Basic Logic  semantics:
The semantics of a formal system are the assumptions and the theorems
that can be proved with their help that explain how the terms of a formal system
represent and how the statements of a formal system come to be true or not true.
This continues Basic Logic by explaining
the fundamentals of formal semantics. There are various novelties in the present
approach, notably  but not only  that Basic Logic is used to express its own
semantics.
1. Truthfunctional semantics
2. Denotational semantics
3. Probabilistic semantics
4. Modal semantics
1. Truthfunctional semantics
In BL we have functions, and so we can use it to define a function to assign
values to statements:
v maps the statements of BL to {0,1} i.e.
Function(v, statements of BL, {0,1})
Now we can write "v(P)=1" i.e. "the value of
(the statement) P equals 1" and "v(P)=0" i.e. "the
value of (the statement) P equals 0" etc. We take "0"
as "not true" and "1" as
true, but could instead
have used v to map to {T,F} or
{true, false}. The reason to rather use {0,1} is
that it makes more sense mathematically and later, when also
probabilities are
used.
Now we can use v to write a number of assumptions and definitions:
A0.  1≠0
A1.  v(P)=1 IFF P
A2. 
v(~P)=1 IFF v(P)=0
A3. 
v(~P)=0 IFF v(P)=1
A4. 
v(P&Q)=1 IFF v(P)=1 & v(Q)=1
A5. 
v(P&Q)=0 IFF v(P)=0 V v(Q)=0
A6. 
v(PVQ)=1 IFF v(P)=1 V v(Q)=1
A7. 
v(PVQ)=0 IFF v(P)=0 & v(Q)=0
A0 is the explicit assumption that 1 is unequal to 0, and A1 is the explicit
assumption that one may write P in BL precisely if the value of P equals 1.
The assumptions A27 are assumed equivalences that may be taken as
definitions, that lay down for "~", "&"
and "V" what it is to be true and not true in terms
of their component statements.
These are also all just what one would expect intuitively if one makes  as
we do  the assumption of truthfunctionality:
 Whether a statement with "~", "&"
or "V" has
the value true (or not) depends on the values of the statements that make up
the statement.
This assumption is in many ways the simplest sort of assumption one can make
about the truthvalues of statements.
One can summarize the yield of A2 to A7 in English as
 A denial of a statement is true precisely if the statement is not true.
 A conjunction of two statements it true precisely if both conjuncts in the
statement are true.
 A disjunction of two statements it true precisely if some disjunct in the
statement is true.
In BL we also assumed explicitly two rewriting rules, and we need
truthvaluational assumptions for these as well:
A8.  v(P)=1 IFF v(P)=1
A9.
 v(P)=0 IFF v(P)=0
A10.
 v(P=Q)=1 IFF v(~PVQ)=1
A11.  v(P=Q)=0 IFF v(~PVQ)=0
Thus, in terms of truthvalues being able to be written in BL coincides with
being true. And in terms of truthvalues that one may write Q if one has written
P coincides with it being true that notP or Q is true.
There is more to rewriting rules than truthvalues, namely that they are
permissions to write and assert in BL, but in so far as truthvalues are
concerned the assumed equivalences make intuitive sense.
This may be illustrated for A10 and A11.
Suppose it is true that notP or Q is true. Then if one may write that P is
true, it follows from the properties assumed for "V" that Q is true, which one
then accordingly may write. Thus the Right Hand Side  RHS  of A10 states a
property that the LHS of A10 should have, if this is to be a rewriting rule that
leads from truths to truths.
Suppose it is true that notP or Q is not true. Then one can show, using
other assumed rules, that in fact P is true and Q is not true. But in that case
a rewriting rule that leads from truths to truths should not be true, and indeed
it is by A11.
Now we note the following fundamental and interesting fact, we in fact used
in the last two paragraphs:
 We can use the rewriting rules for
& and V and
~ we have assumed with BL to validate these rewriting
rules that are truthfunctionally defined in terms of & and
V and ~.
And indeed we can now also use BL and the assumed truthvalues for the
logical connectives to prove a number of statements about BL, such as the
following
MT1.
The PLpart of BL is logically valid.
MT2. The PLpart of BL entails standard classical propositional logic.
By MT1 is meant that each of the assumptions for PL = Propositional Logic in
BL is a statement that has the value 1, and each inferencerule of the form
A =
C that was assumed has the property that if v(A)=1 then
v(C)=1.
