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 non-human 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 truth-conditions, 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:

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:

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 well-known 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 non-Catholics, 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 self-evidently 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 bi-valent 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 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 or-problem 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 self-identical", 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.