scientific
landscape
| modeling |
(models, semantics)
| semiotics |
reasoning
| categories |
| relations |
| objects |
| wholes & levels |
| meta |
glossary
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formalization & symbol manipulation |
| ontology |
| truth & reality |
|
the scientific landscape |
| representation (models, semantics) |
| logic & reasoning | complexity & metrics | math. objects | |||||||||
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| meaning
| linguistic glossary |
| components & datatypes | relations | ||||||||||
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| algebras | ||||||||||||||
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Location: http://www.cs.mun.ca/~ulf/gloss/meaning.html.
By Ulf Schünemann since 2001.
Please mail any comments.
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NB: There are smaller units than symbols/signs:
Source of symbol meaning:
«One view holds that being about something ... must ultimately be rooted
in human usage. ... [O]nly human agents have the primary semantic
authority, as it were, to simply attach a meaning to a symbol. ...
[The] second position regards a symbol as acquiring its meaning
from its syntactic context. According to this view, meanings are determined
by the knowledge-representation frameworks in which they are embedded,
so that the meaning of a predicate symbol used in a system is determined
by the set of axioms, rules, assertions or whatever occurs in it.
The standard account is that it can mean anything that it means
in any possible interpretation of that set of axioms,
so that a richer collection of axioms more fully constrains what it
might possibly mean» [APKR].
On the first account, second-order quantification
can have its classical meaning and its full power to exclude non-standard models,
on the latter account it can only be given impredicative meaning.
1923, C K Ogden and I A Richards, in "The Meaning of Meaning", «demonstrated the widespread confusion regarding the meaning of "meaning" and they sought by systematic analysis to discard meaningless conceptions and to make proper distinctions among valid modes of interpretation. They discussed and evaluated sixteen major definitions of meaning, and formulated a new theory of signs in which the functions of language were were reduced to two, namely, the referential and the emotive» [Realms].
The terms "7+5" and "12" are not synonyms, i.e., they mean different things linguistically. (Kant: "7 + 5" expresses that 7 and 5 are supposed to be added [DvR] whereas "12" does not express this). But they refer to the same Platonic number, they have the same meaning in the (mathematical) world.
| terminology from [HSem 92] and other sources | carrier of meaning | meaning in the language and/or meaning mentally | meaning in the world: | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Aristotle's sema | phôné | pathêmata | prágma | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Stoicists' sema | somainon | lekton | tychanon | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Augustinus's signum | vox sonus | significatus | res | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Modists' dictio | vox | conceptio | res | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Locke's sign | word | idea | (reality of things) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Degerado's signe | sensation | ideé | - | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Gottlob Frege's logic (1892) [>] | Ausdruck (expression) | Sinn (sense) (an abstract object) | Bedeutung (reference) ((sets of) things)|
Husserl | Ausdruck (expression) | Bedeutung | thing |
Betrand Russel | proposition | meaning | denotation |
Alonzo Church [>]
| expression (names and forms)
| sense 'expressed' by name, 'concept of' a denotation
| denotation 'denoted' by name |
Rudolf Carnap [>]
| expression | intension (a set of attributes)
| extension / nominandum,
designates [DvR] (things, properties, facts, etc.) |
Reinhardt Grossmann | description expression | definite description (a state of affairs) | (an entity) |
[OOP?]:
The classical notion of a concept has the following elements:
| name: denoting the concept,
| intension: the properties characterizing the phenomena covered by the concept, and
| extension: the phenomena covered by the concept.
