Propositional attitude: Intentional
relation between a person or personalized entity and a proposition. Two
basic propositional attitudes are believing and desiring.
1. The ubiquity of
propositional attitudes
2. Some fundamental
logical problems with propositional attitudes
3. Grammar for a
logic of propositional attitudes
4. Introduction
to LPA
5. Logical foundations for LPA
6. On iterated attitudes and attitudes in
general
1.
The ubiquity of propositional attitudes
When you think about
it, you notice that human beings think about things, including
themselves and other people, in terms of statements that are made up of
names for things or persons, names for attitudes like thinking,
noticing, believing, desiring etc., and statements (that may but
need not be propositional attitudes).
Thus, one says about
oneself that one desires it rains, others say about one that they
believe that one desires it rains, and so on. All such talk seems
derived from human talk about the capacities of human beings, but often
is also attributed, truly or falsely, to nonhuman beings, as in "the
dog believes his master commanded him to sit" and "this taperecorder
said that this taperecorder can speak English" and "that parrot screams
in Spanish that you are the son of something or other but I believe
neither it nor I understands fully what it says".
It is not difficult
to give an informal sketch of the statements we shall refer to as propositional
attitudes:
In English they are
normally made up of an expression that is a name for some thing, like
"Adam", or for some things, like "the inhabitants of London", followed
by an expression that names an attitude, like "asserts", "believes",
"tries to cause", "desires", "experiences", "imagines", "remembers"
etc. followed by an expression that is a statement, possibly
itself a propositional attitude, as in "Adam believes that Eve desires
that Adam laughs".
2.
Some fundamental logical problems with propositional attitudes
It is an interesting
fact that so far there is no adequate formal logic for propositional
attitudes. The main reasons for this fact are not difficult to indicate.
First, there is the
problem of intentionality that arises for propositional attitudes
in the following two simple related forms:
Even for simple
tautologies, like "he believes p or ~p, whatever proposition p is",
that seem valid because what is attributed to you is a logical
tautology, and thus as valid as "p or ~p", there is the problem that you may never have thought about many propositions. Thus,
most people would reject "he believes the pope loves duckbilled
platypuses or the pope does not love duckbilled platypuses", on the
ground that the pope may never have
believed anything about the pope's love for platypuses either way.
Similarly, true
identities, like "Scott = the author of 'Waverley'", even if they are
necessary, like "tan x = sin x/cos x", may be unknown to many people,
either because they do not know all terms occuring in the identities,
or because while they do know all the terms occurring in the
identities, they don't know the stated identities are true, as in "The
King knows Scott" and "The King knows 'Waverley'", but "The King does
not know Scott = the author of 'Waverley'".
Second, there is the
problem of distribution over "or". This arises as follows:
It seems intuitively
valid that "you believe p and q IFF you believe p and you believe q",
which is to say that the expression "you believe" distributes over "p
and q". This would give us a neat and simple entry to propositional
attitudes analogous to propositional logic, if it would likewise
hold for other binary logical connectives.
Now it might seem at
first as if it also is intuitively valid that "you believe p or q IFF
you believe p or you believe q". But when one thinks about it, this is
intuitely invalid in both directions, for two different
reasons.
If "you believe p or
~p" is true, e.g. because you know about p
and are inclined to believe logically valid statements, it still does
not folllow that, then, "you believe p or you believe ~p", for you may
well deny both, on the ground that you simply do not know whether to
believe p or to believe ~p. Hence, distributing the attitude "believes"
over a disjunction is not intuitively valid.
Conversely, if it is
true that "you believe p" it follows by standard logic that it is true
"you believe p or you believe q". But it still does not follow that
"you believe p or q" if in fact you have never heard of or thought
about q.
Third, there is the
problem of negation, that arises as follows:
There is an ambiguity
in "it is not true that you believe p", for this seems true in two
quite different cases: if in fact "it is true that you believe it is
not p" or if in fact "it is true you never thought about p at all".
These three problems
exist apart from the problem of
quantification, that goes beyond propositional logic, which is
that e.g. "there is an x such that he believes x is God" and "he
believes that there is an x such that x is God" differ in
truthconditions, since the first seems to affirm there is such an x
whatever he believes, and the second doesn't, since it merely affirms
he believes there is such an x (and he may well be mistaken).
This is called the
problem of
quantification, because in standard quantificational logic, inferences
of the form ( Fa  (Ex)Fx ) (when suitably conditionalized) are valid
 but not necessarily so, as the previous paragraph illustrated, if
(Fa) is in fact a statement of a propositional attitude.
3. Grammar for a logic of
propositional attitudes
Now it is easy to
state a formal grammar for a logic of propositional attitudes I call
briefly LPA:

Grammar of LPA


If p is a formula of PL, then
p is a formula of LPA.
If a is a name for a person or thing, A is an attiude, and p is a
formula, then aAp is a formula of LPA.