By MT2 is meant that the PLpart can be used to derive the axioms of any
standard classical propositional logic, such as that in HilbertBernays, which
consists of the following axioms (and I follow Wang, 'Logic, Computers and Sets'
p. 308):
HB1a. P > (Q > P)
HB1b. (P > (P > Q)) > (P > Q)
HB1c. (P > Q) > ((Q > R) > (P > R))
HB2a. (P&Q) > P
HB2a. (P&Q) > Q
HB2c. (P > Q) > ( (P > R) > (P > (Q&R)) )
HB3a. P > (PVQ)
HB3b. Q > (PVQ)
HB3c. (P > Q) > ( (Q > R) > ((PVQ) > R)) )
HB4a. (P IFF Q) > (P >Q)
HB4b. (P IFF Q) > (Q >P)
HB4c. (P > Q) > ( (Q > P) > (P IFF Q) ) )
HB5a. (P > Q) > (~Q > ~P)
HB5b. P > ~~P
HB5c. ~~P > P
The above classical system has several virtues, but the reader who doesn't
know about this (and also those who do) may try to derive the above statements in BL,
to verify MT2.
Now let us consider the truthfunctional semantics we have.
One way to look upon it is as  given the truthfunctional assumption we made
 the analysis of logical connectives in terms of the truthvalues of the
statements they contain. And this is not so much a definition or analysis of the
notion "true" or "truth" as a definition or analysis of the properties of
logical connectives given that one makes the truthfunctional assumption.
More specifically: A truthfunctional semantics does not analyse the notion
of "true" but takes it as given and uses it to analyse the properties of logical
connectives given the assumption of the truthfunctionality of statements.
2. Denotational semantics
But it is also possible to use the tools BL provides to state and explain
another sort of semantics for the statements and terms of BL than a
truthfunctional semantics.
This is a denotational semantics, that explains that a statement is true by
considering whether what the statement represents is or is not in some supposed
domain of things (facts,
structures, events  whatever).
Such a denotational semantics can be added to a truthfunctional semantics,
by adding to start with a function
d maps the statements and terms of BL to the structures
and things in some supposed domain D
This function d has the additional benefit of also allowing the analysis of
predicates, relations, nouns and names in an intuitive way.
Namely  as we shall see later in some detail  by associating with a
predicate the set of things it is true of; by associating with a relation 
twoplace predicate  the set of pairs it is true of; by associating with a noun
the set of things the noun stands for, and by associating with a name the thing
named.
However, here should be first interposed a warning:
Although we suppose a domain D, we are still confined to language, unless D
itself is a language, and thus what d really does, in so far as we can write it
out, is relate the statements of terms of BL to statements and
terms that
represent the structures and
things in D.
The domain D and the language representing D are often confused, also in the
logical literature, but it should be clear that one should avoid this confusion
 or explicitly use it, when one assumes D is a language, such as BL itself, or
some other language.
Furthermore, what makes all this feasible and intuitive, is that one in fact
uses the domain D in a way that is analogous to the
truthfunction that assigns 1. Indeed, for PL there is
A12.  v(P)=1 IFF d(P)≠Ø
Note that by earlier assumptions A12 comes with its associate
v(P)=0 IFF d(P)=Ø,
and by A2 v(~P)=1 IFF d(P)=Ø.
These assumptions do explain truthfunctions to some extent:
 P is assigned
the truthvalue 1 under v precisely if
P is not assigned the void set under
d.
Note this
amounts to using '≠Ø' as an analog of 'exists'.
In fact, we have used this approach with d(.) already when formulating BL in case of
the quantifiers, though without appealing to a function
d. In fact, we can coordinate the notions of
abstraction and the interpretation (or denotation) function
d along the lines of
A12 and A13:
A13.  v(x: Z[..x..]≠Ø)=1
IFF d(Z[..x..])≠Ø
A14.  v(x: ~Z[..x..]≠Ø)=1
IFF d(~Z[..x..])≠Ø
This directly extends to the quantifiers by earlier assumptions. What it says
in English is that the value of the statement that the things that are Z is not
nothing is true precisely if the denotation of the formula about the things that
are Z is not nothing.
However, there is a difference in d(.) for propositions and abstracts:
Abstracts may be true of different things, and accordingly both
x: Z[..x..] and x: ~Z[..x..] may
represent some existing things.
For identity we have the following:
A15.  v(x=y)=1 IFF d(x)=d(y)
which has the merit of explaining identity in terms of identity: It is true
two terms are identical precisely if the denotations of the terms are identical.
One may at this point ask what is the use for a denotational semantics,
apart from perhaps adding some clarification.
The point lies precisely in the addition of the domain D that contains
whatever one's terms are supposed to be about, for once one has a domain D one may impose all manner of
properties or restrictions on D, and thereby vary the interpretation of what one
may or does mean by one's terms.