|
Charles Sanders Peirce's sign (1867)
| representamen | interpretant (mental concept)
| represented object
|
Ferdinand de Saussure's signe (1906/11)
| signifier (form component of sign, non-material unlike Morris)
| signifié, signified | (content/mental component of sign) chose (not linguistics)
|
Ogden, Richards (1923) | symbol | thought or reference | referent
|
Charles Morris's semiotics (1938)
| SYNTACTICS
| study of the relationships between signs on form and content level [KL, ch.4] SEMANTICS
| study of the reference relationship between signs in language and objects in world sign carrier (material, unlike Saussure)
| designate, significate | denotation
|
modern linguistics | form | meaning, content, | valeur (relations to other words) reference (to a relatum)
| MORPHOLOGY, SYNTAX | study of word form and sentence form SEMANTICS | study of a sign's content and the relationship between signs on the content level not linguistics
|
structuralist linguistics | surface structure | deep structure | semantic networks
|
|
computer science | program (concrete syntax) | abstract syntax ?
| operational semantics ?
| denotational semantics ?
|
modeling | diagrams | model (instance of the meta model)
| in the domain[^] | do models mean real things? | |||||||||||||||||||||||||||||||||
2. Alonzo Church
considers the formalization of natural language
as a norm to which every-day linguistic behavior is an imprecise approximation.
He defines a formalized language as the augmentation of
a logistic system aka. deductive system or calculus
[>]
(defined by vocabulary, formation rules, transformation rules, and axioms)
by semantic rules.
To Church, a formalization of natural language, as detailed below,
«the semantical rules must include at least the following:
(5) Rules of sense, by which a sense is determined
for each well-formed expression without free variables ...
(6) Rules of sense-range, assigning to each variable a sense-range.
(7) Rules of sense-value, by which a sense-value is
determined for every well-formed expression containng free variables
and every admissible system of sense-values of its free variables ...
[A]s derived semantical rules rather than primitive, there will be also:
(8) Rules of denotation, by which a denotation is determined for each name.
(9) Rules of range, assigning to each variable a range.
(10) Rules of value, by which a value is determined for every form
and for every admissible system of values of its free variables.
By stating (8), (9), and (10) as primitive rules, without (5), (6), and (7)
there results what may be called the extensional part
of the semantics of a language. The remaining intensional part
of the semantics does not follow from the extensional part. ...
On the other hand, because the meta-linguistic phrase which is used
in the rule of denotation must itself have a sense, there is a certain
sense ... in which the rule of denotation, by being given as a primitive rule
of denotation, uniquely indicates the corresponding rule of sense.
Since the like is true of the rules of range and rules of value,
it is permissible to say that we fixed an interpretation
of a given logistic system, and thus a formalized language,
if we have stated only the extensional part of it»
[AESA].
«A statement of the denotation of a name, the range of a variable,
or the value of a form does not necessarily belong to the semantics of a language.
For example, that `the number of planets' denotes the number nine
is a fact as much of astronomy as it is of the semantics of the English
language ... On the other hand, a statement that `the number of platents'
denotes the number of planets is a purely semantical statement about
the English language. And indeed it would seem that a statement of this kind
may be considered as purely semantical only if it is consequence of the rules
of sense, sense-range, and sense-value, together with the syntactical rules
and the general principles of meaning»
[AESA].
Church's semantic rules are based on Frege's theory.
«[T]he theory of Frege seems to recommend itself above others for its
relative simplicity, naturalness, and explanatory power--or, as I would advocate,
[a modified version].
This modified Fregean theory may be roughly characterized by the tendency
to minimize the category of syncategorematic notations--i.e.,
notations to which no meaning at all is ascribed in isolation
but which may combine with one or more meaningful expressions
to form a meaningful expression--and to reduce the categories
of meaningful expressions to two, (proper) names and forms,
for each of which two kinds of meanings are distinguished in a parallel way.
A name ... has first its denotation, or that of which it is the name.
And each name has also a sense--which is perhaps more properly called
its meaning, since it is held that complete understanding of a language
involves the ability to recognize the sense of any name in the language,
but does not demand knowledge beyon this of the denotation of names.
(Declarative) sentences, in particular, are taken as a kind
of names, the denotation being the truth-value of the sentence,
truth or falsehood, and the sense being the
proposition which the sentence expresses.
