be a polymorphic procedure
translating values from a range of types to strings;
and f uses it to translate a heterogeneous datastructure y
to a string representation (a case of rank-2 polymorphism).
This case the nominalist cannot explain away as schema instantiation--
there has to be something in the runtime system to be passed to f
which represents M itself, and not just instances of it.
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- Extensional meaning of nominalized predicates:
«The addition into impredicative second-order logic of nominalized
predicates as a abstract singular terms was the move that Frege made in
going from the system of his Begriffsschrift to that of his
Grundgesetze. ... This addition was important and fundamental to
Frege's development of arithmetic as part of logic, by which he meant only
extensional logic. That is why the objects he took nominalized predicates
to denote in his system were classes (Begriffsumfange)
or value-ranges (Wertverläufe) [emphasis added],
rather than the intensional objects
that we take them to be in conceptual realism» [KRCR].
Cf. Frege's explanation of universals as the class of objects belonging to it,
and the treatment of this class as an object itself
[history].
- Conceptual Realism «goes even further [than conceptualism]
in its recognition of the contents of our concepts
as intensional objects (denoted by nominalized predicates)
and of the possibility of there being
properties, relations, and natural kinds in nature corresponding to
some, even if not all, of our concepts.» [KRCR]
«What a nominalized predicate denotes, if it denotes anything at all, cannot
be the concept that the predicate stands for in its role as a predicate--
because concepts, as we have said, are unsaturated cognitive structures,
and as such they are not objects. In conceptual realism what a nominalized
predicate denotes is an intensional object [emphasis added]
--namely, the intension or content of the concept that the predicate stands for
in its role as predicate. Here, by the intension or content of a concept I mean
a hypostatization, reification, or projection into the domain of objects of
the truth conditions determined by the different possible applications of
that concept. It is by means of such a projection, or conceptual
nominalization, that we purport to denote the intension or content of a
concept as if it were an independently existing real Platonic form. Thus,
for example, not only do we predicate of someone that she is kind and wise,
or of a box that it is green and rectangualr, but also, through conceptual
normalization of the concept that we predicate, we purport to denote the
properties (in the Platonic sense) of being kind and being wise,
or the properties of being green and being rectangular--i.e.
the properties kindness, wisdom, greenness, and
rectangularity. It is through concept nominalization (as a product
of cultural evolution) that we hypostatize or reify a concept and are able
to grasp its content or intension as an (abstract) object by starting out
from the ceontp[?] as a cognitive capacity. The assumption that conceptual
nominalization (reification) leads to real abstract objects (as emergent
products of cultural evolution) is an ontological posit the goes beyond
conceptualism proper, and it forms part of what I mean by conceptual
realism as a formal ontology. This part of conceptual realism is called
conceptual intensional [as opposed to natural] realism.» [KRCR]
[MB4 39] «There are three possible doctrines concerning wholes:
holism, atomism, and systemism.»
Holism [MB4 39]
«Holism is the ontological view that stresses the
integrity of systems at the expense of their components and the mutual actions
among them. It is characterized by the following theses.
H1 The whole precedes its parts. ...
[Before you can cut some off your hair (part), you need to have your whole hair.
Bunge's objection: But to have hair you must have grown it - piece by piece.]
H2 The whole acts on its parts.
For example, it will be said that the needs of the organism (or the society)
as a whole dictate the functioning of its parts. But of course there would
be no whole were it not for the coordination
of its parts. There is no action of the whole on ist[!] parts;
rather, there are actions of some components upon others.
Thus the vibration modes of any single particle in an elastic body
are influenced by the motion of the other particles; likewise the behav
of any persion is partially determined by that of his fellow society members.
In all these cases we have not the whole acting on its parts but some or
even all of the remaiing components of the system acting on the given component,
or the behavior of the latter being partly determined by the place it
occupies in the system, in particular by its function or role.
