Foundations for
Propositional Attitudes:
<Note: This is in effect an intermediate research report.
>
Sections
1. Introductory assumptions
2. General and schematic ideas and issues
A. Sets
of statements as sets of beliefs
B. Four basic possibilities for attitudes
C. Conditional logical inference
D. Typical conditional logical inference
E.
The problem of proper epistemic relativization
F.
The problem of the basic propositional attitudes and their
rules
3. General assumptions about speakers and languages
4. General assumptions about speakers and meanings
Propositional attitudes are intentional
relations between a person or personalized entity and a proposition. Two basic
propositional attitudes are believing and desiring.
What follows is an outline of a general theory of propositional attitudes,
called LPA (for: Logic of Propositional Attitudes), that is surrected using
Set Theory (ST), which it presumes, as it
presumes Propositonal Logic
(PL) and FirstOrder Predicate Logic (FOPL).
1. Introductory assumptions
The way this is presented here is to presume at the present point
PL, FOPL and ST, all of which the reader may assume to be standard and
classical. Some familiarity with appropriate Axioms, Theorems, Rules of
Inference and adequate Notations are also presumed here. Given these, the
necessary assumptions for a general LPA are stated in both English and ST
(that comprises PL and FOPL).
To start with, let it be noted that PL, FOPL and ST are supposed to be
standard, and that the reader is supposed to be familiar with them, and to
have some favourite schemes of notations for them, and a set of favourite
axioms and rules of inference for them.
In the present note all of this is presupposed and taken for granted, but
it should be noted what kinds of terms are supposed:
 Terms for propositions: Normally p,q,r possibly with suffixes
 Terms for individual things: Normally a,b,c
possibly with suffixes
 Terms for sets: Normally A,B,C
possibly with suffixes
 Terms for persons: Normally α, β, γ
possibly with suffixes
 Terms for propositional attitude: B and others to be introduced
Here are basic assumptions, which are surrected about a language L and a set
of speakers of L, both of which are for the moment presumed:
 A1. Every person has a set of beliefs.
 A2. A proposition is a pair made up of a
statement and an idea represented by the statement.
 A3. A belief is a proposition that is supposed to be true.
 A4. A proposition is true iff the idea it expresses represents something real.
 A5. Two sets represent each other if there is a correlation of the sets
and their individuals that preserves relations.
Formally, this can be written as follows:
 A1. (aePerson)(EX)(X=Ba)
 A2. qePROP IFF (E"q")(E'q')(q=("q",'q')
& "q" is a statement and 'q' an idea & "q" means 'q')
 A3. qePROP  qeBa IFF aBq
 A4. qePROP  q IFF 'q' represents something
real
 A5. R(A,B) IFF (Ef)(f : A > B & (x)(H)(Hx e A IFF f(H)f(x) e B)
Here the following clarifications should be added:
The set of beliefs Ba of person a is a set of propositions, that is: It is
both supposed to be a set; it is personal, in that different persons may have
different sets of beliefs; and it consists of propositions.
Propositions are defined as pairs, namely of a statement in some language
and the idea the statement means or expresses, regardless of its truth.
Beliefs are propositions that someone supposes to be true (for whatever
reasons, good or bad), and thus a statement of propositional attitude, like "a
believes q" is taken to be true precisely if the proposition q is an element
of a's beliefs. Thus the idea "aBq" expresses is that "q" is one of the
beliefs of a. (Note "aBq" is true or not regardless of the issue whether "q"
is true or not: It depends on a but not on "q".)
A proposition q is said to be true precisely if the idea the proposition
expresses represents something real (i.e. in set theory: something other than
the Empty Set). And a set A represents  written R formally  a set B
precisely if there is some map from A to B that preserves properties and
relations. This means that one can infer properties and relations of B from
properties and relations of A. A useful way that corresponds to the normal
course is to presuppose a given f and use R(f,A,B) = df
f : A > B & (x)(H)(Hx e A IFF f(H)f(x) e B): f
represents B with A.
