**
Formal explanation of truth**:
I defined truth as follows:
(*) A statement is true
iff what the statement means
represents a
fact. Accordingly, the **truth** is whatever exists in
reality, whatever is real.
This lemma is given to explaining this formula formally, presupposing
some basic logic. The main
point of the whole exercise is not its formality, but that using
some formalities one can give a fairly full and precise
explanation and
listing of precisely which assumptions enter here.
What we want to explain are
subject-predicate
statements of this form
(1) Fa
in which something named a (with the term "a") is asserted to have a
property named F (with the term "F").
The present explanation of "truth" differs from standard ones like
that on Tarskian lines in insisting that an adequate definition of truth
refers to **both** ideas and
reality, but conforms in this respect to
Aristotle:
"Spoken words are the symbols of mental
experience and written words are the
symbols of spoken words. Just as all men have not the same writing, so
all men have not the same speech sounds, but the mental experiences,
which these directly symbolize, are the same for all, as also are those
things of which our experiences are the
images.
(..)
As there are in the mind thoughts which do not involve
truth and falsity,
and also those which must be true or
false, so it is in speech. For truth and
falsity imply combination and separation. Nouns and verbs, providing
nothing is added, are like thoughts without
combination or separation; 'man' and 'white', as isolated terms, are not
yet either true or false." (p. 7-8, "**Aristotle - selections**",
W.D. Ross Ed.)
These Aristotelian intuitions will be followed below, except that the
the mental experiences of different men or of the same man at different
times in response to hearing or reading the same term of which they know
the meaning are supposed to be
similar.
The answer to the question how to define "is true" for a statement
like (1) in brief is this:
(2) "Fa" in
language L is true in domain D IFF r(d,m,L,D) and
d(m("a")) e d(m("F"))
which is in words:
The sequence of
terms "Fa" that is a statement in language L is
true in domain D precisely if language L
represents
domain D using functions d (denotation)
and m (meaning) - abbreviated: r(d,m,L,D)
- and the denotation of the meaning of the term "a" belongs to the
denotation of the meaning of the term "F".
Here the quoted part in red is a statement in the language L for
which we provide the truth-definition, and the rest of claim (2) and
indeed all of it is part of **natural language **(here: English)
enriched with variables and enriched
with set theory, which occurs in (2) in green.
The main task now is to explain this notion of
**representing**. Here is
a definition - fairly lengthy, but explained below:
r($,d,m,L,I,D) IFF (ae$)(Iea)(T_{i}eL)(T_{j}eL)(D_{i}i inc D)(D_{j} inc D)
( $ is a society of speakers of language
L
&
a is a speaker of
L
&
I is the set of ideas
of
a
&
T_{i} is a term of
L
&
T_{j} is a term of
L
&
D_{i} is a subset of
D
&
D_{j} is a subset of
D &
d : Terms of L |->
powerset of
I
&
m : powerset of I |-> powerset of
D
&
m(~T_{i})=powerset of
I-m(T_{i})
&
m(T_{i})=m(T_{i} & T_{j}) U m(T_{i} &
~T_{j})
&
d(-D_{i})=powerset of
D-m(D_{i})
&
d(D_{i})=m(D_{i} O D_{j})
U m(D_{i} O -D_{j}) )
In words this comes to the following, where the
powerset of the set is the set of all
subsets of the set,
and the domain is some set to the
elements and subsets whereof the terms of language L refer:
We say that language L used in society $ represents domain D for the speakers of L
precisely if
- for every speaker a of L and a's
ideas I
- for every term T
_{i} and every term T_{j}
of L
- for every subset D
_{i}
and every subset D_{j}
of the
domain
- there is a function m (for
meaning) that maps the terms of L to the powerset of ideas of a
- there is a function d (for
denotation)
that maps the powerset of ideas of a to the domain D
and such that
- the
meaning of
**the negation of a term**
is the powerset of I except that part that is the meaning of the term
- the meaning of
**a term** T_{i}
is the union of the meaning of the conjunction of Ti and T_{j}
and the meaning of Ti and the negation of T
- the denotation of the
**complement of
a set** is the powerset of D except that part that is the denotation of
the term
- the
denotation
of
**a set** Di is the union of the denotation of the intersection of D_{i}
and D_{j}
and the denotation of the intersection of D_{i}
and the complement of D_{j}
The first five of these points list the wherewithall for the last four
assumptions, which are mostly what matters.