H3 The whole is more than the sum of its parts. As it stands the thesis
is hardly intelligible. It becomes intelligible if by `sum' one means the
juxtapos (physical sum or association °+) that we met in Sec. 1.6, and if by
`more' one means that the whole, provided it is a sys, has emergent propoerties
that its components lack (cf. Sec. 3.2). ... [but] not every totality is a
system; thus the mere aggregation of things need not result in an integrated
whole or system (cf. Sec. 4.1.). What makes a whole into a sys are precisely
the actions exerted by some of its parts upon others. ...
H4 Wholes emerge under the action of agents that transcend both the
actions among the components and the environmental influences. For example,
morphogenesis is guided by an entelechy, or élan vital,
or morphogenetic field external to the components. ...
H5 Totalities cannot explained by analysis: they are irrational.
This thesis is trivially true if by `analysis' is meant only decomposition
into parts, since then only the components of a system but not its structure
is revealed. If the latter is left out then of course it becomes impossible
to account for the systemic or gestalt properties of a totality.
But the physicist does not claim that water is just an aggregate of
H2O molecules, and the socilogist does not assert that society
is just a collection of persons. In either case the links among the components
(hydrogen bonds, work relations, or what not) must be disclosed or hypothesized
to understand the formation, coehsion, and eventual breakdown of a totality.
Such an analysis is the conceptual basis for any effective synthesis or
upward building, as well as for the effetive analysis or disintegration of a system.
H6 The whole is better than any of its parts. ...»
Systemism [MB4 41] «Whatever truth there is in holism
- namely that there are totalities, that they have properties of their own,
and that they should be treated as wholes -
is contained in systemism, or the philosophy underpinning
systemics or the general theory of systems (cf. Bunge, 1977d).»
Atomism - [MB4 41f]
«the thesis that the whole is somehow contained in its
parts, so that the study of the latter should suffice for understanding the
former. Cartain wholes, to wit systems, do have collective or systemic
properties not possessed by their components, and this is why they must be studied
as systems. Consider the celebrated though ill understood example of a so-called
contingent entity, namely Water = H2O. ... (... the correct
statement is this: For any water body w, C(w)
The set of H2O molecules.)
Moreover, specifying the composition of a system does not suffice to
characterize it as a system: we must add a description of the system structure.
And it so happens that water, as a system composed of myriads of H2O
molecules, has properties that none of its components has -
e.g. transparency, a high dielectric power (hence a high dissolving power),
freezing at 0°C, and so on and so forth. ...
... [T]o account for the behaviour of [a body of water], we need not only all
the knowledge we have about the individual H2O molecule
but also a host of hypotheses and data concerning the structure of water
(i.e. the relative configuration of H2O molecules in the lattice)
as well as hypotheses and data about the dynamics of water bodies
- hypotheses and data which vary of course according to whether water is in the
gaseous, liquid, or solid phase. In sum, to describe, explain or predict the
properties of water we use both microlaws and macrolaws.»
Reductionism <= Atomism
[MB4 42] «Atomism, an ontological doctrine, is usually, though not necessarily allied
with reductionism, the epistemological doctrine according to which the study
of a system is reducible to the study of its components. ...
The reductionist will claim of course that we may use macrolaws and, in general,
laws of systems, as a convenience, though in principle we should be able to get
along with microlaws (or laws of components) alone, since the former are reducible
to (deducible from) the latter. This thesis contains a grain of truth but is
not the whole truth. No theory T2 of water as a body follows solely from
a microhpysical theory T1 of the H2O molecule - not even by
our adjoining what some philosphers call the bridge laws relating macrophysical concepts
(e.g. pressure) to microphysical ones (e.g. molecular impact). Much more
than this must be added to the primary or reducing theory T1 in order to
obtain the secondary or reduced theory, namely hypotheses concering the
interactions among the system components.
... [This is "strong reduction"].
In general we must resort to a more complex strategy, namely weak reduction,
or deduction from a primary theory in conjunct with a set of conjectures and data
congenial with but alient to the former. ...
The subsidary hypotheses constitue a model of the composition and structure
of the system. Since this model, though couched in the language of T1,
is not included in T1, we are in the presence not of straight (or strong)
but of partial (or weak) reduction. ...
