Propositional attitudes revised

Marie Duží, Pavel Materna

(Duží) The problem of propositional attitudes has been dealt with by many philosophers (see, e.g. [Aho 1994], [Bauerle 1989], [Carnap 1947], [Church 1954], [Richard 1990], and many others), and has been also discussed in the Logica conference [Duží 1996], [Duží 1998], [Duží 1999]. Let me summarise my opinion. Since our proposed solution is based on Transparent Intensional Logic (TIL) [Tichý 1988], I have first to repeat basic necessary definitions of TIL. The key notion of TIL is the notion of construction. An informal comment on this notion should precede the definition. Constructions are ways, 'paths' that lead to an indentification of an object (or, sometimes, in well defined cases, fail to identify anything); they can be viewed as sequences of some intelectual steps. The word 'sequence' is not probably the best choice. Sequences of steps are frequently understood as sequences subsequent in time. The result of Tichý's explication is a timeless abstract entity. Similarly as algorithms (as sequences of instructions) are timeless and spaceless, and only their performing is a time consuming process or their recording is located in space, a construction itself is timeless and spaceless, and only its using, as a mental process, is a time consuming sequence of steps. Formally, constructions remind terms of the (typed) lambda calculus, but there is a significant difference: whereas lambda calculus is a formal language which has to be interpreted, no such 'round-away' way is used here. Introducing a symbolic apparatus for handling constructions we introduce, no doubt, an artificial language. Yet using this language we speak directly about the entities which the expressions of that artificial language are about. The formalistic view would require that first a formal language were given and then an interpretation proposed. No such way is used here. 'Looking through' the necessary symbolic expressions we speak directly about what is encoded by them.

Def. 1 (Constructions)

Atomic constructions are variables. For every type (see Def. 2) let a countably infinite number of variables be at our disposal. A variable is an incomplete construction which constructs a definite object of the respective type dependently on valuation (i.e. a total function associating any variable with just one object of the given type). We say that variables v-construct, where v is a parameter for valuations.

If X is an object whatsoever (it may be even a construction), then ⁰X is a construction called trivialization. ⁰X constructs simply X without any change.

If X is a construction which v-constructs a function with values in a set α whose arguments are n-tuples of elements of types β₁,...,β_n, and X₁,...,X_n are constructions which v-construct elements of β₁,...,β_n , respectively, then [X X₁ ... X_n] is a construction called composition. If the function v-constructed by X is not defined on arguments v-constructed by X₁,...,X_n, the composition does not v-construct anything: it is v-improper. If any of X₁,...,X_n is v-improper, the composition is v-improper. Otherwise the composition v-constructs the value of the function v-constructed by X on arguments v-constructed by X₁,...,X_n.

Let x₁,...,x_n be distinct variables ranging over the sets β₁,...,β_n, respectively. Let X be a construction that v-constructs members of a set a . Then [λ x₁...x_nX] is a construction called closure (or traditionally abstraction); it v-constructs the following function F: Let <b₁,...,b_n> be an n-tuple of the members of β₁,...,b_n, respectively. Let v' associate x_i with b_i (1 < i < n) and be in other respects the same valuation as v. Then if X is v'-improper, F is undefined on <b₁,...,b_n> . Otherwise, the value of F is the object v'-constructed by X.

Anything is a construction only due to i) - iv).

Now the question on the type of objects denoted by sentences of the form

(*) X B's that P

where B is an attitude verb like believe, know, doubt, etc. can be answered only after another question is answered, namely what is the meaning of 1the sentence P, for these objects are relations-in-intension of the individual X to the meaning of P. Before answering this question, let us formulate some intuitions:

Meanings should be structured entities

Meanings should be objective, independent of a particular language

Meanings should be context independent (transparent approach)

Meanings should (try to) identify the object denoted by the expression

Roughly, to say that meanings are structured is to say that they are complex entities, entities having parts or constituents, where the constituents are bound together in a certain way. So this view allows that different meanings may have the very same paths or constituents, but have them differently bound together. Our propositions are 'flat' mappings from possible worlds ... o_tw -objects (or sets of possible worlds if you wish), hence they cannot serve as meanings of expressions. There are many philosophers who nowadays reject possible-worlds semantics of propositions and advocate for the view that propositions are structured [Soames 1987], [Salmon 1986]. But we do not have to give up a very natural view of propositions as mappings ..., for we can see that all the above demands are met by the above defined constructions. Hence meaning of an expression can be conceived as a (closed) construction, and the objects denoted by sentences of the form (*) are of the type (oi*_n)_tw .

