Bernard Bolzano: Study of science – Truths and consequences

 After propositions, Bolzano moves to discuss truths in themselves or objective truths that may or may not be known by anyone. He considers this a field not yet investigated by anyone and finds even his own results very preliminary.

Like with propositions, Bolzano begins with characteristics common to all truths, the first one being the familiar one that truths do not actually exist, making it thus incorrect to say that some truths exist eternally and some temporarily. He notes also that all truths deal with objects, so that their subject is a representation that refers to at least one object. Bolzano admits that some truths appear to have propositions as their subject, but thinks that the subject in these cases is actually a representation of a proposition. Similarly, he continues, truths with imaginary objects as their subject (e.g. round square does not exist) usually have a representation of an imaginary object as the real subject (in the example, the truth  is actually “the representation of a round square refers to no objects”). Sometimes, Bolzano adds, such truths are actually hypotheticals: “golden mountain is barren” means actually “if a mountain is golden, it would be barren”.

If the subject of a truth must be a representation referring to some object, Bolzano says, its predicate must be a representation referring to such a characteristic that every object referred to by the subject has. Furthermore, he continues, all truths must be compatible with one another (they are true at the same time), and every truth is derivable from infinitely many other truths, while infinitely many truths are derivable from every truth. Finally, every truth and every collection of truths is a part of another truth (at least the truth saying that all these propositions are true).

Just like with propositions, Bolzano goes on to describe different types of propositions. Actually he mentions only one possible division of truths, namely, that of analytical and synthetical truths. Analytical truth, Bolzano defines, has as its constituent at least one representation that can be switched with any other representation so that the proposition remains a truth, just as long as the proposition (that it, its subject) refers to an  object; any other truth is synthetical truth.

Evidently, Bolzano thinks, there are analytical truths (his example is “every A, which has b, has b”). Then again, he thinks, synthetical truths are a bit more difficult to find. Bolzano notes that there are characteristics that belong to all objects referred to by a representation without being constituents of that representation, but this is not enough to establish the existence of synthetical truths. For instance, the characteristic of the sum of the angles of a triangle equalling two right angles is something featured in all triangles, although this characteristic is not a constituent of the representation of a triangle. Still, Bolzano surmises, the truth “this triangle has angles equal to two right angles” is still analytical by his definition, because in switching the representation “this” with another representation that makes the subject objective (in other words, by changing the object referred to by the word “this” to some other triangle), the proposition remains always true. 

Bolzano notes that the problem just described can be circumvented, if we just pick a subject that is either simple or then of the type “something that has x”. The predicate of the synthetical truth we are searching for, he adds, should not be such that holds of all things when some of its constituents is changed (any characteristic of the type “being x or not-x” is of this kind. Bolzano suggests using either simple predicates or combinations of simple predicates of the kind “x and y and z…”. Finally, the constructed proposition must also be true – Bolzano’s own example is “A is actual” where A is meant to be some intuition. 

Bolzano thinks that there are clearly also negative synthetical truths (his example is “A has no b”, where A is intuition and b is a characteristic that A does not have). Furthermore, he notes, “representation of something, which has characteristics a, b and c, refers to an object” is always a synthetical truth, because if the characteristics are switched to such that are incompatible with one another, the result is not true. Finally, Bolzano points out that “representation of something, which has characteristics a and b, is imaginary” is synthetical, if it is true, because then a and b are incompatible, and when switched with compatible representations, make the resulting proposition untrue.

Bolzano considers it evident that there are both analytical and synthetical truths that are either intuitive propositions or conceptual propositions. Thus, “each triangle is a figure” is a conceptual analytical truth and “this triangle is a figure” is intuitive analytical truth. Similarly, the aforementioned “A is actual” is an intuitive synthetical truth, while Bolzano’s other examples of synthetical truths are conceptual.

The rest of Bolzano’s investigation of truths discusses what he considers the most remarkable relation between them, that is, the already mentioned relation of ground to its consequence relation. He again notes the example of a thermometer rising, which is at most a ground for us knowing that the summer is coming, but not an objective ground for the truth of the summer coming, where the consequence relation would hold independently of our thoughts. This example shows also that the consequence relation is not the same as the relation of derivability: “summer is coming” is derivable from “thermometer is rising”, but not its consequence.