Note that we are not stating that the properties of water, or of any other
macrosystem, are mysterious. One the contrary, they can be exaplained at
least in outline. For example, the exceptionally high bouling pout and evaporation
heat of water are explainable in terms of the hydrogen bonds linking together all
the H2O molecules in a water body, bonds which are in turn explained by the
composition and the structure of the H2O molecule. But the point is
that the intermolecular hypdrogen bonds do not occur in the study of the indivual
H2O molecule. In other words, althogh water is composed of H2O
molecules it does not reduce to H2O ...»
|
a. Mereological Null Individual(s)
[MB3 51, emphasis added]
"The null individual, or nonbeing, is best characterized in terms
of either association or assembly, i.e. either as
(i) that which, associated with any substantial individual, yields the latter,
or as (ii) that which, juxtaposed to any thing leaves it unchanged and,
superposed to it, annuls it. In either case [] [the null individual]
is in S [the furniture of the world] and is part of every member of S.
(The claim that [] belongs to every set, even the empty set, leads to contradition.)
All these are definite properties of []. That is ... it is false to say
that the null individual has no properties - as false as to say that it has
substantial (physical, nonceptual) properties. ...
What is true is that the null individual simply does not exist physically:
it is a fiction introduced to get a smooth theory. We need this pretence
not only in ontology but also in the foundations of science - which is not
surprising because the two intersect. Thus the vacuum, i.e. the null field,
is assigned a number of properties in field theories - such as a refractive
power equal to unity. Moreover it is convenient to introduce several null
individuals, one for every natural class. For example, in optics we need
the void, or null body b0, and darkness, or the null light
field l0. The reference class of a statement about a light
beam l in vacuum is {l, b0}, and one about the dark
void concerns {b0, l0} (...)
We need the null entity to think about real entities but we cannot use
it to build the latter. ... [E]ntities are not constructible out of the nonentity."
Bunge [MB3 278ff] describes the three main views on the nature of space & time:
- Spacetime + matter dualism -
«The container view: space and time constitute the fixed
stage where things play out their comedy. Physical objects exist
in space and time, which in turn are not physical objects. Moreover
a physical object may be defined as anything occupying a region of
space and a stretch of time. ... Not being a physical thing or a
relation among physical objects, the supreme container cannot be
described in physical terms: space and time have got to be described
in purely mathematical terms without even the assisteance of concept-fact
bridges or semantic assumptions. Thus instead of stating that a
certain [mathematical] geometry represents physical space we must say
that the two are identical. ...
The container view of space and time is strongly suggested
by the way science uses these concepts most of the time. Thus one
calculates or measures the place or time at which something
happens ...»
- Critique:
[MB3 327] The container view «... presupposes the autonomous existence
of the spatiotemporal framework, which in turn would be a non-physical object.
... We do not accept the container view because our ontology does not assume
any nonphysical entity - except of course as a fiction.
In our view [the relational view, see below]
things do not float in a given spatiotemporal framework
but hold spatiotemporal relations among each other
- just as persons do not swim in a community but constitute it
by holding connections with one another [emph. added].
Things come with their own (changing) spatiotemporal relations,
and the latter are just relations (but not bonds)
among things and their changes.»
-> cf. systems are constituted by bonding relations between components.
- Geometric monism -
«The prime stuff view: spacetime is the elementary substance
of which every physical object is made. Whatever is part of the
world is a chunk of spacetime, and every substantial property
is a property of a chunk of space. Hence everything physical
ought to be explained in spatiotemporal terms ... Physics becomes
geometry and moreover a geometry in no need of physical interpretation,
as it generates its own interpretation. Geometric monism is substituted
for the spacetime-matter dualism of the container view:
space and time are not just experimentally and epistemologically
prior (as for Kant) but also ontologically primary (as for Alexander).
Indeed, every physical object is regarded as a local wraping of space
(or of spacetime). ...
... This fascinating theory is unfortunately semantically ...
too simple to be true ... [T]he mathematical formalism is declared
to be in no need of semantic hypotheses ("correspondence rules").
Thus instead of saying that a certain figure in spacetime
represents a black hole one simply calls
that surface a black hole:
definitions take the place of semantic assumptions, hence the
border line between formal science and factual science disappears ...»
