Semantics
The most
useful simple definition of "semantics" is "theory
of meaning". In general terms, it is the end of a semantics to
explain how a string of text (or anything else supposedly
meaningful) can represent something that's normally neither text nor
contained in the text.
There are
three immediate problems with any semantics:
1.
the differences involved in meaning and denotation
2. the difficulties involved in treating truth linguistically and in
stepping outside language
3. the meanings of representing and meaning

These
problems are as follows:
1. the differences involved in meaning and denotation:
There are
intuitively two different though related meanings of "meaning",
which can be illustrated by considering the meanings of the terms
"elephant" and "mermaid": Both terms have a meaning
according to any qualified speaker of English, in the sense that any
qualified speaker can mentally imagine elephants and mermaids, but the terms
differ in that there really are elephants while there really are no mermaids.
This is
intuitively a quite clear distinction, but it is hard to spell out precisely,
for one thing because there may be terms one just doesn't know
whether there are any real things it stands for. It is also connected with a
number of oppositions such as "meaning" and "denotation"
to mark the difference between representing mere ideas ("meanings")
that do not represent anything real and representing ideas that do represent
something real ("denotations"). And indeed I will use the
terms "meaning" (intuitively:
the idea a term represents) and "denotation" (intuitively:
whatever things an idea represents) to mark this opposition.
2. the difficulties involved in treating truth linguistically and
in stepping outside language:
There is an
intuitively correct rendering of what it is to be a true statement that was
first clearly stated by Aristotle:
A
statement is true if and only
if it says what it is the case.
We shall restate this
somewhat as:
(*)
A
statement is true if and only
if
what it represents
is what
represents what is the case.
This
rendering has the great merit of explaining why humans would be interested in
believing true statements: It provides them knowledge
about their real environments and themselves.
True statements represent ideas that represent some real fact(s).
The problem
with this intuitively correct rendering is that for most true statements one
knows what is the case is not linguistic or a part of language.
Therefore, to spell out in language what
is truth one needs some way to represent linguistic
terms in language and some way to represent what linguistic terms denote and mean in
language, and not mix up the differences.
In what
follows the wellknown means of quotationmarks is used to mark the
differences between terms for terms, terms for ideas (i.e. what terms mean),
and terms for things (i.e. what ideas refer to). To indicate that a term T is
a term for terms, T will be included in double quotemarks, and "T"
will be read as "the term T". To indicate that a term T is a term
for ideas, T will be included in single quotemarks, and 'T' will be read as
"the idea of T". To indicate that a term T is a term for things, T
will be unquoted, and T will be read as "T" or "the thing
T".
3. the meanings of representing and meaning:
The best
systematic general way to think about meaning and denotation is in terms
of representing, as e.g. a map of England represents the territory of England.
This is an
analogy which is helpful in many ways, including that:
·
the map is usually not the territory (even if it is part
of it)
·
the map does usually not represent all of the territory but only
certain kinds of things occurring in the territory, in certain kinds of
relations
·
the map usually contains legenda and other instructions to
interpret it
·
the map usually contains a lot of what is effectively
interpunction
·
maps are on carriers (paper, screen, rock, sand)
·
the map embodies one of several different possible ways of
representing the things it does
·
the map usually is partial, incomplete and dated  and
· having
a map is better
than
having no map at all to understand the territory the map is about
·
maps may represent nonexisting territories and include guesses
and declarations to the effect "this is uncharted territory"

The problem here is to find a useful central definition of representing that
can be used to explain meaning and reference systematically. Using the tools introduced
above:
Maps, representations and simulations


f maps F

IFF

(x,y)eF iff f(x)=y


repr(X,Y,f)

IFF

($F)(f maps F & (X,Y) inc F)


sim(X,Y,d,e)

IFF

(repr(X,Y,d) & repr(Y,X,e))