Is consequence still always derivable from its ground? Bolzano thinks this is not so, because of the following example. All practical truths, he says, must be founded in a truth of the form “we should do A”, which with some propositions of the form “doing A requires doing X” is the ground of all practical truths. Now, Bolzano suggests, this highest law of ethics appears to be founded in the truth that A is something that we can do, but there seems to be no rule of inference that would account for this unique consequence relation, since they can prove obligations only from obligations.

As we already saw with the example of the ground of practical truths, the complete ground of a truth is usually a collection of truths, while the individual truths in that collection are then just partial grounds. Bolzano ponders the question whether the rules of inference used for deriving the consequence from this collection of truths should be regarded as part of the ground. He comes to the same conclusion as Lewis Carroll that this would lead to an infinite series of ever new rules of inference.

Bolzano then considers the question whether the consequence relation is the same as causality. No literally, he answers, because causes and effects are actual objects, while grounds and consequences are truths, which are not actual. This leaves still the possibility that causes and effects could still be constituents of grounds and consequences. Indeed, some consequence relations involve causes and effects (e.g. “God is” could be a partial ground of “world is”). Still, Bolzano thinks, not all consequence relations involve existing objects, such as consequence relations involved with mathematical truths. Thus, he concludes, concepts of cause and effect must be defined by concepts of ground and consequence, as he has already suggested, but the latter cannot be defined by the former.

Bolzano does not find ground and consequence to be simple concepts, because they are concrete and least have “something” as their constituent (that is, ground is something that “grounds” something else and consequence is something “grounded” by something else). He has no idea whether this abstract notion of “grounding” has a further definition. All the definitions suggested by earlier philosophers, Bolzano says, are either circular or then faulty – for instance, grounding cannot be defined as determining, because an equation determines a magnitude, but is not its ground.

Since we cannot define the consequence relation precisely, Bolzano suggests describing its features as much as possible, in order to distinguish it from other similar relations. Thus, he says, derivability holds between propositions, whether true or false, while consequence holds only between truths. Sometimes we do say that a consequence relation holds between falsities, Bolzano admits, but then we are actually speaking of derivability or then we consider cases where switching some representations in these false propositions and thus making them truths, creates also a consequence relation between them (Bolzano’s example is when we say that “monkey can speak” is a consequence of “monkey has distinct representations”).

Bolzano notes some other similar confusions. Sometimes we confuse causality with consequence and say e.g. that God is the ground of the existence of the world. Sometimes we speak as if mere representations could be grounds, for instance, when the ground of our fear of death should be our representation of it – actually, it is then not representations in themselves, but our thoughts of them we are speaking and they are not grounds, but again, causes. Bolzano also notes a curious case, where the ground of a proposition being true or false are said to be its constituent representations. He thinks that in this case, the ground lies actually in the truths involving the constituents of the proposition.

Comparing further derivability and consequence relation, Bolzano points out that a proposition can be derivable from itself, but nothing is a ground of itself. Then again, derivability can hold between collections of propositions, and Bolzano thinks that the same is true of the consequence relation. We have already noted that while a ground can be a single truth (thus, A is a ground for “A is true”), it is often a collection of propositions and the individual propositions in that collection are just partial grounds. Similarly, individual propositions based on some ground are just partial consequences, and the collection of all these partial consequences is the complete consequence of the ground.

The notion of total consequence reveals yet another difference from the notion of derivability. While we can derive several propositions from one proposition, the total consequence – consequence in the proper sense of the word, Bolzano thinks – is always unique, as defined by all the partial consequences. It seems that a consequence can then have many grounds, but these are all just partial grounds, Bolzano argues, but the total ground is always unique.