-
Bunge's proposal of a relational definition of spacetime
is based on a ternary interposition or betweenness
relation x|y|z [MB3 284].
This relation must satisfy
the following properties for any thing u, v, x, y, z:
- It holds only between three different things,
or on a single thing (ie. x|x|x).
- It is symmetric in the outer parameters:
x|y|z => z|y|x.
- Transitivity of sorts: A thing y is between u and v
if y is between some x and z
which are between y and,
respectively, u and v:
u|x|y &
x|y|z &
y|z|v => u|y|v.
- There can be nothing "between" a whole and its parts.
- Any "basic" thing y is surrounded by some basic things x and z.
- The world is dense: If x and z are mereologically unordered
then some "basic" thing y is between them.
- Effective: If x|y|z and x acts on z
then there is one event of x acting on y
which preceeds all the events of x acting on z.
|
Things are primary: «The relational view: space and time are ...
a network of relations among factual items - things and their changes.
What is assigned mathematical (topological, affine, or metrical)
properties is not spacetime itself but the set of things - atoms,
fields, etc. - and their changes. No changing things, no spacetime.
...
... The relational view is that spacetime is the basic
structure of the totality of possible facts. But, unlike
the prime stuff view, which eventually developed into a a full
fledged theory (geometrydynamics), the relational view has
heretofore remained at the heuristic stage.»
[MB3 296] «Ordinary space, then, is just as real as any other real
relation. (Actually space is not a relation but a set of relatedd things
or, if preferred, a relational structure.
But, not being a thing, physical space has no causal efficacy. In other words,
the spactial relations are nonbonding relations
rather than bonds or couplings.
... That is, just as things do not act upon space (since space is not a thing),
so space does not react back on things. ...
... According to our theory neither space nor things exist by
themselves. Only mutually spaced things exist.
Moreover, the separations or spacings among things may alter with
changes in the things themselves. Hence real space is as much in flux as are
things. Real space is then a dynamic structure of the collection of things.
Ordinary space may be pictured as an elastic net, or fluctuating lattice,
the nodes of which are things.»
- Compare some latest approaches to reconcile quantum theory and relativity theory
[Bild der Wissenschaft 12/2003]:
- String theory - all 4 forces are explained by one GUT:
Elementary particles are activation states of swinging strings.
Requires 9-10 dimensions of space.
Space is not quantumed.
- Quantum geometry with spin networks: also space is quantumed
- developed by Abhay Ashtekar:
Space and time are not fundamental but constructed from 'spin networks'
(the term goes back to Penrose's Twister theory of the 1970s).
«"The spin networks exist not in space. Their structures create the space"
says Smolin», Ashtekar's collegue.
It makes no sense physically to ask
what is between the 1-dimensional edges of the network.
Loose ends of the network appear as the fermions (quarks and leptons),
from which all matter is constructed,
and as the hypothetisized Higgs-bosons, which create the mass.
Different activation states of the edges
appear as the bosons (photons, gravitons, ...)
which transfer the natural forces.
Time is created only by changes in the activation states
and in the connection structure.
Predecessor theory 1992: Quantum loop theory,
with space made of loops the size of one Planck-size
= 10-33cm = 20 decimal orders below the radius of a proton
= 16 decimal orders below the resolution of the best particle accellerators
= as tiny compared to an atom, as a human cell to the milky-way.
At this scale, relativity theory does not work.
- Conceptual or de dicto possibility of a formula p.
Formulas are not possible in themselves--they just are;
what is possible or not is some relational property
like exemplification, proving it, etc. [MB3 168].
Conceptual possibility is reducible to conceptual actuality and thus redundant [MB3 165-7]:
- p is logically possible relative to a subset A of knowledge
iff A does not entail "not p".
Eg. any noncontradictory proposition.
p is logically necessary relative to A
iff A entails "p".
- p is mathematically possible relative to a body K of knowledge
iff in K there is a model for p.
Eg. "x2 = -1" is mathematically possible (complex numbers!)
p is mathematically necessary relative to K
iff p is satisfied in every model in K
- p is epistemically possible relative to a body K of knowledge
iff p contradicts no knowledge in K.
