(models, semantics)
logic &
complexity & metrics math. objects
wholes & levels
& symbol
components & datatypes relations
truth & reality
Location: By Ulf Schünemann since 2001. Please mail any comments.

Meaning [of Symbols]

Computer -> Meaning? «In light of the centrality of the claim that computers are symbol manipulators, it is curious that virtually nothing has been written about how computers may be said to store and manipulate symbols. It is not a trivial problem from the standpoint of semiotics. Unlike utterances and inscriptions ..., most symbols employed in real production-model computers are never directly encountered by anyone, and most users and even programmers are blissfully unaware of the conventions that underlie the possibility of representation in computers» [SCI 129].
«Suppose someone claims that a computer program is thinking about a tree. What could this possibly mean? Different accounts are given of how some computational activity might be said to be 'about' some world or thing, or to 'refer' to it, or have it as its 'semantics'. These accounts sometimes differ profoundly, and some are only coherent under presuppositions which others implicitly reject. The resulting intellectual tension has produced several different schools of thought in the development of representational notation. ...
The various ideas about the meaning of meaning differ along dimensions which are orthogonal to most of the discussion in the knowledge-representation field. They are not attached to particular notations - logics or conceptual graphs or semantic networks or diagram - or particular kinds of formal semantic theory, for example. One can take any position in any of these technical debates without having yet committed to a position in these essentially ontological and metaphysical discussions. ...
... There are alternative positions which can quite rationally be taken on such questions as what does a word mean, questions which are often taken to be so blindingly obvious that it is not worth even asking them; and it is important to understand that these alternatives are possible, in order to avoid misunderstandings which seem to be technical disagreements but which actually reflect such different fundamental positions» [APKR].
Languange -> Meaning -> Ontology «Anyone who deals with the semantics of natural language and, therefore, with the explanation of the relation between language and the world, is driven to ask questions which lead him into the field of ontology or metaphysics. In order to analyse the information carried by linguistic utterances the inquirer has to discover what the expressions they contain refer to. For this purpose, he needs a theory of the world and, particularly, of what basic categories of entities there are, what fundamental properties the entities have and how they are related.» [OntDom 785]
-> see also
  1. The Carriers of Meaning
  2. The Domain of Meaning
  3. Content & Denotation
  4. Logical Models of Language
  5. Logical Parts of Speech (and their denotation)
  6. Meaning of Physical Formulae
  7. Definitions
  8. Definition =/= Reduction
[SuB] Norbert Bischof: Struktur und Bedeutung: Eine Einführung in die Systemtheorie; Verlag Hans Huber 1995.
[OntDom] Johannes Dölling: Ontological domains, semantic sorts and systematic ambiguity; 785-807: Int J Human-Computer Studies 43; 1995.
[OntRed] Reinhardt Grossmann: Ontological Reduction; Indiana Univ. Press 1973.
[APKR] Pat Hayes: Aristotelian and Platonic Views of Knowledge Representation; in Tepfenhart, Dick, Sowa (ed): Conceptual Structures: Current Practices (ICCS'94); LNAI 835; 1994.
[SCI] Steven W Horst: Symbols, computations, and intentionality: a critique of the computational theory of mind; Univ. of California Press 1996.
[KL] Bernhard Imhasly, Bernhard Marfurt, Paul Portmann: Konzepte der Linguistik: eine Einführung; (3. Auflage) Aula Verlag 1986.
[EML] John Lyons: Einführung in die moderne Linguistik (7th unchanged ed.); C H Beck 1989.
- original: Introduction to Theoretial Linguistics, Cambridge Univ. Press 1968.
[RTDU] Harald Søndergaard, Peter Sestoft: Referential Transparencey, Definiteness and Unfoldability; Acta Informatica 27; Springer 1990.

The Carriers of Meaning

Meaning implies that something stands for, is a sign for, represents, something else. This page is concerned with the meaning of symbols, ie., with what symbols represent. The more inclusive term is signs.

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.