Derivability is a transitive relation (that is, if A is derivable from B and B from C, A is derivable). Due to the just mentioned uniqueness, Bolzano points out, the consequence relation cannot be transitive. Similarly, while A can be derivable from B and B from A, the same sort of reciprocity does not hold for grounds and consequences: we do speak of e.g. reciprocal causality, Bolzano admits, but then it is actually a case of the interacting things having forces that cause some changes in the other thing – not both things literally causing one another. Then again, just like what is a derived conclusion can in another context be a premiss, so a consequence of something can be a ground of something else.

Derivability always survives summing up collections of propositions (e.g. if A and B are derivable from C and D and E and F are derivable from G and H, then A, B, E and F are derivable from C, D, G and H). With consequence relation this is not so easy, Bolzano states, since one collection might contain consequences of the other collection, which would cause difficulties, if the collections contained same truths (e.g. if A and B are consequence of C and D and C and D are consequence of E and F, the collection of A, B, C and D is not the consequence of C, D, E and F). Even if we avoided the cases where the collections contained same truths, the sum might still not work, Bolzano thinks, because the consequence relation requires a more intrinsic connection between the truths (e.g. the collection of “God is almighty” and “isosceles triangle has two equal angles” is not a ground of the collection of “the actual world is the best possible” and “the equilateral triangle has all angles equal”).

Collections of premisses and collections of conclusions appear to have no hierarchical order, and same seems to be true of the collection of partial grounds and the collection of partial consequences, Bolzano thinks: although there may be a preferred order how to present grounds in a scientific treatise, this is just a subjective preference. A more difficult question, Bolzano thinks, is whether just like with premisses, there could be partial grounds, one of which is a consequence of the other. Surprisingly, he suggests that consequence relation does resemble derivability in this respect: “every truth is something immaterial” is ground for the truth “the proposition that every truth is immaterial is itself a truth” and both of them are partial grounds for the truth “the proposition that every truth is immaterial is itself immaterial”.

Every proposition is both a premiss and a conclusion, but is every truth a ground and a consequence? According to Bolzano, every truth is a ground for further truths (at least for the truth that the truth is true). Then again, he thinks it probable that some truths are not consequences of anything (his example is the truth that something exists). Bolzano calls such truths fundamental. He suggests that there must be several fundamental truths, because it seems impossible that all others could be consequences of just one truth.

Bolzano considers what will happen if we ask for all partial grounds of a truth and then for partial grounds of these new truths and so on, creating a tree-like structure. If we reach fundamental truths, he notes, this process ends. Still, Bolzano suggests, there may be truths, where this search for further and further partial grounds continues indefinitely: his example is that of variable states of created substances, which seem to depend on an infinite series of causes.

Bolzano also coins the term auxiliary truth to refer to any truth in the just described tree-like structure, created from a given truth. Unlike the notion of ground, the notion of auxiliary truth is transitive, but like with ground, no truth can be auxiliary truth of itself. Then again, Bolzano notes, the same truth can be auxiliary for another truth in many different places of the same structure: just look at mathematical proofs, where we have to use some truth many times.

Bolzano notes that truths involving only pure concepts are grounded in other truths involving only pure concepts, although they might be known through experiences. Furthermore, pure conceptual truth that is a ground of another pure conceptual truth can never be more complex than the consequence (Bolzano thinks that this is not true of intuitive truths, since they are often grounded in general truths that are more complex). Since Bolzano has assumed that there is only a finite number of simple concepts, he can also assume that there is only a limited number of fundamental conceptual truths. Thus, at least with conceptual truths, the consequence relation is ordered so that the greatest number of consequences are based on the smallest number of fundamental truths.

Bolzano closes the discussion by noting that sometimes grounds are derivable from consequences and sometimes not: as an example of former we have “every A is B” and “every A is C”, which together form the ground of “every A is B and C” and are derivable from it, while as an example of latter we have “what has a, has b” and “what has b, has c”, which form the ground of “what has a, has c”, but are not derivable from it. Just like with causes, Bolzano defines condition in a wider sense as an auxiliary truth of some truth that is derivable from this truth.