Now we can be a little more precise about beliefs by surrecting some very
general assumptions about them:
 A6. p is an element of the set of a's beliefs iff ~p is not an element of a's
beliefs
 A7. aBp Sentails aBq iff p entails q and S
is true of a, p and q
 A8. If x equals y is in the set of a's beliefs and x is in the set of a's beliefs
and T is true of a, x and y then y equals x is in the set of a's beliefs and y is in the set of a's beliefs
Formally this may be written as
 A6. p e Ba  ~( ~p e Ba)
 A7. p  q & S(a,p,q) IFF aBp _{S}
aBq
 A8. x e Ba & y e Ba & T(a,x,y) IFF _{T}
aBy=x e Ba
Here A6 asserts a version of consistency of beliefs: If a believes q then a
does not believe notq, in effect.
A7 formulates a kind of logical implication relativized to a statement S
about the person a and the propositions p and q, namely that a believes that p
Sentails that a believes that q precisely if p entails q and the statement S is
true of a, p and q. This gives the schematic form of many possible conditions
to qualify what a believes and its implications. Note that {p,q :
aBp _{S} aBq} is a subset of
{p,q : p  q}.
A8 formulates a condition on identities that amounts to the thesis that a
person believes an identity only if the person knows all the terms involved in
the statement of identity. This may be weakened along the lines of P2, by
adding some condition.
Then there are the following basic assumptions involving beliefs:
 A9. q belongs to a's beliefs iff q is
element of a's beliefs or ~q is element of a's beliefs or qV~q is element of
a's beliefs
 A10. Either a believes q and a does not believe ~q and
q belongs to a's
beliefs or a believes ~q and a does not
believe q and q belongs to a's beliefs or not a believes q and not a does
not believe ~q and q belongs to a's beliefs or not a believes q and not a does
not believe ~q and q does not belong to a's beliefs
Formally
 A9. q e Ba IFF (q) e Ba V (~q) e Ba V
(qV~q) e Ba
 A10. (q e Ba) & ~(~q e Ba) & q e Ba V
(~q e Ba) & ~(q e Ba) & q e Ba V
~(q e Ba) & ~(~q e Ba) & q e Ba V
~(q e Ba) & ~(~q e Ba) & ~q e Ba
The first of these statements defines a useful sense of a person's a being
acquainted with a proposition q: If a believes something that involves it. It
should be noted that if we presuppose classical logic all the way this
simplifies to: q e Ba IFF (qV~q) e Ba. This is
explained in the following remark.
The second statement outlines the four basic possibilities for any person
a's beliefs and any proposition p: Either the person is acquainted with it,
and believes the proposition to be true, or false or neither true nor false,
or else the person is not acquainted with it, and also therefore does not
believe the proposition is either true or false.
Of these four possibilities, the third and fourth are special and somewhat
subtle. One way of agreeing with the third possibility,
~(q e Ba) & ~(~q e Ba) & q e Ba is of holding
that the proposition is merely probable, and 0<pr(q)<1. But so far no
room has been made for probabilities. In any case, disjunction is problematic
because this third possibility is tenable and possible and logically distinct
from the fourth and the other two.
Next for quantifiers and beliefs:
 A11. a believes there is an x that is F iff there is an x such that x belongs
to a's beliefs and a believes x is F
 A12. Any term z belongs to a's beliefs iff a believes there is an y that is z or there is no y that is z
Formally
 A11. aB(Ex)Fx IFF (Ex)(xeBa & aBFx)
 A12. zeTERM & zeBa IFF aB(Ey)(y=z) V ~(Ey)(y=z)
The point of the first assumption is to avoid problems of inference, and it
is much like qeBa IFF qeProp & aBq, for both
reduce the logic that is involved to that of being an element of a's beliefs:
person a believes that something x is F precisely if a believes something
about x and a believes that x is F. Thus, a given person a may and does
believe that there is a triunite divinity precisely if there is a term in a's
beliefs about which he believes that it is a triunite divinity. As before,
whether there is also such a thing in reality is another question entirely.
Indeed, A11 allows also to express the relevant difference: It is that between
(Ex)(xeBa & aBFx & xeRW) and (Ex)(xeBa & aBFx & ~(xeRW)). In the first case
there also is something in the real world conforming to a's belief, but this
is not so in the second case.
The second assumption does something similar for terms as p did for
propositions: A term is an element of the beliefs of a person a precisely if a
believes either that there is something identical to it or else a believer
there is nothing identical to it.