Now there is no paradox in somebody's knowing (believing, ...) that P without his/her knowing (believing, ...) that P', though P and P' being L-equivalent sentences, i.e. sentences denoting the same propositions. A student can easily believe that 'it is not true that if he/she studies hard he/she will pass an exam' without believing that 'he/she will study hard and won't pass the exam anyway', for the meanings of the embedded clauses are different: They are different constructions (first of the 'form' [ (AÉB)], second of the 'form' [A& B]). Similarly many people know that '2 + 2 = 4' without knowing that arithmetic is incomplete.

(Materna) Now I agree with what you proclaimed in connection with attitudes. If the dependent clause is a non-empirical claim (mostly an arithmetical sentence), there is simply no other possibility than to represent this claim by a construction - otherwise the paradoxes like that of omniscience are inevitable. I only say that as soon as the dependent clause is an empirical sentence the situation is not that simple. The argument that also in this case the attitude relates an individual with a construction is supported by the fact that a sufficiently complex L-transformation of the dependent clause A leads to a very complicated construction, so that the believer, knower etc. may change his/her attitude when the same proposition is denoted by the L-equivalent sentence B. Yet this 'empirical case' differs from the 'non-empirical case' in one respect: we can interpret this attitude to the 'state of the world', independently of the way the respective proposition has been constructed. So the believer etc. can possess this attitude without knowing that he/she does. If somebody believes that Mt Everest is higher/lower that Mt Blanc, then his/her attitude concerns the proposition itself, and if this proposition is represented by a very complicated equivalent sentence (expressing, therefore, a much more complicated construction), then dissenting with this sentence cannot influence his/her conviction that ... . So we can see that the attitude expressions could be systematically ambiguous, denoting either (oi*_n)_tw -, or (oio_tw)_tw - objects. This hypothesis is relevant from the viewpoint of deduction. Can you see it?

(Duží) Well, to summarise your point, in the case of empirical embedded sentences, what is believed, known, etc., is the state-of-affairs referred to by the embedded sentence, i.e. a proposition. This would lead to a dualism: the type of propositional attitudes to mathematical (analytical) clauses would differ from the type of attitudes to empirical clauses. This dualism was defended in [Materna 1984] and rejected in, e.g., [Duží 1996] and [Materna 1998], and I don't fully understand why to come back to it. First, when accepting this dualism, we would come to a necessary consequence that somebody can believe something without knowing about that, or know something without believing that, etc. Isn't it somehow counterintuitive? I am convinced that I know what I know, believe, etc. Second, I think that the substitution test is quite convincing. As M. Richard in [Richard 1990, p. 17] says: "In general, if it is just possible that X believes that A be true while X believes that B not be true, then we have to assign the terms 'that A' and 'that B' different things." Now it is not only possible but quite frequent that when I ask a student 'Do you believe that it is not true that if you study hard, you will pass the exam?' he/she answers 'yes', and when I ask 'Do you believe that you will study hard and won't pass the exam anyway' he/she answers 'no'. How is this possible? Why do I have to teach the student that the two clauses are L-equivalent? You might probably answer that the student doesn't answer correctly because he actually believes the second clause as well (or does not believe the first one?) but does not know about it, right? How could this be possible? The only explanation could be that he/she does not understand the clauses. But understanding is knowing the meaning, and if we agree that meanings are structured, i.e. our TIL logical constructions, then he has an attitude to the meaning, i.e. a construction. And we are back again.

(Materna) Don't forget that defining attitudes as relations to constructions/concepts determines without any change just that relation which can be verified/falsified through the 'substitution test'. The other relation, that one which relates individuals with propositions, cannot be confronted with substitution tests. One of the consequences of introducing this alternative interpretation of 'attitude verbs' consists in admitting that having an expression of the form

X believes etc. that A

we can raise the claim that X's belief etc. concerns unambiguously the construction/concept underlying A, but we cannot be sure whether X does or does not possess his/her conviction, doubt etc. as regards the state of the world given by the proposition which is identified by the concept underlying A. This proposition is the result of such a 'decyphering' the given concept which is done by an 'ideal' language user. Drawing possible conclusions from the sentence of the above form when assuming that the attitude verb has been interpreted in this 'propositional' sense can be realised only when a further premise is given, viz. that X's decyphering has been done rightly, i.e., that X has understood the sentence A.