The notion
of mapping in Basic Logic is defined using functors: A map is a relation the last terms
of which are given by a characteristic functor for
their first terms. This makes mappings functional without making functions
necessarily mappings, since it may happen that a given function comes without
functors for its values. Note that functors were
introduced in the grammar of BL as "x(y)" and correspond to the
English phrase "the x of y" or "y's x" and that a
characteristic functor f for a functional relation F is one that allows
the finding of all y such that (x,y)eF for any x using the functor f.
So functors
are things that are capable of responding systematically to other things
(when these are presented to them in an appropriate way), while real things
in general may be presumed to be functorial in certain ways in that real
things respond to other real things in some definite determinate ways.
Maps can be
seen as functions with an extra: both f(x) and y are explicitly included, and
indeed one can conceive of a map as a functor for a function: Having the
functor f enables one to find all second terms of F given first terms of F
and presenting these to f. Thus, if F is the relation 'fathered', e.g.
'father of' may be taken as a functor for it. That is: one has with (F =
fathered) for each x one y that is the father of y and one has for each x the
f that somehow finds or produces the father of x whomever he may be. (In
actual practice such f may be a passport or a DNAtest.)
One often
helpful way of thinking about this is that f(x) is the
program to find y. This makes sense already for simple functions like
doubling, squaring or taking the logarithm: As soon as the numbers grow
largish the actual calculations  say: the double, square and logarithm to 10
places of 1234.567.890 (which cannot be found in a lookup table (as is
in fact presumed by a mere function!) and have to be somehow
calculated, and to calculate these we do in fact apply a paper or electronic
algorithm or program (that takes a certain time, and itself
is a definite entity or thing). And indeed an ordinary handheld calculator
contains programs (functors in BL terms)
to find values for inputs which it displays once calculated.
And it also
should be noted that there are in nature and in
experience very many quite ordinary things that work like functors,
from lightswitches and other switches to chemical reagents that allow one to
test whether a certain substance is gold (or helium or etc.) to software and
hardware for many kinds of tasks . Also, one is oneself functorial in many
ways, including laughing and blushing, both of which are functorial processes
that allow others to see that one is effected in a certain way by something.
The notion
of representation given here uses mappings.
The terms X and Y in it are any terms for classes or sets, and
"(X ,Y) inc F" abbreviates "(x)(y)(xeX & yeY > (x,y)
e F)". Note that the same X and Y may be included in different
mappings by another g and G, and that what is mapped in X to what in Y
depends on the mapping that is chosen. The general point of a representation
of Y by f on X is that one can infer structures and
relations in Y from structures and relations in X by way of f and F,
just as with an ordinary paper map, where one can infer the lay of the land
from a map about it.
Also, with
paper maps one has a rather clear instance of "f(x)" in the
legenda. Thus, on ordinary political maps one can infer the approximate
number of inhabitants of a city from its color or shape on the map, where
this this shape or color then encodes 'the number of inhabitants of'.
The notion
of simulation amounts to mutual representation. Convenient names for the
functors d and e
are resp. "denotes" and "encodes". Note that by the rules for equality
we have d(e(y))=y and e(d(x))=x, given that d(x)=y and e(y)=x. Alternative
names for this relation of simulation are: "similar",
"isomorphic", "analogies" and "mapping". Here
is a simple sketch illustrating the general idea of simulating:
If we
suppose the above sketch depicts Ideas on the left and a World on the right,
one may add a linguistic representation thus: B0(B1, B2(B3)).
The terms
"maps", "represents" and "simulates"
represent the structures fit and used for mapping, representing and
simulating things and relations  in short: structures  linguistically. And
especially simulations are formal analogies: If X and
Y do simulate each other  are analogies, similar
structures, mappings of each other 
using d and e
this means that certain kinds of things and relations in X correspond to certain kinds of things and
relations in Y and conversely, so that when
one knows the denoting functor d one can
infer aspects of Y from aspects of X and if one knows the meaning functor e one can infer aspects of X
from aspects of Y. (Note that while these
undo each other formally, intuitively they differ: The entity a term or idea
denotes is related differently to the term or idea than an idea or thing is
related to the term or the idea that represents it.)
But none of
this explains the differences between means and denotes. Now intuitively,
what is meant by a statement or term is some idea of some speaker, if only an
idea of the speaker himself. and what is referred to or denoted is some
thing(s) or structure(s) in some world, real or fictional, but supposedly
such as diverse speakers can find evidence about and come to agreements about
with other speakers about what is and is not in it.
For
semantics we shall use "[q]_{(L,M,D,m,d)}=1"
which may be read as "q is element of L that means something in M by d
that is meant by q via m and denotes something in D by d that is meant
by what q means via m", where "[q]" is a conveniently
brief variant of a mapping term: "[q]" is read as "the
truthvalue of q". One reason to introduce language, ideas,
worlds and mappings by subscripts is that they are often left out as
"understood from the context".
It should be
noted that this notation comprises quite a lot in a compact way that may be
also spelled out somewhat differently  as will indeed be done in later
chapters. Here I shall merely sketch the general foundations.
Now suppose
that a language L, ideas I and things T are given such that L and I simulate
each other and I and T simulate each other, in both cases by the same
functors d and e that involve the relations of denoting
and encoding that comprise (L,I) and
(I,T) i.e. (x)(y)(xeL & yeI > (x,y)eD & (y,x)eE))
and (x)(y)(xeI
& yeT > (x,y)eD & (y,x)eE)). Note that by earlier conventions
and assumptions this means that there are things or processes that are denote
and mean.
This general
assumption may be fleshed out and qualified in several ways, but the
general idea is that one has the functors meaning and denoting that relate
both linguistic items and human ideas and human ideas and entities in some
world or domain, in such a way that the structures that are related are
similar.
Semantics