2. General and schematic ideas and issues
Now let us take stock for a moment, and articulate a general and
schematic idea that underlies what so far was assumed, and what follows.
The general idea is that implicit in A1A5 and A9 and A11: That we can deal with the
beliefs of a person a as with a set of propositions and so on paper (and the
computer screen) as a set of statements.
 A. Sets of statements as sets of beliefs
Thus with every person we associate a set of beliefs, and
a set of statements that express these beliefs, that is rendered as Ba and
read as "the beliefs of a". It is the presence or absence of q in Ba that
determines respectively the truth or falsity of aBq.
Therefore (and therewith) it may also be the case that, as A10 is concerned
with, that ~(q e Ba) & ~(~q e Ba) & q e Ba V
~(q e Ba) & ~(~q e Ba) & ~q e Ba. And this means in particular that
any of the following statements in the disjunction may be true:
aB(qV~q) & aB(q) V
aB(qV~q) & aB(~q) V
aB(qV~q) & ~aB(q) & ~aB(~q) V
~aB(qV~q) & ~aB(q) & ~aB(~q)
or more briefly
(q)eBa
(~q)eBa
qeBa
~(qeBa)
for we may define
aB(qV~q) =df qeBa
aB(q) =df (q)eBa
aB(~q) =df (~q)eBa
Therefore what we have is in effect
 B. Four basic possibilities for
attitudes
(q)eBa IFF aB(qV~q) & aB(q) & ~aB(~q)
(~q)eBa IFF aB(qV~q) & aB(~q) & ~aB(q)
qeBa IFF aB(qV~q) & ~aB(q) & ~aB(~q)
~(qeBa) IFF ~aB(qV~q) & ~aB(q) & ~aB(~q)
Note that (A) effectively allows us to deal with the beliefs of another
person and with claims about the other person's beliefs in terms of the set of
statements that is supposed to represent the beliefs that the person has. This
is an important simplification in principle, as it allows us to decide in
principle the truth of any statement of the form "aBq", for all we need to
know is whether or not "q" e Ba.
Now one obvious problem is what for any given person a does follow from the
beliefs a has, according to a. Here we have to make some assumptions, since
different persons may have different beliefs about what implies what, and
indeed also may change their mind about relations of inference, but we also
need assumptions sufficiently extensive to allow us to treat different persons
different ideas about reasoning, and to allow us to infer at least
conditionally something about their beliefs.
Here the basics are given in A6A8:
 C. Conditional logical inference
All reasoners are minimally consistent and all reasoners use some condition
to infer propositions from propositions and identities from terms, but in
any case the main condition for the set of inferences allowed to a person is
that whatever the conditions used by the person, the inferences are
logically valid, in the sense that if aBq _{S} aBr then at least q
 r.
Note there are very many possibilities here in principle, and that person a
may be mistaken in attributing certain conditions for inference to person b.
General forms that A7 and A8 may take are:
 D. Typical conditional logical
inference
p  q & p e Ba & q e Ba IFF aBp _{S}
aBq
i.e. S(a, p, q) iff p en Ba & q e Ba
x e Ba & y e Ba & x=y IFF _{T} aBy=x e Ba
i.e. T(a,x,y) iff x=y
These assumptions may be precisified by precisifying the meaning of what is
in a person's a consciousness at a given time, but keeping this in mind we may
say that typically a person is taken to be conscious of all logically valid
consequences that follow from what is in his consciousness.
In reality, persons may not live up to this ideal (and normally won't) but
then one has the point that they should, in as much as the consequences
are logically valid and the terms that lead to them are supposed to be
believed by the person.
Finally, there are two general problem that may be formulated as follows.
First, there is this:
 E. The problem of proper
epistemic relativization
When providing a logic of propositional attitudes,
we need to articulate both what is or should be involved for any user of any
language in which the attitudes occur, what thus is believed by each user of
the language to hold for all users of the language, and also what is or
should be up to the personal choice or idiosyncracy of any user of the
language.
What we need to have to have anything worthy of the name "logic" is a
system that allows us to reason about any reasoner using a given language,
while leaving some room for more personal uses and beliefs about what follows from
what.