This merely supplies
syntactical structures, like aAp, aA(pV~p), aAbBp, aA(aB(pVq)) etcetera
(where I presume a certain liberality in the use of brackets).
Here is a list of
some basic attitudes I shall use:

Basic Attitudes


A  Asserts
B  Believes
C  Causes (tries to cause)
D  Desires
E  Experiences
F  Feels
I  Imagines
R  Remembers
S  Senses
(as in: Sees, Hears, Smells, Tastes, Touches ..)

This gives a simple
yet rather rich and intricate formulas, some of the simplest examples
are "John asserts the car is stuck", "Peter believes the car is stuck",
"John tries to cause the car is not stuck", "Peter desires the car is
not stuck" and so on, briefly jAs, pBs, jC~s, pD~s and so on, as in
pI~s, jR~s, pB(jD~s).
4. Introduction to LPA
The basic motivation
of LPA is to set up a logical system to reason with statements of
propositional attitude. By "attitudes" I mean verbs like "asserts",
"believes", "desires", "knows" and many more that are used in English
to relate a person to a proposition or to the
idea the proposition states.
To have a logical
system that is adequate to translate English statements of propositions
is a pressing demand for philosophy, psychology
and linguistics, not to speak of logic, because very many of the
statements people assert and defend are statements involving
propositional attitudes, for it is difficult to speak of persons
without speaking of their beliefs, desires, assertions, hopes, feelings,
fears, actions, fantasies,
illusions, wishful thinking and so on.
Sofar there are no
logical systems for propositional attitudes that even begin to be
minimally adequate. Part of the reason are wellknown difficulties with
quantifiers and substitution, illustrated by the following two
problems, of which the first goes back to antiquity:
(1) Suppose the man
standing before Antigone is her brother Orestes, but Antigone doesn't
recognize him and thinks he is a stranger. Clearly then, Antigone will
believe that the man standing before her is a stranger. But we agreed
Orestes = the man standing before her, and it is widely assumed
identities are characterised by the property that either term in an
identity can be substituted for the other in any context. However, in
the case of propositional attitudes doing so immediately yields the
obvious falsity that Antigone believes Orestes is a stranger.
(2) Suppose the
Pope believes there is one God, and He is a Trinitarian mystery. Does
it follow that, if so, there is one God and the Pope believes He is a
Trinitarian mystery? Clearly not, especially for nonCatholics, and in
more general terms it doesn't follow from someone's belief that there
is a thing of a certain kind that indeed there is a thing of a certain
kind, and so there is a problem about quantification and propositional
attitudes, for it would seem that by ordinary rules of quantification
the move from "The Pope believes God=God" to "There is an x such that
the Pope believes x=God" would be validated, which would make theology
much easier than it is and ought to be  and it should be noted,
incidentally, that most people accept the inference from "you believe
1=1" to "There is an number 1 such that you believe 1=1".
These are serious
problems, but they presuppose identity and quantification and also an
underlying system of propositional logic enriched with terms for
attitudes and persons.
Now it is easy to
show that even on a merely propositional level, apart from identity and
quantification, there are fundamental logic problems with propositional
attitudes.
This can be shown
after explaining a few notational conventions we shall continue to use,
and which are very helpful, simple and intuitive.
Basic
Grammar of LPA

Names of persons etc: a,b,c etc.
Names of attitudes:
A = Asserts
B = Believes
C = tries to Cause
D = Desires
E = Experiences
I = Imagines
R = Remembers
X = an arbitrary attitude
Names of propositions: p,q,r etc.

whence for example:
"aAq" = "a asserts q"
"bB~q" = "b believes not q"
"cDdCaBcBq" = "c desires that d tries to cause that a believes
that c believes that q"
a.s.o.