The Domain of Meaning

Semantic equations [[x]] = x, [[X]] = X can be interpreted three ways [APKR]:
  1. Realist (or Aristotelian) answer: x is an object in the world, and X is a set (collection) of objects in the world.
  2. Platonic answer: x is an abstract mathematical object, and X is a mathematical set of mathematical objects. «On this view, we have to distinguish carefully between the physical world of everyday life and the Platonic world of mathematical abstractions. Mathematics is about abstract 'worlds' containing such things as integers, algebras, graphs, categories and the like, and sets belong with these. These categories are completely disjoint from collections of physical things such as people or cans of corned beef. On this view, talk of the set of objects in my fridge commits a category error» [APKR].
  3. Piercian answer: x is an abstract object called a meaning, and X is a set of meanings.
Sets of things? «The Platonic and Piercian answers share a distaste for the Realist's blithe confidence on the talking of sets of things ... The physical world indeed does not come alread separated into individuals, and the task of individuation is nontrivial. ... A Piercian will emphasis[!] that in trying to apply set-theoretic talk to the actual world we often come across difficult borderline cases. If we say that the concept of 'chair' denotes the set of chairs, should we take that to include a reclining armchairs or a spindle-back stool, a royal throne or an executioner's electric chair?»
The Realist replies that his set language «does not imply that one particular, correct, method has been decided upon to carve the world at its joints; it simply assumes that some parts of the world are being individuated. There may be infinitely many ways to do it. ... One way to interpret 'chair' includes the electric chair, another does not. Both are possible, but in each case there is a set of objects involved.
Both the Realist and the Piercian may wish to avoid getting involved with these complex philosophical issues of how people decide on which parts of reality count as things. The Piercian joins with the Platonist in refusing to talk of reality. The Realist escapes the responsibility by observing that sets fit onto the world in any way whatsoever ...
It seems that the differences between the Realist and Piercian positions are less concerned with the mathematical contents of their semantic theories than with what they take to be their philosophical implications. But most of their goals are identical: both advocate the use of model-theoretic approaches to the analysis of meaning, both agree that individuation is a complex matter, and bot urge that our current semantic theories should not be hampered by this complexity.»

Content & Denotation

«Representation is a philosophically puzzling relation. To take one simple example, 'x represents y' seems to express a relation between two things. But while the existence of a relation between two things trivially entails that they exist, this is not true for the relation of representation: a picture, thought, or sentence can represent the Judgement of Paris even if there was in fact no such event. Yet who can deny that all representations do in fact represent something?» [x]

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 meaningmeaning in the language
and/or meaning mentally
meaning in the world:
Aristotle's semaphôné pathêmata prágma
Stoicists' sema somainon lekton tychanon
Augustinus's signumvox sonussignificatusres
Modists' dictio vox conceptio res
Locke's sign word idea (reality of things)
Degerado's signesensation 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 propositionmeaning denotation
Alonzo Church [>] expression (names and forms) sense 'expressed' by name, 'concept of' a denotation denotation 'denoted' by name
Rudolf Carnap [>] expressionintension (a set of attributes) extension / nominandum, designates [DvR] (things, properties, facts, etc.)
Reinhardt Grossmanndescription expressiondefinite 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) representameninterpretant (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)symbolthought or reference referent
Charles Morris's semiotics (1938) SYNTACTICS
study of the relationships between signs on form and content level [KL, ch.4]
study of the reference relationship between signs in language and objects in world
sign carrier (material, unlike Saussure) designate, significate denotation
modern linguisticsformmeaning, content,
valeur (relations to other words)
reference (to a relatum)
study of word form and sentence form
study of a sign's content and the relationship between signs on the content level
not linguistics
structuralist linguisticssurface structuredeep structuresemantic 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?

cf. reference as a relation

Notes on Denotation

Example: All terms for fictious entities (``abstract objects'') like "unicorns" or "Sherlok Holmes" have the same denotation: none. That terms like "unicorn", i.e., horse-like animal with a single straight horn on its head (this is the content of "unicorn"), have no denotation is not a property of the logic or language but requires observing the real world for such animals. (In other words, the statement "the denotation of `unicorn' is empty" is an a-posteriori truth.)
-> cf. Meinong on the properties of golden mountains and round squares [x].
-> cf. David Lewis's doctrine of modal realism: Pegasus and the like have full-blooded existence, and differ from non-fictious objects only in residing in other possible worlds. [x]
-> cf. Edward Zalta and his philosophy of abstract objects.


The ``structural method in semantics'' defines content (sense) without reference to the notion of "concept" [EML 453]: «W.r.t. the empiricial study of the language structure, the sense of a lexical unit can be defined as not only depending on, but morevover being identical with the set of relations, that exist between the unit in question and units in the same lexical system». This structural notion of content is also called valeur [ch.4, KL]: A sign has a valeur by virtue of its relationships with other signs in a langue. E.g. "Sheep", "mutton" (Engl.) and "mouton" (French) have different valeur, still in some situations "mouton" and "sheep" have the same meaning, in others "mouton" and "mutton".