Sim(L,I,m) & sim(I,T,d)


[q]_{(L,I,T,m,d)}=1

IFF

qeL & m(q)≠ø & d(m(q))≠ø


[q]_{(L,I,T,m,d)}=0

IFF

qeL & ~( m(q)≠ø & d(m(q)) ≠ø
)


[q]_{(L,I,T,m,d)}=1 V [q]_{(L,I,T,m. d)}=0

IFF

($ta..tk)($t1..tn)[taeL & .. & tneL
& (ta..tk)(t1.. ta .. tk .. tn) = q)


[p>q]_{(L,I,T,m,d)}=1

IFF

[p]_{(L,I,T,m,d)} <= [q]_{(L,I,T,m,d)}


[x=y]_{(L,I,T,m,d)} =1

IFF

xeL & yeL & m(x)=m(y) & d(m(x))=d(m(y))


_{ }[E!x]_{(L,I,T,m,d)} =1

IFF

($x')($x'')(x'eI & x''eT & xeL & m(x)=x'
& d(x')=x'')


[($x)(t.(z))]_{(L,I,T,m,d)} =1

_{ }IFF

xeL & teL & zeL & m(xt(z))eI &
d(m(xt(z)))eT


_{ }[($x)~(t(z))]_{(L,I,T,m,d)} =1

_{ }IFF

xeL & teL & zeL & ~[m(xt(z))eI &
d(m(xt(z)))eT]


_{ }[(x)(t(z))]_{(L,I,T,}_{m,d)} =1

_{ }IFF

xeL & teL & zeL > m(xt(z))eI &
m(d(xt(z)))eT






Pos[q]_{(L,I,m,d)}=1

IFF

($T)(qeL & m(q)eI & d(m(q))eT)


Nec[q]_{(L,I,m,d)}=1

IFF

("T)( qeL > m(q)eI & d(m(q))eT)






S = STRUCTURES_{(L,I,T,m,d)}

IFF

S = (t)(teL & m(t)eI)


T = THOUGHTS_{(L,I,T,m,d)}

IFF

T = (q)(qeL & m(q)eI)