In terms of these conventions, it seems intuitive to assume that the
following is true
(1) aB(p&q) iff aBq & aBq
where we assume for
the moment that (1) is an extension not of EPL but CPL. Taking this for
granted (there may be some problems when the times of these beliefs
differ, but these we shall disregard, and maintain that by and large
(1) encodes a principle that must hold) it may seem at first blush to
also assume that the following is true, still presuming we are writing
a kind of classical
propositional logic to which terms for
attitudes have been added:
(2) aB(pVq) iff (aBp V aBq)
But it is easy to see
that intuitively both implications may be false:
First, consider
aB(pV~p) e.g. "you believe that (it rains in Reykjavik or it doesn't
rain in Reykjavik)". Now most speakers of English will insist that they
are quite capable of believing this while it may be true that they
don't believe it rains in Reykjavik and while it is also true that they
don't believe it doesn't rain in Reykjavik, for the simple reason that
they are not in Reykjavik and don't know the weather there, although
even so they are quite willing to affirm that it either rains or
doesn't rain in Reykjavik. So (2) fails to be a valid implication from
the left to the right.
Second, consider e.g.
"aBp" for "you believe Rome is in Italy". Now by standard logic  the
rule for disjunction introduction  if it is true you believe Rome is
in Italy, then it is true that you believe Rome is in Italy or you
believe the Pope raped your grandmother (since in standard logic if p
then pVq, and by parity of reasoning if aBp then aBp V aBq). So if (2)
holds it also follows that therefore aB(pVq). But nearly all speakers
of English will resist that it follows from "you believe (Rome is in
Italy)" that "you believe (Rome is in Italy or the Pope raped your
grandmother)". And one simple reason they may give is that they never
even contemplated any sexual relations between the Pope and their
grandmothers. So (2) fails to be a valid implication from the right to
the left.
This example shows
that already on a propositional level, regardless of difficulties with
quantifiers and predicate logic, classical propositional logic cannot
be selfevidently or at all be extended to account properly for
attitudes, where "properly" means at least "preserves many intuitions
about valid implications and doesn't contradict basic intuitions about
valid implications".
There are various ways to provide
logical or algebraic foundations of EPL and LPA, e.g. in my M.A.
thesis. This is an alternative one, that is in quite a few respects
simpler:
We assume a standard bivalent
propositional logic PL, and extend it grammatically to LPA by adding
the rule
If p e PL, then p e LPA
If p e LPA, then aBp e LPA
This gives us propositions of form
~aB~p, aB(pVq), aBaBp, aB(aBp&p) etc. To get a logical grip on this
we assume the following two axioms
Here pea for a being
acquainted with p, that is, apart from what a believes about p's truth:
A1 ( (p>q)
> aB(p>q) if pea & qea
A2 ( aB(p>q) > (aBp>aBq)
A1 says a believes all
logical implications in propositions a knows.
A2 says that
if a believes p implies q then a believes p implies a believes q.
This is like necessity in modal logic, supposing Np > NNp. And
note the s: The theorems of
PL extended with LPA.
T1
aB(p&q) > aBp
T2 aB(p&q) > aBq
The
first from PL (p&q)>p and the second from (p&q)>q
and A1 and A2. Likewise
T3
aBp > aB(pVq) if qea
T4 aBq > aB(pVq) if
pea
Next
an important theorem:
T5 (pIFFq) >  (aBp IFF aBq) if pea and qea
From pIFFq we have p>q and q>p from which aBp>aBq
and aBq>aBp whence T5.
Therefore, if pea and qea and they are logically equivalent a believes
they are equivalent and so they are. This makes a ideally logical, in
principle.
Then we have from (p>p) that aB(p>p) and since
(p>p) IFF (~pVp) we get from T5
T6  aB(pV~p) if pea
We also have
T7  (aBp & aBq) > aB(p&q)
This follows from  (p > (q > (p&q)) and A1 and A2.
Also there is
T8  (aBp V aBq) > aB(pVq) if pea and qea
namely from a proof by cases and T3 and T4.
Now we assume, also in view of the converse of T8
A3 
(aBp > ~aB~p)
that is similar to the modal formula N(p) > ~N~(p)
Then from A3 by PL
T9  ~(aBp&aB~p)
whence
T10  ~aB(p&~p)
by indirect proof using
T1 and T2.
Then there is
T11 p >
aBp if pea
For we have (p) IFF  (p IFF pV~p) i.e. any provable tautology is
provably equivalent to the tautology (pV~p), and indeed any other
tautology. Hence, supposing a is acquainted with p, by T5 p >
aBp IFF aBpV~p and so the conclusion by T6. Note that p must be a
tautology here, but the proof is the same with any number of variables,
as in  ((p&q)Vr) V ~((p&q)Vr).
Now it so happens that we have  for example  from aB(p&q)
> aB(p) that  aBaB(p&q) > aBaB(p), and therefore
iterated attitudes are implied by the formalism and the axioms already.