A good example  also historically correct  is that once one explicitly
supposes that basic arithmetic concerns the positive integers, one rapidly meets
the need to extend that domain of positive integers first by adding the
number 0, motivated by the properties of subtraction and expressions like
77, and then negative integers, motivated again by the properties of
subtraction and expressions like 79, and then also fractions, motivated by the
properties of division and expressions like 2:4.
Having introduced the number 0 and negative integers and fractions, one may again ask
which of one's assumptions remain true if one removes experimentally from one's
domain D all even numbers or all nonprime numbers etc.
Thus introducing a domain and a function to the things in that domain adds
rather a lot of sophistication to one's semantics and may clarify many things in
it, essentially because it forces one to write out the assumptions one makes
about the domain and the relations of one's terms to the things in the domain.
3. Probabilistic semantics
Indeed, one can further sophisticate and clarify semantics, logic and
reasoning by adding to the above denotational semantics such assumptions as
allow one to include probabilities.
This may be done as follows, using d(.) introduced above, and presupposing at
this point some clarifications about numbers and sets, namely by introducing a
relation called 'represents symbolically and numerically' abbeviated
rsn, with the following definition, that I explain below:
 rsn[L,D] IFF L is a Language & D is a set &
(Ed)(E#)
(
d : Terms of L > D*
&
# : D* >
N
&
D = d(T_{i}) U d(~T_{i}) &
d(T_{i}) = d(T_{i}&T_{j}) U d(T_{i}&~T_{j}) &
#(D) = #(D_{i}) + #(D_{i}) &
#(D_{i}) = #(D_{i}OD_{j}) + #(D_{i}OD_{j}) &
#(D_{i}) = #(D_{j}) IFF (Ef)(f : D_{i} 11 D_{j}) )
The star  as in D*  is used to indicate
the powerset of a set. A set can be taken as what is described or
represented by an abstract and is a collection of things, that may be empty if
there are no things as described by the abstract. A subset of a given set
is any set that only has elements in the given set. The powerset of a set
is the set of all its subsets.
Now in the above definition, d and
# are two functions that may be read
respectively as 'denotation of' and 'number of'. The denotation
maps the terms of a language L to the powerset of a domain D, and the number
maps the subsets of D to the real numbers N.
Next, D = d(T_{i}) U d(~T_{i})
stipulates that the domain consists of the denotation of any term together with
the denotation of its complement, which also encodes a desirable property of
denial and negation, namely that what is not so consists of anything that does
not have the property of being so.
Likewise, d(T_{i}) =
d(T_{i}&T_{j}) U d(T_{i}&~T_{j})
stipulates that any term's denotation can be analyzed as being made up of the
union of the conjunction of the denotation of the term and an arbitrary term
together with the denotation of the term and the denial of the arbitrary term.
This also encodes a desirable property of denial and negation, and conforms to
one's intuitively valid claims to the effect that e.g. the term woman stands for
the union of redhaired and nonredhaired women.
Turning to # that encodes the notion of
number, we can see that #(D) = #(D_{i}) + #(D_{i})
encodes a desirable and intuitive property of the complement of a set: The
number of any set consists of the sum of any subset in the set together with the
complement of that set. Thus, the number of all the French equals the number of
French men plus the number of all French who are not men.
Similarly, #(D_{i}) = #(D_{i}OD_{j})+#(D_{i}OD_{j})
encodes the desirable and intuitive property of sets that the number of
things a set comprises is made up of the sum of the number of the set's
intersection with an arbitrary set plus the number of the set's intersection
with the complement of the arbitrary set. Thus, the number of French women
equals the number of French adult women plus the number of French nonadult
women, and also the number of French blond women plus the number of French
nonblond women.
The last assumed property of # in the
above definition was already assumed earlier, and embodies Hume's Postulate:
Two sets have the same number precisely if there is a 11 mapping between
them. This postulate is sufficient  together with other assumptions  to derive
the usual theorems about numbers (as Frege first showed in detail).
Having explained in some formal detail what one means by 'represents
numerically and symbolically', one may add two assumptions that at the same time
encode intuitions about proportion
and enable one to surrect probability theory in Basic Logic:
A16.  p(D_{i}) = #(D_{i}) : #(D)
A17.  p(D_{i}D_{j}) = #(D_{i}OD_{j}) : #(D_{j})
The first of these lays down that the proportion of a subset equals the
number of the subset divided by the number of the set it is a subset of. The
second of these lays down that the proportion of a subset in another subset
equals the number of the intersection of both subsets divided by the number of
the other subset.
These assumptions enable the derivation of the standard Kolmogorovaxioms for
probability, that accordingly is taken as a kind of proportion and
derived from the cardinal numbers of sets (that are represented by
#).