Second, there is this:
 F. The problem of the basic
propositional attitudes and their rules
We need to lay down which propositional attitudes are basic, and what
are the basic rules of inference that are used for them.
First note that part of the problems involved here have been solved in
principle, by our assumptions about logical inference: It must be logically
valid in any case, but may be restricted by some condition that depends on the
person or the terms used.
But this leaves the problems what are the basic propositional attitudes
unsettled, and likewise it doesn't settle problems about how these
interrelate, and what to do with iterations of attitudes, like aBbBq and
aBaBq.
Here is a convenient approach to these problems: Make initially all
attitudes beliefs or contained in beliefs, and assume about the given formulas
that
aBbBq  q e Ba
aBaBq  q e Ba
that is: If a believes that b believes that q then at least a is acquainted
with q, but if a believes that a believes that q then (also) a believes q.
Note that the first formula has the useful implication that a person needs to
be acquainted with the ideas and statements that he attributes to others.
3. General assumptions about speakers and languages
One way to deal with the above problem E of proper epistemic relativization
is to frame a set of postulates for a logic of propositional attitudes in
terms of general assumptions of what speakers of a language must know and
agree on to speak the language.
Since clearly knowing is a verb of propositional attitude, we need either
to assume or define it, and shall define it as simply as we can:
(a knows q) IFF (a believes q) & q
One is said to know what one believes truly, regardless from whether one
knows one knows or knows the reasons for one's beliefs. Presupposing this
definition, we may then frame the following assumptions
For every (suitably qualified) language L
and every
(suitably qualified)
speaker a of L  or formally
(LeLan)(aeSp(L)):
1.
speaker a knows that every speaker b of L knows that there
is a speaker of L:
aK (beSp(L)) (bK (Es)seSp(L)),
2.
speaker a knows
of every speaker b of L that b knows that for every speaker
s of L there is a set of beliefs of the speaker that is
said to belong to the speaker:
aK (beSp(L)) (bK (seSp(L))(EX)(Xes & X=Bs).
3.
speaker a knows of every speaker b of L that b knows that
for every functional term f f maps a class A to a class B iff for every
a in A there is one y in B such that the f of x is y:
aK (beSp(L)) (bK (f)(f:A=>B) iff (xeA)(E1yeB)(f(x)=y)).
4.
speaker a of L knows of every speaker b of L that b knows
that the ideas of L is the set of ideas that speakers of the language
have:
aK (beSp(L)) (bK I(L)={q:(EseSp(L))(
qeBs)}).
5.
speaker a of L knows of every speaker b of L that b knows
that meaning is a function from L to the ideas of L and denotation
a function from the ideas of L to reality R:
aK (beSp(L)) (bK m:L=>I(L) & d:I(L)=>R).
6.
speaker a of L knows of every speaker b of L and every
statement q in L that b knows that there is something that is the reality R
and q is true iff the denotation of the meaning of q is in R:
aK (beSp(L)) (qeL)(bK (EX)(X=R) & (True(q) iff d(m(q))eR).
7.
speaker a of L knows of every speaker b of L and every
statement q in L that b knows that q iff q is true and the meaning of
(b believes q) belongs to b's ideas and the meaning of q belongs to b's ideas:
aK
(beSp(L)) (qeL)(bKq iff True(q) & m(bBq)e[b] & m(q)e[b]).
8.
speaker a of L knows of every speaker b of L and every
statement q in L that b knows that if q is a theorem of L then a knows
that b knows q if a knows that the meaning of q is in [b] and also b knows
that a knows q if b knows that the meaning of q is in [a]:
aK (beSp(L)) (qeL)(bK {(=q)eL>aK(m(q)e[b]>bKq)
& bK(m(q)e[a] > aKq)}).
9.
speaker a of L knows of every speaker b of L and every
statement or term t in L that b knows that the meaning of t is t1 iff b
experiences that something is t iff b experiences that something is t1:
aK (beSp(L)) (teL)(bK(m(t)=t1) iff (bX(Ey)(y=t) iff bX(Ez)(z=t1)).
10.
speaker a of L knows of every speaker b
of L and every statement or term t in L that b knows that t2 is the denotation
of t1 iff whenever b experiences something is t2
then b experiences something is t1:
aK (beSp(L)) (teL)(bK(d(t1)=t2) iff bX(Ey)(y=t2)>bX(Ez)(z=t1)).