A4
 aBaBp > aBp
that conforms to the modal  NN(p) > N(p) simplifies and can be
read as affirming that one's beliefs about one's beliefs are correct
(which does not mean one's beliefs must be correct).
MT. The individuals
that satisfy the axioms A1, A2 and A3 are logical: they believe all
logical theorems in propositions known to them; they are
reasonable: their beliefs allow logical inferences; and they are
consistent: they do not believe both p and ~p. They follow and believe
the rules of logic.
These
individuals are not real human beings, but rather idealized logical
reasoners: If a person would reason logically, and the above axioms
hold, then the above theorems are true of the person.
Now to avoid the problem about or that
was sketched in section 4, we
can use A3 that is equivalent to  (~aBp V ~aB~p) and
define
D1 ?aBp
=df (~aBp & ~aB~p)
We shall turn to interpreting ?aBp in a
moment, but first note that it follows that
T12 aBp
V aB~p V ?aBp
immediately from the definition, since
the denial of T12 is a logical contradiction.
This provides an answer to the problem
about or if we
assume, as we shall do, that ?aBp may be true, and that a may know it
is true, namely precisely in those cases that a knows that a does not
believe p nor does a believe ~p (of which there ought to be many if a
is halfway rational).
Indeed, we shall assume that ?aBp may be
true because we
know that there are many propositions that we know we cannot
confidently affirm or deny, while we are quite willing to affirm that,
nevertheless, these propositions are true or not, as T6 has it.
There also is the following theorem, for
the same reason as T12
T13
aBaBp V aBaB~p V aB?aBp V ?aBaBp
For we have from D1 that ?aBaBp IFF
~aBaBp & ~aB~aBp and therefore T13 must be true since it can't be
false.
And now it follows from T12 and T13 by
A4 that we have just six fundamental possibilities:
T14 There are just 6 basic attitudes (involving a, B
and p), that we may name as
stated:
aBp
& aBaBp
...................................................conscious belief in p
aBp & ~aBaBp&~aBaB~p & ~aB(~aBp&~aB~p)....unconscious
belief in p
aB~p &
aBaB~p...............................................conscious belief
in ~p
aB~p & ~aBaBp&~aBaB~p & ~aB(~aBp&~aB~p)..unconscious
belief in ~p
~aBp & ~aB~p & aB(~aBp&~aB~p)....................conscious
uncertainty about p
~aBp & ~aB~p & ~aB(~aBp&~aB~p)..................ignorance
about p
This can be made to look prettier and a
bit easier by adding a few definitions:
D2 ??aBp IFF (
?aBp & ?aBaBp)....a is unacquainted with p
D3 !aBp IFF
(?aBp & aB?aBp)
We put it as we did, because ?aBp is
ambiguous
intuitively speaking: Either it means a is acquainted with p but
neither believes p nor believes ~p, as D2 has it, or it means a is not
acquainted with p at all, as D3 has it.
So we can now reformulate T14:
T14 aBp
& aBaBp V
aBp & ?aBaBp V
aB~p
& aBaB~p V
aB~p &
?aBaBp V
?aBp & aB?aBp V
?aBp & ?aBaBp
with the same readings, and using ?aBaBp IFF ?aB~aBp as follows easily
from D1.
To return for a moment to the problem
about or: It is
noteworthy that while according to T1 and T2 it is a theorem that
aB(p&q) > (aBp & aBq) the formula aB(pVq) > aBp V aBq
is not generally valid, since it is possible  e.g. if q IFF ~p but
also in contingent cases, as when a believes he will go on holiday to
Paris or Quito, but has't made up his mind yet  that aB(pVq) &
~aBp & ~aBq.
So this seems to be a resolution of the
orproblem with minimal means.
The same description applies to the
following sketch of
a resolution of the problem of quantification, that indeed is a sketch
because I have not formulated principles of quantification, though this
can be done along the lines used for propositional logic.
The principle that will allow us to
resolve the quantification problem turns on introducing the person a's
world a
and the real world W and assuming the following axiom, with Q being
either the existential or the universal quantifier, but in either case
the same in the formula that follows:
A5 (Qxea)(aBFx) IFF
(aB(QxeW)Fx).
That is: a believes all
or some x in W are F is
equivalent with: of all or some x in a,
a believes to be F: There is something in a's world that a believes
to be F (e.g. divine) iff a believes there is something in the real
world that is F (e.g. divine).
Note both sides of iff are
about the real world, that
contains both a and a's beliefs, regardless of the truth of these
beliefs, and that neither side implies that a
is right that x is F (though this can be added by adding the conjunct
(QxeW)(Fx) in case a is right.
Note how this resolves
the problematic statements mentioned in section
4:
"The
Pope believes there is an x such that x=God" turns into "There is an x
in the pope's world that the pope believes to be God and
selfidentical", which ought to be acceptable to both the pope and