A name is said to denote its denotation
and to express its sense,
and the sense is said to be a concept of the denotation--although
this use of the word `concept' has no analogue in the writings of Frege,
and must be carefully distinguished from Frege's use of `Begriff.'
Thus anything which is or is capable of being the sense of some name
in some language, actual or possible, is a concept.[fn:
... In logical order, the notion of a concept must be postulated
and that of a possible language defined by means of it.]
The terms individual concept, function concept,
and the like are then to mean a concept which is a concept of an
individual, of a function, etc.
A class concept may be identified with a property,
and a truth-value concept ... with a proposition.
Names are to be meaningful expressions without free variables,
and expressions which are analogous to names except that they contain free
variables, we call forms ...
Each variable has a range, which is the class of admissible
values of the variable. And analogous to the denotation of a name,
a form has a value
for every system of admissible values of its free variables.
The assignment of a value to a variable,
though it is not a syntactical operation,
corresponds in a certainway to the syntactic operation of substituting
a [name] for the variable. The denotation of the substituted [name]
represents the value of the variable. And the sense of the substituted
[name] may be taken as representing a sense-value of the variable.
Thus every variable has, besides its range, als a sense-range,
which is the class of admissible sense-values of the variable.
And analogous to the sense of a name, a form has a sense-value
for every system of admissible sense-values of its free variables»
[AESA].
Church assumes the following principles for his formalized language:
3. Carnap. In logic, «"intension" indicates the internal content of a term or concept that constitutes its formal definition; and "extension" indicates its range of applicability by naming the particular objects that it denotes. For instance, the intension of "ship" as a substantive is "vehicle for conveyance on water," whereas its extension embraces such things as cargo ships, passenger ships, battleships, and sailing ships. The distinction between intension and extension is not the same as that between connotation and denotation.»
Cf. judegement vs. proposition
«At the risk of oversimplification, we will affirm that the logical statement
is composed of three parts:
(Objects are however absolutely crucial to make any use of physical formulae:
«If my favorite ice-hockey puck shots of the edge of my kitchen table,
at which point will it land? -- The point that is determined by the laws of
physics. In formaluae capturing the law might use
h for the table's height,
g for the accelaration due to gravity,
t for the flight time,
xi for the point below the table'e edge,
xf for the landing point,
and d for the distance.
«It turns out that [these singular terms] are crucially but curiously
ambiguous, in a systematic way, as between a universal and a particular
reading. Moreover, this ambiguity is an essential feature in allowing
the problem to be worked» [OOO 162f].
Also Dedekind did «in no way reduce the sum function to some other entity»
when he gave a recursive definition of it.
«Every recursive definition of a function (relation) is in reality
a description of the function (relation)» [OntRed 58].
«Only if we can show that the expression 'the number x which is the sum of M and 1'
is just another expression for the description the number y which is the successor of M
can we claim that there is no such entity as the sum relation in addition
to the successor relation. Only then can we maintain
that a genuine ontological reduction has taken place.
Again, I think that we do not deal with two expressions for one description,
but rather with two descriptions of the same entity.
If so, then there exist at least two relations for numbers,
namely, the sum relation and the successor relation» [OntRed 59].
«[W]e have the general equivalence that for all x, y, and z,
the sum relation holds among x, y, and z
if and only if the zth iterate of the successor relation
holds between x and y. But this equivalence does not follow
from the acceptance of an abbreviation proposal» [OntRed 60].
The verb 'to be' is important in the proper formation of a statement
because we address the being of the subject
when a predicate is affirmed or denied of the subject
by means of the verb copula. ...
A statement such as 'our linesmen play tough defence' in its proper statement format
would be 'our linesmen / are / players who play a tough defense'.
They are further distinguished into
universal:particular (statement's quantity - derived from subject's quantity), and
necessary:impossible:contingent (matter).
A universal affirmative statement (e.g. 'every human is rational')
is true only when the statement's matter is necessary by the nature of the subject.