What is to be noted here is especially that - as I define it -
truth requires **both** a domain of
ideas of speakers of language L, that can be
rendered as a set, and a
domain of things
that the terms of L denote, that also
can be rendered as a set.
The speakers of L are those who have learned the meanings of the
terms of L, and have learned many of these meanings by learning also the
denotations of these meanings. Thus, they may know that the term
"elephant" represents a large mammal with a trunk, and they may have
seen some real elephants, but they also may know that the term "mermaid"
represents a creature with the upper body of a woman and the lower body
of a fish or dolphin, which they may have seen pictures of, but no such
real things in the real world, because
mermaids do not exist.
The last four of the above points attribute properties to the
functions that were earlier assumed, and these properties are in fact
such that **negations** of terms and **complements** of sets
behave as one intuitively expects (what is not X in domain D is
everything in D except what happens to be X in D, for example, and the
same for I) and such
that **conjunctions** of terms and **intersections** of sets also
behave as one intuitively expects (what is X in domain D is made up of
what is both X and Y in D plus what is both X and not Y in D, and the
same for I).
And now we can see the point of the whole notion of
representing:
Supposing that for speakers of L domain D is represented by functions
m and d, then the claim that "Babbar is an elephant" is true in D
precisely if the denotation of the meaning of "Babbar" is an element of
the denotation of the meaning of "is an elephant".
Note that as Aristotle insisted in the above quotation, the terms
"Babbar" and "is an elephant" both do have a meaning (if they can be
understood at all) but by themselves no truth, and may have a denotation
(if what they mean represents something real) and thus may be said then
to refer to something that exists, yet
still without being truths, for **truth and falsity arises from the
combinations of terms into statements**.
And we can now also start giving an explanation why the **Dutch** statement
"Babbar is een olifant" would mean the same as the **English**
statement "Babbar is an elephant":
Namely, because "Babbar" in either case is taken to refer to the same
thing, and
"is een olifant" is taken to refer to precisely the same
set as "is an
elephant" - and this regardless of whether there really are any
elephants. (If Babbar dies and all elephants also are exterminated,
still the two statements in the two languages mean the same provided
their constituent terms mean the same, as they well may do regardless of
the existence of Babbar or of elephants.)
Thus, the present explanation of the notion of truth that involves
both ideas and domains undoes a confusion of ideas and domains (e.g. as
propounded by Davidson) that derives from a confused understanding of
model theory.
Also, it should be mentioned here that the domain is arbitrary as
long as the speakers of the language L agree what set of things - what
universe of discourse - the language they presently use is supposed to
be about.
It makes sometimes sense to say that "In Conan Doyle's fictional
England, Sherlock Holmes lived in London" - supposing from the start
that what one chooses to speak about is Conan Doyle's
fictional England. Thus, one may also
insist "In Conan Doyle's fictional England King Arthur did not live in
London" - and indeed, both claims in this paragraph gives some
information about what Conan Doyle's
fiction is about.
Apart from that, usually and mostly natural language (if not used
postmodernistically) is supposed to
refer to such things as are in the
everyday natural reality every human
being lives in, and that is best if not fully charted by
science.
Finally, it is not difficult to extend the above definition of
represents in (3) to one that also allows reckoning with numbers and
probabilities. This is sketched in some detail in **
**__
The measurement of reality by truth and
probability,__ and also in
representing
formalized. |