Logical Models of Language

1. Frege. The phenomenon of referential opacity proves that truth or falsehood of a sentence is not only determined by denotation. So there must be more to meaning that the denotations of its parts (if ``meaning'' is to be understood as that which determines truth). «Frege accounted for this by introducing another notion into the theory of meaning, which he called 'Sinn', usually translated 'sense'. The sense of an expression is, intuitively, not what is referred to by an expression, but the way it is referred to. Each sense determines one reference, but to one reference there may correspond many senses ... Central to Frege's view is that senses are abstract objects, not ideas in people's minds» [x, underlining added]. Frege: "morning star" and "evening star" have a different sensce but the same reference (the planet Venus). «The reference of an expression is the entity it stands for: referring expressions stand for objects, predicates stand for functions (in the mathematical sense, which Frege called 'concepts'), and sentences stand for truth-values. Referring expressions and predicates combine to form whole sentences, whose references are a function of the references of their parts» [x]. «All true statements have the same reference: the truth; all false sentences have the same reference: falsehood» [DvR].

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» [

Church assumes the following principles for his formalized language:

  1. [No ambiguity:] (a) Every name has a unique concept as its sense. (b) A form has a unique concept as its sense-value for any assignment of admissible sense-values to each free variable in it. (c) Every concept is a concept of at most one thing.
  2. [Sense-Range:] Every variable has a non-empty class of concepts as its sense-range.
  3. [Reduction denotation/range/value to sense/sense-range/sense-value + concept-of:] (a) The denotation is that of which its sense is a concept. (b) The range of a variable is the class of thing of which the members of the sense-range are concepts. (c) The value of form F for value assignment x1=a1, ... xn=an is the thing A such that the sense-value of F for sense assignment x1=s1, ... xn=sn, where the si are concepts of the si, is a concept of A.
  4. [Sensual transparency:] If C' is obtained from name C by replacing a particular occurence of a name c (form f) by name c' (form f') with the same sense (with the same free variables and sense-value for every admissible system of sense assignments), then C' is a name with the same sense as C.
  5. [Referential transparency >:] If C' is obtained from name C by replacing a particular occurence of a name c (form f) by name c' (form f') with the same dentoation (with the same free variables and value for every admissible system of value assignments), then C' is a name with the same denotation as C.
  6. [Renaming bound variables:] Renaming the (bound) variables in name C from xi to yi with corresponding sense-ranges (ranges) results in a name C' with the same sense (denotation).
  7. [Substituting free variables:] A name results from substituting for all free variables xi of a form a name ni with a sense in the corresponding sense-range.
  8. [Assignment reduces to substitution:] The sense of form F with names ni substituted for variables xi is the same as the sense-value of F with sense assignment xi = sense of ni.
  9. Similarly, rules for substitution in forms, and for substitution of forms, variables, and names for free variables.
«To those who find forbidding the array of abstract entities and principles concerning them which is here proposed, I would say that the problems which give rise to the proposal are difficult and a simpler theory is not known to be possible.[fn: At the present stage it cannot be said with assurance that a modification of Frege's theory will ultimately prove to be the best or the simplest. Alternative theories demanding study are: the theory of Russell, which relies on the elimination of names by contextual definintion to an extent sufficient to render the distinction of sense and denotation unnecessary; the modification of Russell's theory, briefly suggested by Smullyan (...), according to which descriptive phrases are to be considered as actually contained in the logistic system rather than being ... "mere typographical conveniences," but are to differe from names in that they retain their need for scope indicators; and finally, the theory of Carnap's Meaning and Necessity.] » [AESA, underlining added].

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.»

Logical Parts of Speech (and their denotation)

In a logical analysis of parts of speech (by philosophical logic) there are names (denoting objects), predicates (denoting universals/concepts/classes), sentences or prepositions (denoting truth-values), and the operators and connectives (always? denoting truth-functions) and quantifiers (denoting ?). The latter form complex prepositions[x] and sentences[x] out of simpler ones. «A simple sentence typically concatenates a single name with unitary predicate, as, for example, in 'Mars is red'. (Relational sentences involve more names, as in 'Mars is smaller than Venus', but a sentence like this is still regarded as simple.)» [x].

Cf. judegement vs. proposition

«At the risk of oversimplification, we will affirm that the logical statement is composed of three parts:

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'.
... The elements of the statement ... by themselves, are neither true nor false. Only when the composite expression (the statement) affirms or denies the predicate of the subject does truth enter in» [AUOOP].
Compound statements are combined from categorial statements by logical connectives, like the conditional, conjunctive, disjunctive [AUOOP]. Categorial statements predicate one thing (the predicate) of another (the subject) by means of the verb copula [AUOOP]. They are divided (by quality) into 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].