O = OBJECTS_{(L,I,T,m,d)}

IFF

O = (t)(teL & m(t)eI & d(m(t))eT)


F = FACTS_{(L,I,T,m,d)}

IFF

F = (q)(qeL & m(q)eI & d(m(q))eT)

Here are
some brief comments:
This assumes
in effect that languages simulate ideas and ideas simulate things, and that indeed the same functors
are involved that are converses of each other. This may be an idealization,
but it is convenient (and in fact can be achieved formally in any case). In
both cases, the main import of "simulates" is "has the same
structures".
Note that
what is explicit here in semantics is usually left out: There is a domain of human ideas that is denoted by
linguistic structures and that denotes things in some possible world or
domain.
[q]_{(L,I,T,m,d)}=1

IFF

qeL & m(q)≠ø & d(m(q)) ≠ø

[q]_{(L,I,T,m,d)}=0

IFF

qeL & ~( m(q)≠ø
& d(m(q)) ≠ø )

Formally,
all semantics adds is terminology for explicitly formulating (supposed)
truths about some presumed world or domain, but it does so by fitting this within
linguistic structures that represent ideas of things.
For
intuitively to state a truth is to represent linguistically some idea that represents
something in some presumed domain of things of some kind, and this is precisely
what the first assumption says, just as the second says that to state a falsehood is to
represent linguistically something that does not represent an idea that
represents something in the presumed domain. And indeed two reasons to
be explicit about both ideas and domains next to languages is
that different languages may be used to describe parts of the same domain,
and different domains  fact, fiction, guess, past, future, possibility,
indeed anything whatsoever  may be described by the same language, and
indeed in any case one describes a domain one does so by describing one's
ideas about it.
Note that
the import of the first equivalence is that a statement is true iff it both denotes an idea and that idea denotes a fact, all in the presumed domains of
human ideas and world spoken or thought about. Therefore in the second
equivalence, that stipulates when a statement is not true this may be so for
several reasons: There is no idea the term represents, or there may be
no thing the idea represents, or both. That is alternatively: [q]_{(L,I,T,}_{m,d)}=0 IFF qeL & [m(q)=Ø V d(m(q))=Ø] that expresses that either the term q denotes
nothing ("is not meaningful") or else that while the term denotes
some idea that idea denotes nothing in the world T spoken about ("is not
true").
[q]_{(L,I,T,m,d)}=1 V [q]_{(L,I,T,m,d)}=0 IFF (Eta..(Etk)(Et1..tn) [taeL & .. & tneL
& (ta..tk)(t1.. ta .. tk .. tn) = q)]

[p>q]_{(L,I,T,m,d)}=1

IFF

[p]_{(L,I,T,m,d)} <= [q]_{(L,I,T,m,d)}

The first
of these assumptions says in effect that any
statement is true or not precisely if it is in fact an abstraction.
Indeed, this makes statements kinds of abstractions, and this seems a sound
insight most people tend to miss most of the time: Even the truths one knows
are partial, abstract and selected, and normally what one states even if true
states only a small part of what is truly going on.
The second
of these assumptions uses the fact that truthvalues were assumed to be numbers and in effect defines an implication to be
true iff its consequent is true if its antecedent is true. (This recourse to
numbers not necessary but very convenient, since having numbers one has
known structures for and properties of numbers one can use. This also is
convenient when logic is extended to probability, as will be done in
a later chapter.)
Here and elsewhere
in formal semantics the reader should realize that a considerable part
of any good linguistic analysis of a concept consists in defining it in terms
of earlier presumed notions, and that definitions take the form of
equivalences.

[x=y]_{(L,I,T,m,d)} =1

IFF

xeL & yeL & m(x)=m(y) & d(m(x))=d(m(y))


[E!x]_{(L,I,T,m,d)}
=1

IFF

($x')($x'')(x'eI & x''eT & xeL & m(x)=x'
& d(x')=x'')

These two
postulates state when an equality is true (iff
its two terms denote the same idea that denote the same thing) and when a
term for an entity exists (iff it denotes some
idea that denotes something in the domain).