11.
speaker a of L knows of every speaker b of L and every
statement q in L that b understands q iff there are antecedents and
consequents of q that b knows (that differ from q, though this is not explicit
here):
aK (beSp(L)) (qeL)(bUq iff (Et)(Es)(bK(t>q) & bK(q>s)).
12.
speaker a of L knows of every speaker b of L and every
statement q in L that if b asserts q and every statement in q is understood by
a then a understands q:
aK (beSp(L)) (qeL)(bAq & (teq)(aUt) > aUq).
Now one of the useful things of aKq is that if this is true we know that q
is true. Hence the following statements may be presumed to be implied by the
above assumptions
For every speaker b of L b knows that
(Es)seSp(L)
(EX)(Xeb & X=Bb & beR)
(f)(f:A=>B) iff (xeA)(E1yeB)(f(x)=y)
I(L)={q:(EseSp(L))(qeBs)}
m:L=>I(L) & d:I(L)=>R)
True(q) iff d(m(q))eR
bKq iff True(q) & m(bBq)eBb & m(q)eBb
{(=q)eL>(aeSp(L))(beSp(L)(aK(m(q)eBb>bKq) & bK(m(q)eBa > aKq)}
bK(m(t)=t1) iff (bX(Ey)(y=t) iff bX(Ez)(z=t1))
bK(d(t1)=t2) iff (bX(Ey)(y=t2)>bX(Ez)(z=t1))
bUq iff (Et)(Es)(bK(t>q) & bK(q>s))
(beSp(L)) (qeL)(bAq & (teq)(aUt) > aUq).
or in words:
For every speaker b of L b knows that
there is at least one speaker of L
b is a part of reality and has a set of beliefs Bb
functional terms relate the terms in their domain to
unique terms in their range
the ideas of a language are the ideas of the speakers
that are beliefs
meaning is a function from language to ideas and
denotation is a function from ideas
to reality
a statement is true precisely if the denotation of its
meaning is real
b knows q if q is true and b believes q
q is a theorem if any speaker who knows the terms of q
also knows that q
b knows that the meaning of t is t1 iff
b experiences something
is t iff b experieces something is t1
b knows that the denotation of t1 is t2 iff
b experiences something
is t2 only if b experiences something is t1
b understand q iff b knows some antecedents and
consequents of q
b understands q iff b understands all terms of q.
Note these are all consequences of the assumptions that were chosen, and that
these assumptions both involve and express beliefs about language, reality,
beliefs, functions, ideas, meanings, denotations, truth, knowledge, theorems,
knowing facts, and understanding terms and propositions.
The assumptions that were made may be not wholly correct (for all speakers
and all languages) or may be incomplete, but it seems that something like them
is involved in both the acquisition and the use of language, and that we
cannot do without their articulation if we try to give a foundation for a
logic of propositional attitudes.