(Duží) Now I don't understand your answer exactly. Could you express your idea in a more intelligible way? Anyway, let us presuppose X's perfect understanding the sentence A. When accepting the 'propositional interpretation' of attitudes, we have to draw a necessary consequence:

But how should we conceive the notion of understanding the embedded sentence A? We can make the following explication:

Understanding the sentence A means X's ability to discover the respective procedure corresponding to the construction which is the sense of the expression A assigned to it on the conventional basis of the given language.

Now discovering, knowing the construction, does it mean that X has also the ability to handle it? In other words, does it mean that X is able to draw all the logical consequences, to discover all the logically equivalent constructions, etc.? Why then do I have to teach students logic, if they know it everything (though without their knowing that they know)? I am afraid that we have thus got into somehow peculiar schizophrenic state, don't we?

Anyway, let us now peruse another example, the example that can seemingly serve as an argument against my opinion. I mean the well-known Quine's example [Quine 1956] that has been a subject of much dispute [Zalta 1988], namely the problem of Ralph's spy, when an obviously valid inference:

(1) Ralph believes that the tallest spy is a spy

hence

(2) Something is such that Ralph believes it to be a spy

seems to fail in such a situation when Ralph does not have a slightest idea who the tallest spy is. The solution is obvious: Of course, Ralph's belief cannot concern a definite individual. It concerns instead an individual office, an i_tw -object, constructed by [λw λt [⁰Tall_wt ⁰S_wt]] (S - a property of being a spy, an (oi)_tw-object, Tall - a (ι(oi))_tw-object, an intension which associates with every world-time couple the function that selects the tallest individual (if any) from any class of individuals). Hence we can paraphrase (2) as follows:

(2') There is an individual office such that Ralph believes that its holder is a spy.

Now there is no failure of inference here. From (1), (2') certainly follows. There is an individual office, namely the office of the tallest spy, constructed by [λw λt [⁰Tall_wt ⁰S_wt]], such that ... . Well, now according to our above discussion, there are two possible analyses of (1) and (2'), 'constructional' one and 'propositional' one. Let us consider first the 'propositional' case:

(1') λw λt [⁰B_wt ⁰R [λw lt [⁰S_wt [λw λt [⁰Tall_wt ⁰S_wt]]_wt ]] ]

(2'') λw λt $c [⁰B_wt ⁰R [lw λt [⁰S_wt c_wt ]] ]

(where B(elieve)/ (oio_tw)_tw , R(alph)/ ι , S, Tall as above, variable c ranges over i_tw ).

The analysis seems to be quite simple, why then there is a great non-trivial problem with quantifying into 'intentional contexts' [Materna 1997]? I claim that this analysis is not correct, for the variable c occurring "inside" the context cannot be so carelessly quantified. It is not used here but mentioned, which is not rendered by the above construction.

Using 'constructional' approach to the attitudes, we get:

(1'') λw λt [⁰B_wt ⁰R ⁰[λw λt [⁰S_wt [λw λt [⁰Tall_wt ⁰S_wt]]_wt ]] ]

where B(elieve) is now an (oi*₁)_tw -object.

An attempt to perform the analysis in an analogical way as (2'') now fails:

(2''') λw λt $c [⁰B_wt ⁰R ⁰[lw λt [⁰S_wt c_wt ]] ]

Variable c occurring inside the context is here o-bound (bound by the outer trivialisation), hence it is mentioned rather than used and cannot be quantified, which is correct. But it does not mean that the above correct inference cannot be solved within the 'constructional' approach. We just need two additional functions - Sub(stitution)/ (*₁, *₁, *₁, *₁) and Tr(ivialisation)/ (*₁ ι_tw) [Materna 1997], and we obtain the correct analysis of (2):

(2'''') λw λt $c [⁰B_wt ⁰R [⁰Sub [⁰Tr c] ⁰c ⁰[λw λt [⁰S_wt c_wt ]] ]]

This construction 'functions' in the following way: Trivialisation of the office constructed by c is substituted for c in the construction [lw λt [⁰S_wt c_wt ]].

(Materna) Well, first of all I would like to state that I need not choose any "more intelligible formulations", for you demonstrated by your reaction that you understand what I try to say.

Second, the situation is not that schizophrenic: we teach logic just because the students do not know that they know... . Logic makes their knowledge explicit. Imagine the following situation. The student knows that

each of his friends has a girl-friend.

One of the consequences of this fact is that

none of the girl-friend-less men is his friend.