Eg. evolution theory.
p is epistemically necessary relative to K
iff K implies "p".
- p is methodologically possible relative to a body K of knowledge
iff there is no known method m in K
"such that tests run with m disconfirm p relative to K"
[does he realy mean "no" + "disconfirm" ?].
Eg. predictions derived from scientific theories, bec. their negates are refutible in principle.
p is methodologically necessary relative to K
iff all test with known methods m in K confirm p relative to K.
- Real or de re possibility - cf. modes of being
Fact x is realy possible according to a theory T and a body E of empirical knowledge
iff T & E fails to entail the negation of the proposition p describing the fact x [MB3 178].
- causal disposition: happens whenever the circumstances allow it
- chance propensity: eg. excited atom
may change state to any lower state
Because conceptual possibility of formulae is a relational property,
«a statement of the form "It is possible that p"
is not at the same level as p itself but is a metastatement ...
Not so states of affairs: these are either actual or possible absolutely,
in the sense that their possibility does not depend upon any body of knowledge.
Propositions such as "That atom may disintegrate within the next minute",
and "This organism is viable", are object statements not metastatements;
they involve the notion of real possibility not that of conceptual possibility» [MB3 168].
Immanuel Kant introduced distinction of statements (truths) in the two dimensions
of epistomology (a-priori vs. a-posteriori)
and of logic (analytic vs. synthetic).
Much philosophical debate is about whether
there is something like synthetic a-priori truth
(if not, the two dimensions would coincide).
|
| statements which are true |
in our logic
"logically"
(it rains or it doesn't)
| in our language
"per definitionem"
(unicorns have one horn on their head)
| in our model of the world(?)
- Kant: 7 + 5 = 12
- Hoare: consequences of program execution
- Poincaré: induction principle
- Wittgenstein: prime number distribution
| in our physical world
"de facto"
(all dogs eat bones)
|
Immanuel Kant | analytic truth (logic dimension)
= obtained by truth-preserving deduction?
| synthetic truth (logic dimension)
= obtained by other methods (observation, semantic, reduction, probabilistic)?
a-priori truth (epistemic dimension)
= obtained without observation of the physical world
| a-posteriori truth (epistemic)
= obtained by observations of the physical world
|
Rudolf Carnap | L-true | ? | does not exist,
"7 + 5 = 12" is analytic
| F-true
| |
necessity | logical (narrow sense)
| logical (wide sense) | ? | metaphysical (Kripke) | (no necessity)
| |
Grossmann on identity statements [OntRed]
| trivial | with abbreviations
| with synonymy | informative
(not explicit compared to a-priori / a-posteriori)
|
my construal
| computable properties
| non-computable properties of abstractions
eg. among the functions from TMs to {true,false}
there must be one, call it f, such that "f(m) <=> m halts" is true.
The truth of "f(m) <=> m halts" is independent of the physical world (a-priori)
and at the same time cannot be computed (not analytical)
|
| | | | |
A. Strong synthetic a-priori truths/knowledge
- mathematical-logical analysis
can bring us knowledge about reality:
- Immanuel Kant's example: "7 + 5 = 12".
"7 + 5" expresses that 7 and 5 should unified.
It does not define what the result of this unification is.
- C A R Hoare:
«Computer programming is an exact science in that all the properties
of a program and all the consequences of executing it in any environment
can, in principle, be found out from the text of the program itself
by means of purely deductive reasoning»
[Hoare 1969, p 576, quoted from PV2 249, emphasis added].
This can be so in principle, because he seems to believe that
one can (in principle) establish with mathematical certainty
the correct functioning of a physical system:
«When the correctness of a program, its compiler, and
the hardware of the computer have all
been established with mathematical certainty,
it will be possible to place great reliance on the results of the program,
and predict their properties with a confidence limited only
by the reliability of the electronics»
[PV 207, emphasis added].
B. Weak synthetic a-priori truths/knowledge
- our axiomatically defined conceptual constructs
have some properties that are undeducible:
|
- Analytic:
«According to Kant, an analytic statement (or judgement) is one
in which the concept of the predicate is already contained, or thought,
in the concept of the subject
- an example would be the statement that a vixen is a female fox -
whereas a synthetic statement is one in which this is not so,
for instance, the statement that foxes are carnivorous.