A universal negative statement (e.g. 'no animal is a stone')
is true only when the statement's matter is impossible by the nature of the subject.
Both the particular affirmative (e.g., 'some people have red hair')
and the particular negative can be true simultaneously
when the matter said contingently (or accidentally) of the subject.
[AUOOP].
Names
«According to some theories [of reference in philosophical logic],
a name refers to a particular thing by
virtue of its being associated with some description which applies
uniquely to that thing. Other theories hold that the link between name
and thing named is causal in nature. (Theories of either sort are
intimately bound up with questions concerning
identity[x].
and individuation[x].)»
[x].
The issue of plural reference
«In order to minimize the ontological basis scholars working in Montague
style semantics have commonly restricted themselves to few kinds of
primitives. For instance, Montague's mode structures make use of three sets:
the set of entities or individuals forming the universe of discourse, the
set of possible worlds and the set of moments in time. All other thinks
taken into consideration are modelled by set-theoretic constructs over
these ingredients. The orthodox adherents of the strategy of modelling
especially presuppose a homogeneous domain of individuals. In this manner,
distinctions between categories of individuals are represented by means of
discriminations within the hierarchy of sets (or functions). It appears
that such an approach is inadequate in several respects» [OntDom].
«A term is semantically singular if it designates one object, and
semantically plural if it designates more than one object. `Socrates' is
semantically singular, and `Lennon and McCartney' is semantically plural. A
term which is either semantically singular or semantically plural we call
referential. A term which is not referential, i.e. one which does
not desinate anything at all, we call empty. In normal use,
syntactically singular and plural terms are intended to be semantically
singular and plural respectively, but both kinds may be empty, as `Pegasus'
and `Holmes and Watson' testify. ...
What is important is the recognition that designation may be not merely a
function, as in classical logic, or a partial function, as in free logic,
but in general a relation. The term `the authors of Principia
Mathematica' designates both Russell and Whitehead,
that is, it designates Russell and it designates Whitehead, and
no one else. It is not true of each of them ...
nor does it designate the set {Russell, Whitehead} as this is normally
understood. How can a set write, or co-operate in writing a book? A set is
an abstract individual, and cannot put pen to paper or exercise any other
causal influence. At best, its members can do that»
[Parts 143f].
«(Boolos 1984) "It is haywire to think that when you have some
Cheereos you are eating a set--what you're doing is: eating THE
CHEEREOS" ... but if the plurality the Cheerios is not a set, then
some alternative account of its nature has to be given.
Predicates
«[W]here a predicate may be thought of as
what remains when one or more names are deleted from a sentence -
these are variously held to carry reference to
universals[x]
concepts[x], or
classes[x].
Thus the predicate '... is red', formed by
deleting the name from a sentence like 'Mars is red', is held by some
philosophical logicians to stand for the property of redness, by
others to express our concept of redness, and by yet others to denote
the class of red things. Monolithic theories of reference are
unpromising, however. Even if some names refer by way of description,
other names and name-like parts of speech - such as demonstratives and
personal pronouns - plausibly do not. And even if some predicates
stand for universals, others - such as negative and disjunctive
predicates - can scarcely be held to do so»
[x].
Prepositions
In philosophical logic,
the denotation of prepositions (sentences)
is taken to be a truth-value.
«Truth[x]
and falsehood - if indeed they are properties at all - are
properties of whole sentences or propositions, rather than of their
subsentential or subpropositional components. Theories of truth are
many and various, ranging from the robust and intuitively appealing
correspondence theory[x]
- which holds that the truth of a sentence or proposition
consists in its correspondence to extra-linguistic or extra-mental
fact[x]
- to the redundancy theory[x]
at the other extreme, according to which all talk of truth and falsehood is,
at least in principle, eliminable without loss of expressive power.