«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

  1. As mapping to set
    «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].
    «Traditionally, one assumes that a definite plural NP like the pupil denotes the same set of indivudals as the common noun pupil, whereas the definite singular NP the pupil has an individual as its denotation. This gives rise to the semantic difficulty that many verbal phrases need various interpretations depending on whether they are combined with an individual-denoting or set-denoting phrase. Thus, for instance, walked has to be semantically represented by a first-order as well as a second-order predicate. In order to overcome this difficulty, some authors suggest that not only plural, but also singular, NPs refer to sets of individuals. Both proposals, however, are ontologically defective: sets as abstract entites outside of space and time are turned into bearers of qualities characteristic of concrete obejcts. In addition, a multitude of entities cannot be grasped in such a manner. For instance, it is evident that mass objects, i.e. quantities of matter, must not be represented as sets for the simple reason that their identity conditions exclude this possibility. In this way, a semantic analysis relying on set-theoretic representation of individuals fails to correspond with natural language ontology» [OntDom].
    As a consequence, the attitude according to which differences between kinds of individual are to be reconstructed as differences between kinds of set has increasingly been retracted. In fact, there are genreal modifications performed in the methodology of formal semantics in favour of a recognition of the "genuine" ontological structure reflected in the way we talk about the world. Suggesting an algebraic semantics of plural, Link (1983) has had a crucial influence on this development [see 'plural reference 3' below]. The main points of this approach may be summarized as follows: no set-theoretic model is given for pluralities; rather, the universe of discourse is considered to comprise ordinary individuals along with particular plural individuals. ... Thus, ... the definite singular NP the pupil as well as the definite plural NP the pupils denote individuals, albeit of different sorts. ... Taking into account certain analogies between plurals and (so-called predicative) mass terms, a combined theory is supplied for both phenomena. ...» [OntDom]
  2. As relation with many instead of mapping to many-as-one
    «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].
    «If the linguistic phenomenon of plural reference is relatively unproblematic, a more difficult question is whether there are plural objects, objects that are essentially not one thing but many things» [Parts 144].
  3. As mapping to mereological sum instead of mapping to set
    «(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.
    «(Lewis 1991) "... if we accept mereology we are committed to the existence of all manner of mereological fisions. But given a prior commitment to cats, say, a committment to cat-fusions is not a further commitment. The fusion is nothing over and above the cats that compose it. It just is them. They just are it ... If you draw up an inventory of Reality accordning to your scheme of things, it would be double counting to list the cats and then also list their fusion." ... "Most of all, it is the axiom of Unrestricted Composition that arouses suspiscion. I say that whenever there are somethings, they have a fusion. Whenever! It doesn't matter whether they are all and only the satisfiers of some description. It doesnt matter wheter there is any set, or even any class, of them ... There is still a fusion. So I am committed to all manner of unheared-of things: trout-turkey, fusions of individuals and classes, all the worlds styrofoam, and many, many more. We are not accustomed to speak of think about such things. How is it done? Do we really have to?
    It is done w the greatest of ease. It is no problem to describe an unheared-of fusion. It is nothing over and above its parts, so to describe it your need only describe its parts. Describe the character of the parts, describe their interrelation, and you have ipse facto described their fusion. The trout-turkey in no way defies description. ... it is part fush and part fowl. It is neither here nor there, so where is it?--Partly here, partly there. That much we can say, and that's enough. Its character is exhausted by the character and relations of its parts."» [AlgSem 780].


«[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].


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.
There are other ways of forming complex sentences than by connecting simpler ones, the most important being through the use of quantifiers[x] - expressions like 'something', 'nobody', 'every planet', and 'most dogs'. The analysis and interpretation of such expressions forms another major area of philosophical logic. An example of an important issue which arises under this heading is the question how existential propositions[x] should be understood - propositions like 'Mars exists' or 'Planets exist'. According to one approach, the latter may be analysed as meaning 'Something is a planet' and the former as 'Something is identical with Mars' (both of which involve a quantifier), but this is not universally accepted as correct. Another issue connected with the role of quantifiers is the question how definite descriptions[x] - expressions of the form 'the so-and-so' - should be interpreted, whether as referential (or namelike) or alternatively as implicitly quantificational in force, as Russell held» [x].

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.
Modal expressions give rise to special problems in so far as they often appear to create contexts which are non-extensional or 'opaque' (extensionality[x]) - such a context being one in which one term cannot always be substituted for another having the same reference without affecting the truth-value of the modal sentence as a whole in which the term appears. For example, substituting 'the number of the planets' for 'nine' in the sentence 'Necessarily, nine is greater than seven', appears to change its truth-value from truth to falsehood, even though those terms have the same reference. (No such change occurs if the modal expression 'necessarily' is dropped from the sentence.) How to handle such phenomena - which also arise in connection with the so-called propositional attitudes[x], such as belief - is another widely studied area of philosophical logic.