The Logical Positivists, adopting the linguistic turn,
held that an analytic statement is one which is true or false
purely in virtue of the meanings of the words used to make it
and the grammatical rules governing their combination.
This definition has the advantages that it does not have application
only to statements of subject-predicate form and
avoids either reliance on the obscure notion of 'containment'
or appeal to psychological considerations»
[xrefer].
- L-true: A truth by logical necessity in the narrow sense,
i.e., one «that follows from the laws of logic alone
(though there is some debate over what those laws are).
Thus, a statement like 'Either it will rain or it will not rain'
expresses a logically necessary truth,
because it is an instance of the law of excluded middle.
Again, 'If all men are mortal and Socrates is a man, then Socrates is mortal'
expresses a logically necessary truth, because in standard logic
if we may deduce the consequent of a conditional from its antecedent,
then the truth of the conditional follows from the laws of logic alone»
[x].
Truth is determined solely by interpreting the logic symbols "or" and "not".
The meaning of "(it) rains" and the state of the world are irrelevant.
«Thus 'Either it will rain or it will not rain'
expresses a logically necessary truth
because it has the logical form 'Either p or not p'»
[x].
In case of identity statements: a = a
is a trivial truth by the ontological law of self-identity [OntRed 34]
(«[S]imilar considerations hold for sentences other than identity statements»)
- Wide sense:
The meaning of analytic is not as clearly and generally acceptably
specified, as one would wish [DvR 269].
One definition (better: term explication)
of analytically true statement is that it is a statement
which can be transformed into an L-true statement
by replacing its constituents with ``synonym'' terms [DvR 258]
(i.e., substitutions of terms with the same denotation,
c.f. referential transparency).
This is the wider sense of 'logical necessity':
«[I]n a wider sense a sentence
may be said to express a logical necessity if, although not
itself a sentence true solely in virtue of its logical form, it
may be transformed into such a sentence by replacing
certain terms in it by other, definitionally equivalent terms
... In this wider sense, logically necessary truths
are often identified with analytic truths»
[x].
Quine's example: "no bachelor is married".
This statement can be analyzed by expanding the meaning of of the "bachelor" in the English language:
"no (human being who is male & adult & not married & not a widower) is married".
This transformed statement is an L-truth.
- Identity statements (2): Identity statements with abbreviations
may be true because their abbreviation-expanded versions
are identical by the ontological law of self-identity [OntRed 34].
- Identity statements (3):
Identity statements with or without abbreviations which are true
because their abbreviation-expansions contain only expressions
which are the language's different labels for the same thing
(synonyms).
«These true identity statements are, of course, as uninteresting and
uninformative as those abbreviational identity sentences which reduce
directly to instances of the law of self-identity» [OntRed 34]
De-re vs. de-dicto necessity:
«One broad distinction that is commonly drawn is that
between de re[x]
and de dicto necessity and possibility, the former concerning objects
and their properties and the latter concerning propositions or
sentences. Thus, a supposedly analytic[x]
truth such as 'All bachelors are unmarried' is widely regarded as constituting a
de dicto necessity, in that, given its meaning, what it
says could not be false. But notice that this does not imply that any
man who happens to be a bachelor is incapable of being married -
though should he become so, it will, of course, no longer be correct
to describe him as a 'bachelor'. Thus there is no de re
necessity for any man to be unmarried, even if he should
happen to be a bachelor. By contrast, there arguably is
a de re necessity for any
man to have a body consisting of flesh and bones, since the property
of having such a body is apparently essential to being human»
[x].
"Unicorns have one horn on their head" (or "all dogs are animals")
provides no new information about the world (and thus is an analytic truth),
because it is part of the meaning of "unicorn" to have a single straight horn on the head.
Analytic truth of statements can be determined
by analyzing the meaning of the terms occuring in the statement,
and thus they are all a-priori statements.
It is independent of the real (physical) world -- unicorns are fictional.