[(Ex)(t(z))]_{(L,I,T,m,d)} =1

_{ }IFF

xeL & teL & zeL & m(xt(z))eI
& d(m(xt(z)))eT


[(Ex)~(t(z))]_{(L,I,T,m,d)}
=1

_{ }IFF

xeL & teL & zeL & ~[m(xt(z))eI
& d(m(xt(z)))eT]


[(x)(t(z))]_{(L,I,T,m,d)}
=1

_{ }IFF

xeL & teL & zeL
> m(xt(z))eI & d(m(xt(z)))eT

Here
the truthvalues for quantifiers are given in terms of their satisfying
abstractions: That there is some x that satisfies the things t that are z is
true in T by d iff x and z belong to language L that represents domain T by
functor d while the denotation of freely substituting x for t in z
exists in T. This is similar for there NOT being some x that satisfies the
things t that are z is true in T by d. The assumption for "for
all" involves mere substitution, to accomodate valid inferences like
(x)(y)(x=y > y=y).

Pos[q]_{(L,I,m,d)}=1

IFF

(ET)(qeL & m(q)eI & d(m(q))eT)


Nec[q]_{(L,I,m,d)}=1

IFF

(ET)( qeL > m(q)eI & d(m(q))eT)

Here logical possibility and logical
necessity are defined by means of quantification over domains. The
main reason to include these definitions is to show how modalities like
possibility and necessity may be dealt with plausibly: By quantifying over
several domains (then often called "possible
worlds"  which is somewhat misleading for several reasons, one
of which is that one may also quantify over domains with supposedly
impossible or fictional entities).

S = STRUCTURES_{(L,I,T,m,d)}

IFF

S = (q)(qeL & qePROP & m(q)eI)


T = THOUGHTS_{(L,I,Tm,d)}

IFF

T = (t)(teL & teTERM & m(q)eI)


O =
OBJECTS_{(L,I,T,m,d)}

IFF

O = (t)(teL & teTERM & m(t)eI & d(m(t))eT)


F = FACTS_{(L,I,T,m,d)}

IFF

F = (q)(qeL & qePROP & m(q)eI & m(d(q))eT)

This uses
earlier definitions to define some useful terms: The
structures of language L are precisely the ideas represented by the
statements of L and the thoughts of a language L are
precisely the ideas represented by the terms in L. The
facts of a language L related to a domain T by a functor d are
precisely the statements in L that do represent some structure(s) in T,
while the objects of language L are precisely
the terms in L that represent some thing(s) in T.
Of course,
the structures and thoughts of L may have nonempty intersection, and the
same holds for the objects and facts of L. And in general these distinctions
derive from the language L.
Also, the
things that make up L are structures as well (made from terms) – in fact
anything whatsoever – if it can be represented by language  is supposed to
be a structure, and the statements and terms of a language are structures as
well.
Note that in
each case the reference to the domain and the functor is quite essential:
Without these nothing true or real is conveyed, whatever the
appearances. And note also that the domain and the language may be
 and usually are  known to be simplifications of other domains and
modes of speech. (The models  descriptions, explanations, representations,
diagrams  used in science normally disregard all the possible
facts that are supposedly irrelevant to whatever is modelled.)
As the given
definitions of logical possibility and necessity likewise convey the notion
of a true statement as defined is both relative in
several senses and objectively so: It is
relative to a language, a domain and a mapping relating these, but once these
are given or fixed it is objectively so whether the specified domain does or
does not contain something that corresponds to the statement by the mapping
(even if one does not know whether the domain does contain a structure
represented by a statement).
...................................................
Dec 15,
2009: This is the old file corrected with "(Ex)" for the existential
quantifier, "(x)" for the universal quantifier, and with some adjusting of the
white tables.
It now also
is the start of a new version directly called Basic Natural Logic.
Maarten Maartensz