These two theories are examples, respectively, of substantive and
deflationary[x]
accounts of truth, other substantive theories being the
coherence theory[x],
the pragmatic theory[x],
and the semantic theory[x],
while other deflationary theories include the prosentential
theory and the performative theory (which sees the truth-predicate
'... is true' as a device for the expression of agreement between
speakers). As with the theory of reference, a monolithic approach to
truth, despite its attractive simplicity, may not be capable of doing
justice to all applications of the notion. Thus the correspondence
theory, though plausible as regards a posteriori or empirical truths,
is apparently not equipped to deal with
a priori[x] or
analytic[x]
truths, since there is no very obvious 'fact' to which a truth like
'Everything is either red or not red' can be seen to 'correspond'.
Again, the performative theory, while attractive as an account of the
use of a sentence like 'That's true!' uttered in response to another's
assertion, has trouble in accounting for the use of the truth-
predicate in the antecedent of a conditional, where no assertion is
made or implied»
[x].
Complex prepositions
«One way in which complex sentences can be formed is by modifying or
connecting simple ones; for instance, by negating 'Mars is red' to
form the negation[x]
'Mars is not red', or by conjoining it with 'Venus
is white' to form 'Mars is red and Venus is white'.
Sentential operators and connectives[x],
like 'not', 'and', 'or', and 'if', are
extensively studied by philosophical logicians.
In many cases, these operators and connectives can plausibly be held to be
truth-functional[x]
- meaning that the truth-value of complex sentences formed with
their aid is determined entirely by the truth-values of the component
sentences involved (as, for example, 'Mars is not red' is true just in
case 'Mars is red' is not true). But in other cases - and notably with
the conditional connective 'if' - a claim of truth-functionality is
less compelling. The analysis of conditional[x]
sentences has accordingly become a major topic in philosophical logic, with some
theorists seeing them as involving modal notions while others favour
probabilistic analyses.
The issue of modality
«As for the question how, if at all, we can analyse modal
propositions, opinions vary between those who regard modal notions as
fundamental and irreducible and those who regard them as being
explicable in other terms - for instance, in terms of
possible worlds[x],
conceived as 'ways the world might have been'. (Although this
appears circular, in that 'possible' and 'might' are themselves modal
expressions, with care the appearance is arguably removable.) For
instance, the claim that every man necessarily has a body made of
flesh and bones might be construed as equivalent to saying, of each
man, that he has a body made of flesh and bones in every possible
world in which he exists. However, we should always be on guard
against ambiguity when talking of necessity, because it comes in many
different varieties -
logical necessity[x],
metaphysical necessity[x],
epistemic necessity[x],
and nomic necessity[x]
being just four.
Meaning of Physical Formulae
-> modeling physical reality )
The laws of physics are universal. So «is there any problem with their
"specifying" a particular point? None whatsover. We say that the
point of impact is determined by the laws of physics; what we mean
is that it is determined by two things in combination: the laws, which are
universal, and the launching point, which is particular. The laws
themselves are universal, and they apply universally; but in their
application they are functions from particularity to particularity ...
Even the position terms xi and xf
receive universal ... interpretations on the official view: as
designating the abstract measure of the distance between the two
points in question and some putative "origin."
«The position terms have to be interpreted this way, moreover,
since subtraction, normally defined over numbers, is understood as
extended to arithmetized measures, but is not normally
thought of as applying to locations. In fact it [would]
be meaningless ...; what sense is there in imagining subtracting
the position of the Statue of Liberty, say, from the position of
the World Trade Center? One needs origins, orientations, and
measures for mathematics to get a grip.
The fact that the terms are given in units betrays this
fact of measurement; if they genuinely designated particular
locations, units would not be necessary. Units figure only in the
universal interpretation.»
Definitions
Syntactic definitions, and analogously also semantic definitions,
can be classified by how they are expressed [ZDM]:
... But whether or not a certain expression in a natural language is
realy an abbreviation for another one may turn out to be a difficult
problem.» [OR 31f]
Cf. logical definitions by common accident.