Also the truth of "all dogs are animals" is determined
without observing and comparing
dogs and non-dogs, and animals and non-animals in the physical world.
Since the physical world is irrelevant,
the meaning of the statement's terms
which analytical truth depends on is not the denotation
but the content/sense.
- Beyond linguistic meaning:
Identity statements (4):
Identity statements with or without abbreviations which are true
not only because of self-identity and synonymy
in the abbreviation-expansions.
-- only these are informative identity statements.
[OntRed 34].
-
«All "interesting," all "informative" identity sentences [in languages with
at most one label for every entity] contain at least one expresion for a
definite description. Consider, then,
an identity sentence with two different definite description
expressions. Such a sentence states that an entity uniquely characterized
by a certain property (or set of properties) is the same as an entity
uniquely characterized by a different property (or set of properties).
... [It is not] to say that the two properties are the same or that the two
descriptions are the same.
- ... The discovery of a true informative identity statement may greatly
enlarge our knowledge about the world
» [OntRed 32f]
- For example, Russell's definition of addition as unification of non-overlapping classes
is an informative identity statement [>].
- A-posteriori / F-true: Truth or falsehood of the statement "All dogs eat bones"
can only be determined by observing the world,
and thus it is an a-posteriori statement.
Knowing truth or falsehood of a-posteriori statements
increases our knowledge about the world,
and thus they are synthetic statements.
A-posteriori truth depends
on the denotation, not content/sense,
of the terms in the statement.
The denotation tells us which things in the physical world to look at
to determine factual truth.
- Kripke's metaphysical necessity:
«The notion that there is a kind of objective necessity which
is at once stronger than physical necessity and yet not
simply identifiabe with logical necessity owes much to the
work of Kripke. Logically necessary truths are, it seems,
knowable a priori, but Kripke argues that metaphysical
necessity is, typically, only dicoverable a posteriori - that
is, on the basis of empirical evidence. For instance, Kripke
holds that if an identity statement like 'Water is H2O' is
true, then it is necessarily true - in the sense that it is true
in every [logically] possible world in which water exists. However,
plainly, we can only know that water is H2O on empirical
grounds, through scientific investigation - and we might be
mistaken about this. It is vital, then, not to confuse
metaphysical necessity with epistemic necessity»
[x].
There are two philosphical direction distinguished by the believe whether
new knowledge (i.e., synthetic truths)
can be obtained without observation of the world (i.e., a-priori)
or not.
The main candidate for synthetic a-priori knowledge are the structural sciences of
mathematics, geometry, symbolic logic, cybernetics, computer science, etc.
- Modern Empirists (Logical Empirists) say no.
In order to show that mathematical statements are analytic
they have to be reduced to a logic where they are L-true.
This is done in the logicism program by
Frege, Russell and Whitehead.
For example, a number n is modeled as the set of all sets X with n elements,
formalized as a predicate Pn about
the existence of n different elements xi in each X
and the non-existence of an element in X not equal to any of the xi.
-
Rudolf Carnap refutes Kant's example that
applying a-priori knowledg about geometry to the physical world yields new knowledge.
Instead Carnap says there are actually two kinds of geometry:
the mathematical geometry, which is analytic,
and the physical geometry, which is a-posteriori.
-
Viktor Kraft: "7 + 5 = 12" does not refer to a fact
but describes a transformation of two groups of entities to one group based on rules. ...
The same entities are grouped differently.
- Constructivists or Neo-Intuitionists
(because they regard counting as an original intuition)
say yes.
Kant,
- Immanuel Kant's arithmetic statemets (see above).
- Henri Poincaré's induction principle (see above).
- Ludwig Wittgenstein's prime numbers (see above).
- Paul Lorenzen: arithmetically true statements should be distinguished
from analytically true statements.
Paul Bernay (an axiomatist!): «That a subsumption of arithmetic theorems
under formal logic has been successful, cannot be denied.
That means, it has been achieved to phrase these theorems in purely logical terms
and to proof them based on this phrasing within the logic.
It is questionable, though, whether to consider this result such
that through it an actually philophical understanding of arithmetic theorems
as been obtained.»
Ulf Schünemann 310901, 120503