atheists, while Antigone's belief about the man standing before her who
she does not recognize as her brother likewise gets rendered as a
belief about Antigone's world.
To obtain a fully
formalized LPA some
further specifications need to be added, but these are mostly the
standard ones, next to the above principles.
6.
On iterated attitudes and attitudes in general
There are quite a few things one can do
with the above apparatus, as shown e.g. in the lemmas Cooperation, Power,
Personal
Perspectives, Society in this dictionary
and while much remains to be done (or has been done, but lingers in my
notebooks), it seems this clarifies a number of things, and does so by
what seem to be minimalist assumptions.
In this section only two general points
will be considered, mentioned in the section title.
First, iterated
attitudes
 a believes that a believes p, a believes that b believes q, a
believes that a is undecided about q, b believes that a believes ~p and
so on  make propositional attitudes considerably more complicated; are
necessary if one wants any logical representation of much attitudinal
talk in natural language, where people often have beliefs about the
beliefs of others and themselves; and also provide at least formally a
way to consider and represent conscious reasoning and conscious willing.
The reason for the last point is that by
A4 believing
that one believes p (consciously) implies that one believes p (also
unconsciously), while the converse need not hold at all, as is
factually correct, since one may believe something that one is not 
then and there  conscious of one believes.
This also means that one has  at least
 a formal basis
for consciously creating oneself, namely by coming to beliefs that one
consciously adopts, and thereby, if A4 is true of one, will become
one's beliefs, also if one does not consciously think of them.
Second, attitudes
in general, as introduced in section 3,
where Asserting, Believing, Causing (trying to cause), Desiring,
Experiencing, Feeling, Imagining, Remembering and Sensing were
mentioned, notably because (i) these are attitudes that are very
common and (ii) these are attitudes that seem to be such that any 
sane, not drunk, not drugged, conscious  person can keep apart in his
own experiencing: One knows  qualifications as stated  that one is
remembering, imagining, fantasizing or sensing things when one does,
usually and in most circumstances and conditions at least.
At this point, having in fact drawn up
the outlines of a
logic of propositional attitudes that only dealt with belief, there are
in fact two possible tracks:
First, one can explicitly introduce
further attitudes, perhaps with their own specific axioms, and second,
one can choose
to attribute such attitudes to specific persons (of a certain kind,
say) that these person believe (while others may not), thus making
other attitudes than belief depending on beliefs about them that a
person has, or lacks.
Both tracks have their strengths and
weaknesses, that
can be illustrated by considering desiring. Grammatically, this is a
propositional attitude, but some have denied desiring should be
considered a propositional attitude in logic, and the same holds for
some other propositional attitudes.
A general answer is that if a
propositional attitude, in
the grammatical sense, satisfies the four axioms that have been
formulated for 'Believes', then it makes sense to consider it a
propositional attitude in the logical sense.
A specific answer, here illustrated with
'Desires', is
that some attitudes are  wholly, or perhaps partially, with a residue
left out  definable in terms of others, possibly supplemented with a
special assumption, as for 'Desires', here abbreviated to 'D':
A6
(a)(p)(Ex)( aBv(p)=x ) if pea
i.e. Every proposition has
some value for everyone
A7 (a)(p) (
aB(v(p)>0) IFF aB(v(~p)<0) ) if pea
i.e. the value of any p is positive iff the value of ~p is
negative for everyone
A8 (a)(p)( aDp IFF aBv(p)>0
) if pea
i.e. one desires p precisely if one believes p's value (for one)
is positive
These three assumptions reduce (or
'reduce') desires to
beliefs about values, that as defined are quite
convenient, e.g.
because there is a sort of consistency by A7, without commiting one to
absolute values (so that v(p) may be very positive, while v(~p) may
only be
slightly negative).
Second, the above assumptions for
'Desires', as indeed
this whole Logic of Propositional Attitudes, is offered in a pragmatic
rather than a metaphysical spirit: There is a considerable need
for a
set of simple assumptions to guide and inform one's reasonings with
propositional attitudes, and the present lemma sketched something like
a minimal system that seems adequate to quite a few of one's
intuitions, while avoiding the pitfalls sketched in section 4.