4. General assumptions about speakers and meanings
Next about expressions, ideas, meanings, denotations and speakers:
For every (suitably qualified) language L and
every
(suitably qualified)
speaker a of L:
(LeLan)(aeSp(L)):
1. There are
expressions "x", ideas 'x', things x and speakers a such that the speaker a
knows that the meaning of "x" is 'x' and the denotation of 'x' is x:
(E"x")(E'x')(Ex)(Ea)
(aK[m("x")='x' & d('x')=x)])
2. There are expressions "x", ideas 'x', things x
and speakers a such that the speaker a knows that the meaning of "x" is 'x'
and the denotation of 'x' is x:
(E1"x")(E1'x')(E1x)(Ea)
(aK[M("x",'x') & D('x',x)])
3. There are expressions "x", ideas 'x', things x, and
some speakers and some world W, and some language L such that a believes "x"
means 'x' and 'x' denotes x and in fact a is a speaker of L and "x" means 'x'
in L and 'x' denotes x in L and a is in W and "x" is in W and 'x' is in a and
x is in W and L is in W and "x" is in:
(E1"x")(E1'x')(E1x)(Ea)(EW)(EL)
(aB[m("x",'x') & d('x',x)] & aeSp(L) &
"m("x",'x')"eL &
"d('x',x)"eL & aeW &
"x"eW & 'x'ea & xeW & LeW & "x"eL)
4. The expression "a believes p" is true iff there is an
expression "p", an idea 'p' and a person b such that "p" means 'p' and "a"
means b and 'p' is a belief of b
"aBp"eTrue iff (E"p")(E'p')(Eb)
(m("p")='p' & 'p'eB(b) & m("a")=b)
iff m("p")eBa
5. For any expression "x", idea 'x', thing x, and speaker
a (of L), a knows that the meaning of "x" is 'x' and the denotation of 'x' is
x iff
(a) for every speaker b of L there is one (and the same) idea 'y' such that b
experiences "x" belongs b's language iff b experiences 'y' belongs to b's
ideas while in fact 'y'='x' and
(b) for every speaker b of L and everything z there is one (and the same) idea
'v' such that if x=z and b experiences that z belongs to b's world then b
experiences 'v' belongs to b's ideas while in fact 'v'='x'
("x")('x')(x)(a)(aK[m("x")='x' & d('x')=x)]) iff
(a) (b)(E1'y') (bE"x"eL(b) iff bE'y'eI(b) & 'y'='x) &
(b) (b)(z)(E1'v') (x=z & bEzeW(b) > bE'v'eI(b) & 'v'='x' )
6. In the case of propositions, two different speakers
have the same idea of 'x' iff the speakers agree on both all the consequences
and all the presumptions of 'x', and
(6.1) (E1'x')('x'eI(a) & 'x'eI(b) &~(a=b)) iff ('y')('yx'eI(a) iff
'yx'eI(b)) & ('y')('yx'eI(a) iff
'yx'eI(b)) & (E'y')('yx')
& (E'y')('xy')
(6.2) ('F')('Fx'eI(a) iff 'Fx'eI(b)) & (E'F')('Fx'eI(a) & 'Fx'eI(b)).
7. in the case of terms, two different speakers have the
same idea 'x' iff there is a context 'F' both attribute to 'x' and besides
both attribute all the same contexts to 'x'.
L(a) inc L(b) iff
(x)(y)(xeL(a) & m(a,x)=y > xeL(b) & m(b,x)=y)
8. a's language is included in b's language if all terms
in a's language with a meaning are terms in b's language with the same
meaning.
(t)(L)("t"eL > "t"eW)
9. All terms of any language are elements of the real
world.
W(a) inc W(b) iff (x)(xeW(a) > xeW(b))
W(a) inc W iff (x)(xeW(a) > d(x)eW)
10. Personal worlds are setlike, and are realistic if each
and every idea in a personal world denotes a real thing.
(a)(x)(xeI(a) > xea)
(a)(x)(xeW(a) & aB(Real(x)) > aB(d(x)eW)
11. Ideas are elements of speakers, and speakers who
believe that what they believe is real, believe that the denotation of their
real beliefs are in the real world.
(1) All persons have overlapping personal worlds and ideas, even if they speak different languages.
(a)(b)(~W(a).W(b)=0 & ~I(a).I(b)=0)
(2) All persons have some ideas they believe are really true.
(a)(aB[~W(a).W=0])
(3) Every element of the real world has some consequence in every speaker's world.
(a)(x)(xeW > (Ey)(xy & yeW(a))
(4) If the worlds of a and b are the same, then both believe they believe the same and both believe what each believes is true.
(a)(b)(W(a)=W(b) > aB(bBq) iff bB(aBq) & aB(bBq) iff qeW)
(5) Speakers believe a proposition is true iff speakers believe it is part of the speaker's world and the real world.
aB(True(q)) iff aB(qeW(a) & qeW))
(6) Speakers believe a proposition is part of the real world only if they believe that for every speaker there is some consequence of the proposition which that speaker believes.
aB(qeW) > aB[(b)(Es)(qs & bBs)]
(7) Speakers believe a proposition q is part of the real world only if they believe that for every speaker b there is some consequence s and some condition t such that the condition t entails the proposition s (together with q that's supposed to be true) and b experiences s if condition t is true.
aB(qeW) >
aB[(b)(Es)(Et)(t&qs & tbEs)]