This is a new interpretation of probability, that I call the
cardinal interpretation of probability, and that has the merit of giving
room for both objective chance and personal probability. Namely as
follows: Suppose one has a bag of beans. Then there is an objective
proportion of white beans in the bag, that may be any number between 0 and
1, depending on the actual constitution of the bag, and you and I may have 
quite  different ideas about this proportion, that can be formulated in terms
of our personal guesses about the proportion.
It also allows us to say when a personal probability is true: If it equals
the objective probability. Thus, if you guess that the probability of randomly
drawing a white bean from the bag is 1/2 and my guess is 1/10 and in fact the
number of white beans is 500 and the number of beans 1000, your guess is true
and mine is not. And if in fact the actual number of white beans is 550, your
guess, although not quite true, is more true than mine.
Note also that often we may not know whose personal probability is (more)
true, but that we can also often quote a lot of evidence  such as
random samples from the bag of beans  that allow us to conclude with high probability
what the proportion of the sampled characteristic really is. (See:
Rules of
Probabilistic Reasoning.)
4. Modal semantics
Modalities of statements are attributes of statements like 'is
necessary', 'is possible' and 'is contingent'. Modalities  and there are quite
a few more, and quite a few terms that are like modalities  can be treated and
incorporated into BL in several ways.
The simplest is as follows and merely uses
=
and
Ø
as above, while refining these a little:
A18.  v( Nec P)=1 IFF d(~P)=Ø
A19.  v( Pos P)=1 IFF d(P)≠Ø
A20.  v( Con P)=1 IFF
v( Pos P)=1 &
v( Pos ~P)=1
As stated, this considers modal terms only when used to qualify statements,
but one can also consider modal terms as qualifying the predicates and
relations in statements, and e.g. consider the  supposedly  necessary
features a mammal must have to be a mammal.
The analysis of the modalities 'necessary', 'possible' and 'contingent' that
was just proposed suffices for many purposes, and can be considered as merely
truthfunctional.
There is a related and more sophisticated analysis that is due to Saul Kripke
and that takes in consideration the domains, and indeed the possibility
that one's terms may range over several or many domains.
When this is done, the several domains that are considered often are called 'worlds'
and one may easily find phrases in these treatments to the effect that a
statement is necessary iff it is true in 'all possible worlds'.
This also allows for the possibility that some of these  socalled  worlds
are related in ways to some of these worlds that others of these worlds are not.
Thus, a world in which Gengis Khan has been born and is a happy healthy toddler
holds different possibilities from the same world in which he was aborted, and a
world in which Gengis Khan has wrought havoc as an adult must be in some sense 'accessible',
as the term is, from a world in which he was a healthy toddler, whereas a world
in which Gengis Khan was aborted should not give access to a world in which he
later lived and wrought havoc on his contemporaries.
This usage of 'possible world' etc. has some justification, but is
also sometimes misleading, in that the possibilities one considers are often not
so much different possible worlds as different possible situations,
actions, developments or expectations concerning the world
one lives in, that may run different courses, i.a. dependent on one's own
ommissions and commissions in it.
But this usage, that involves relating modal terms to different possible
domains, situations or worlds one's terms may represent and may be true of, also
has advantages, and explains some relativity of modal terms (namely: to domains)
that cannot be brought out if one considers merely one domain, while it also
enables one to consider how different domains may or may not relate to other
domains, rather like a possible world with Sherlock Holmes involves some
possible world with the parents of Sherlock Holmes, whereas a world without
Sherlock Holmes also can do without his parents.
In any case, the analysis of modalities that also involves domains is along
the following lines, where the denotation function d is made into one that
depends not only on a statement but also on a world, domain or situation the
statement is about:
A21.  v( Nec P)=1 IFF
~(ED_{i})( d(D_{i},P)=Ø
)
A22.  v( Pos P)=1 IFF (ED_{i})( d(D_{i},P)≠Ø
)
A23.  v( Con P)=1 IFF
v( Pos P)=1 &
v( Pos ~P)=1
Thus, in this approach a statement P is necessary precisely if it is
true in all possible domains (worlds, situations); a statement P is
possible precisely if it is true in some possible domains (worlds,
situations); and a statement P is contingent precisely if it is true in
some situations and false in others.
Another approach to modality, or a supplementary approach, is to use the
probabilistic semantics introduced above, and stipulate that a statement is
necessary precisely if its probability equals 1 and possible
precisely if its probability is not 0; and contingent precisely if its
probability is neither 1 nor 0.
See further:
Basic
Logic  extended semantics
