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. Probabilistic foundations of EPL and LPA

 

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 a problem that arises when one introduces quantifiers, 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 affrims he believes there is such an x (and he may well be mistaken).

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.

Here is a link to a more recent probabilistic foundation of EPL and LPA:

• Extended Propositional Logic

and a a more recent  alternative with an algebraic semantics as foundation:

• Extended Propositional Logic - Algebraic Semantics

The last is probably the clearest.