We surely admit that the student need not assent to the sentence that claims this second fact. Logic has to convince him that knowing the first fact he knows the second as well. Considering "knowing" as a propositional attitude, an attitude to the state of the world, we have to be aware of the "impersonal" character of this relation. Observe, however, that the student will be eventually (under 'normal' conditions) convinced that his first knowledge concerns also the state of the world given by the second sentence.

Finally, I do not agree with you that in the propositional case the variable c is mentioned. The non-trivial problem with the quantification into can be stated with Quine if either the variable ranging over individuals is taken into account (Quine could not suspect that a variable could range over individual offices), or if we consider the "not impersonal" attitude, which is subject to the "substitutional test" and which we construe as being a 'constructional' one. Quine rightly felt that something like such a "not impersonal" attitude could be a source of non-triviality of the problem, and this 'mentioning' variable in the dependent clause, conceived of in the linguistic interpretation, was for him one of the arguments against the possibility of purely logical analyses of this kind.

(Duží) As I see it now, we have to distinguish between two kinds of knowing, believing, etc. The first kind is a "normal" (or as you call it 'personal', I'd rather call it explicit knowing, believing, ...), and the second kind is an implicit knowing, believing, ... (or as you call that 'non-personal'). In the first case the attitude verb denotes an object of a type (oi*_n)_tw , i.e. it is an attitude to a construction - meaning of the embedded sentence, in the second case the 'implicit' knowing, believing, ... denotes an object of a type (oio_tw)_tw , i.e. it is an attitude to a proposition. Explicit knowing can 'evolve' in time, while implicit knowing is stable. Anyway, in my opinion, the second case seems to be still rather peculiar, for such intuitive things like that we know what we know, we know what we believe, we believe what we know, etc., do not hold in this case.

Second, back to the Ralph's example: Could you specify in a more exact way what do you mean by using and mentioning? In my opinion, the pronoun it (, its) stands for the variable c. And this pronoun is mentioned in the respective sentence, we speak about it and claim that it is believed by Ralph ... . Hence the respective variable c should be mentioned in the respective construction as well.

(Mterna) Well, the peculiarity you speak about is not too strong. I admit that with the term 'attitude' an assumption is usually connected that attitudes are always accompanied with a conscious state of mind. If so, then my 'propositional version' should be rather called 'propositional relation', which is more 'neutral' and lacks the 'consciousness connotation'. Also, the threat of 'omniscience' (we speak about the empirical case only) is not imminent: accepting (rejecting etc.) a proposition I indeed accept all the consequences thereof, but I need not be aware of this fact (and therefore we need logic, among other things). A special problem with logical analysis arises, however: considering a belief sentence we have then two possibilities of analysis, one constructional, and one 'propositional', and there is no criterion of deciding, which of them is 'the right one', at least when the sentence is considered in isolation. But I set aside this problem here.

As for your question about using and mentioning variables, the distinction can be characterised as follows: using a variable we exploit it as a construction which can 'range' over the given type (free variables) or which by its 'ranging' contributed to the evaluation of the closure (λ-bound variables). If, on the other hand, a variable is (immediately or not) under trivialization, it cannot range; it is only a subconstruction of a construction which is to be taken as a whole, i.e., such that it is not the object being constructed what we are interested in.

Compare from this view point the two analyses of the 'Ralph example'. The 'propositional version' does not block the possibility of 'quantifying into' ? to evaluate the construction we have to let the variable c range (over the class of individual offices).

So have I convinced you that analyses of 'belief sentences' are systematically ambiguous?

(Duží) Well, but you did not answer my question concerning the intuitive feeling we have: The pronoun it is mentioned in the respective sentence. Hence the construction - meaning should respect this fact. And this is not fulfilled by the 'propositional' analysis.

Anyway, accepting the two kinds of knowing, believing, ... , we should called them explicit ('personal') knowing, believing, ..., and implicit ('non-personal') knowing, believing, ... . When distinguishing these two kinds of 'belief sentences', then no ambiguous cases would arise. We just have to explicitly state which of the two notions we have in mind.

(Materna) To the problem of mentioning it I can say two points. First, the problem of using vs. mentioning expressions is not the same problem as that of using vs. mentioning constructions (see my 1988, 5.5). Second, your example is not O.K. The pronoun it is in the given context typically used. It is mentioned in such contexts as "it is a personal pronoun".

As for your second point, you are completely right. One could even formulate a claim:

Which could be a nice close of our dialogue.

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Published (2001): In: Majer (ed.): The Logica Yearbook 2000, 163-173, Prague: Filosofia. (with M. Duzí)