Meaning of Physical Formulae

Physics talks about measures of properties, without mentioning objects. Physics «presupposes, but does not explain, notions of reference, representation, objectivity, and the like ... Ontologically ... it is not clear that physics actually does presuppose the notion of individuality», of discrete individuals, on top of particularity [OOO 176]. In physics, «[t]he primary theoretic identifiers are used to designate the properties and relations ... whereas the particulars they apply to are left wholly implicit» [OOO 173]. «It is not just that no particular object figures in the laws of mechanics or electromagnetism; there are no generic place-holders for objects, either. The equations contain no variables that range over individuals; not even, surprisingly, over space-time points, such as particular locations. Instead, ... the variables range over the measures of properties---for example, over generic measures of the distance between locations and a fixed point of origin» [OOO 177 >].

(Objects are however absolutely crucial to make any use of physical formulae:
 -> modeling physical reality )

«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].

  1. «When we interpret the equations as describing some situation at hand---e.g., as applying to the distance between the table and the wall [of my kitchen]---we take the terms as having particular reference ... So xi and xf would be names of particular locations ... Similarly ... h would also be taken ... as referring to the height of this particular table, in the sense that that names a different thing from the height of some other table ...»
  2. «With respect to the official interpretation of equations, however, the situation immediately changes color. The particular interpretation vanishes, and all terms are instead interpreted as having purely universal interpretation ... They are taken as referring to properties or features---measurable features, ... that could be exemplified by any number of other particular phenomena. Thus d, for example, which works out on the official interpretation to be 1.35 meters, refers to an amount of distance, not to the unique particular spatial spearation between the edge of the table and the point of impact. Variable d does not refer to that located separation of the world, in other words; rather, from this point of view it applies to that separation in virtue of being a measure of that separation, a measure that would be exemplified by any other sepration between two particular locations that were also 1.35 meters apart. Similarly, if, on the particular interpretation, h referred to the height of this table, and h' to that of another table of the same extent, then on the universal interpretation the equation h = h' would not be a claim of the common extent of two particulars, but rather an assertion of identicality between one and the same abstract measure.
    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.»
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 ...

The laws of physics are fundamentally deiectic» [OOO 168].


Organizing the classifications of definitions is a pain. There are several list of definition classifications in this glossary. One is this:

  • A real definition says what (the ``essence'' of) the thing R is to which x refers (which thing R term x refers to is x's nominal definition). These kinds of definitions you find in an encylcopedia (as opposed to a linguistic dictionary), e.g., «An internal combustine engine is ...», «A Christmas tree is ...», ... These are the definitions important in the natural sciences and humantities [ZDM]. Carl G Hempel propose three directions of the classical notion of "real definition" [DvR]:

    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.»
    Cf. definitions in general.
    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]:
    1. Definitions where the specifying difference D within the genus G is in terms of intrinsic accidents:
      1. Essential definitions/relations say what `kind of thing' X is: "Man is a rational animal".
      2. A non-essential definition/relation «tells us ... what belongs to or inhers in the subject in some way. From the viewpoint of definition even these non-essential connections can help us to distinguish adequately one kind of thing (i.e. species) from another» [AUOOP]:
        • Definition by a proper accident (something that belongs to X alone): "man is a tool-forging animal", "gold is the most malleable metal".
        • Definition by combinations of common accidents (each of which alone is not exclusive to the X): "man is a featherless, furless biped with fingernails and toenails" (this definition is by Plato)
    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).
  • Definitions where the specifying difference D within the genus G is in terms of extrinsic accidents:
    1. Causal definition employes agent or end: "Man is an animal who chooses what he sees as good in order to be happy"
    2. Measurement/Operational definition assumes a standard of measuring: "Man is an animal with forty-six chromosomes" (used in experimental sciences)
    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

    Numbers are not classes (as construed by Frege and Russell): «[C]lasses stand in the relation of having members to entities. No number stands in this relation to anything. On the other hand, numbes ... stand in various arithmetical relations to each other. No class stands in such a relation to anything. ... It follows that numbers are not classes. ...
    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].
    «It is claimed that ... arithmetic relations can be "defined" in terms of relations among classes. "The chief point to be observed" says Russell, "is that logical addition of classes is the fundamental notion, while the arithmetical addition of numbers is wholly subsequent." ... But we may have acquired by now a sagacious suspicion of any claim that certain entities do not exist because they can be defined in terms of other entities. ...» [OntRed 55]
    «Consider Russell's definition of addition:
    (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].

    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].

    Ulf Schünemann 080402