Real Definitions Classified by Used Category
These categories are important as the top-level genera,
in which everything is found that logic talks about.
«Any kind of thing for which there is no higher genus
can never be defined logically, nor can it be a subject of logic,
because very definitum requires a genus» [AUOOP].
A "logical definition" (opposed to a "material definition" in AUOOP),
«attends to the concept that is signified by the word. ...
There can be many different kinds of logical definitions,
because of the different ways in which they specify the definitum.»
«Individuals can never be defined logically, nor are they the subject of logic
because their differences are only material rather than a difference of kind.»
Based on the logical categories, statements/definitions
of the form "X is-a D G" about a subject/definitum species X
can be categorized as follows [AUOOP]:
Cf. definitions in general.
The `is-a' relation between an individual and its species
and between a species and its genus is an ``essential relation.''
This is to be contrasted with the notion of `inheritance' (as incremental modification),
which is a non-essential relation based on how one thing was derived from another [AUOOP].
Note that the two are not the same:
«[W]e would not say that `home sapiens is a home erectus'.
... [W]e may say `home sapiens is an incremental modification of home erectus',
but we may not say that `man is an incremental modification of animal'
(and here it is extremely important to understand that animal is an abstract idea, i.e. a genus)
...
[W]e could not say `home erectus is a generalization of home sapiens'»
[AUOOP].
In most object-oriented languages, class inheritance is restricted
to conform to an is-a relation w.r.t. the supported operations,
and it is good style to limit oneself to inheritance
which conforms to an is-a relation w.r.t. the operation's behavior
(aka Liskov principle of substitutability).
Talking about extrinsics is always (often?)
necessary to define classes of human artefacts,
e.g. tables, chairs, or games.
The real:nominal distinction was already used by Aristotle [ZDM].
Definition =/= Reduction
Ii have heard the objection that numbers have members after all
and that my view must therefore be false. They have members, it is said,
because they are identical with certain classes.
Notice that I don not claim that numbers have no members because they are nt classes,
but rather that since they have not members they cannot be classes.
How do I know that they have no members?
I cannot do better than to trust arithmetic,
just as I cannot do better than to trust physics
when I want to know, for instance, whether or not electrons have color.
For the rest, I can only challange my opponent to name a single argument
that speaks for numbers having members-an argument other than that
numbers are classes, of course,
and hence contrary to what we all commonly believe» [OntRed 54].
(R) m+n is the number of a class w which is the logical sum
of two classes u and v which have no common term
and of which one has m terms, the other n terms
... [If 'is' means 'is identical with' then]
(R) is an identity statement with two descriptions.
Numbers are described as sums of numbers and
also as numbers of (the elements of) certain classes.
...
There is no doubt that this identity sentence is true.
The questions is, is it true as a matter or mere abbreviation or not?
Are not only the described entities [extension]
the same but also the descriptions [intension]
themselves? ...
How could [the left-side expression] be an abbreviation,
since the description which it represents does not involve classes at all,
while the description on the right-side expression does?
I do not think, therefore, that this identity statement
reduces to an instance of the law of self-identity.
Rather it is informative; it tells us that a number
which stands in an arithmetic relation to two other numbers
is the same entity as the number of elements of a class
which stands to two other classes in a certain class relation. ...
If so, then there can be no doubt that ...
there exist arithmetic relations among numbers.
[And if 'which' means 'if and only if it' it]
seems to me even more obvious, if anything, that the expression on the
left side of the equivalence sign is not an abbreviation of the expression
on the right side. The two sentences do not represent the same state of affairs.
How could they, since the s/o/a represented on the left side
contains the relation of identity, while the s/o/a on the right side does not?
But if the two sentences represent different s/o/as,
then one cannot claim to have shown that there is no such
arithmetic relation as the sum relation» [OntRed 55f].
Ulf Schünemann 080402