Bernard Bolzano: Study of science – Analysing propositions

Bolzano rounds up his discussion of propositions with the favourite pastime of the later logicist school: that of revealing the hidden logical structure of ordinary linguistic expressions. Remember that Bolzano's ultimate goal is to show that all propositions bend to the universal form of “A has b”. Thus, he uses the rather convoluted terminology that a representation has objectivity, when he wants to say simply that a representation refers to an object. In line with this, Bolzano suggests that “nothing is B” or means actually that the representation of something, which has the characteristic b, has no objectivity, rendering it unnecessary to take the word “nothing” as the subject of this proposition. Similarly, he thinks that “an A is B” means actually that the representation of an A, which has b, has objectivity.

Bolzano moves on to impersonal sentences, using in German the “Es gibt” -structure, translatable often with the words “there is”, which is again another way to say that a representation refers to an object. This is true, he insists, even of weather reports, which form one subspecies of “Es gibt” -sentences (for instance, “it snows” should mean that a representation of snowfall happening now refers to an object). Other expressions with the “Es gibt” -structure have different readings for Bolzano: “they say…” means that there are people who say something, while “it is true that…” means that a certain proposition is true.

Bolzano points out that “some A are B” in its usual sense means that characteristic b belongs to several, but not all and not even to remarkably many A. On the other hand, he states, “many A are B” means that the characteristic b belongs not to all, but to a great number of A. Indeed, Bolzano continues, the word “many” is relative and could mean many things: an absolutely great number of things, like 3 billion, a great number in relation to something else, like the number of stars we can see with bare eyes, in comparison with the number of all of them, and a great relation of numbers in comparison with something else, for instance, when many of our family are sick, although it covers only a tenth, in proportion to other families this is much.

Bolzano goes through some relatively simple cases. “n As are Bs” means either that there are at least n As that are Bs or that exactly n As are Bs, both easily turnable to his favoured form. Comparative proposition “A has characteristic b more than C has” speaks actually of the degree or magnitude of the respective characteristic of the objects, while exclusive proposition “only A alone is B” means that the characteristic b belongs exclusively to A and implicitly to some A (whether it belongs to all of them depends on the context). On the other hand, Bolzano notes, “A is only B alone” does not mean that it would have no other characteristics, since everything has infinitely many characteristics, but that it has of some relevant characteristics only that of b (for instance, “Caius is only good to invent rhymes” suggests that he has no other characteristics of a good poet).

What Bolzano calls a restrictive proposition – “A, as C, is B” – can have many different meanings. Firstly, “Earth, as consisting of many substances, is destructible” suggests that consisting of many substances is a ground for Earth being destructible. Then again, “Caius, as a musician, is wonderful” does not mean that Caius himself would be wonderful, but his musicianship is. Finally “Titus, as a judge, should accept no gifts” does not mean that Titus should accept no gifts, because he is judge, nor that his judgeship should accept no gifts, but points to a special relation between the three concepts – Bolzano does not explicate this relation more precisely, but it is something to the effect that it is a characteristic of his position of a judge that while doing his duty in that position, he should not accept any gifts, which would be considered bribes. Bolzano also notes that a proposition “A as A is B” is equally ambiguous, but usually means that the characteristic b, ascribed to A, is derivable already from concept A.

We have already seen Bolzano deal with the phrase “A is, because B is” which means that the B is a ground or at least a partial ground of A. He adds that often this form is used merely to indicate that A is derivable from B. Bolzano points out that if-then -structure is also often used to describe the relation of derivability, but it is in some cases applied elsewhere. For example, “if digits in a number are arbitrarily switched and the result is subtracted from the original, then the result is always divisible with 9” indicates no derivability, he thinks, because there are no obvious variable representations, and instead it describes certain relations between numbers (that the number differing from a given number only thus that its digits are in a different order, is to this one in such a relation that their difference is always divisible by 9). Similarly “if Caius is silent in this situation, he is unthankful” does not indicate derivability, but describes a general proposition that anyone who is silent in the situation of Caius is unthankful, while “if Caius is dead, then Sempronius is a beggar” deals with relations between Caius and Sempronius, which make the latter necessarily a beggar with the death of the former.

“A determines B” (for instance, “two sides of a triangle and the enclosed angle determine the whole triangle”) means, Bolzano thinks, that certain propositions expressing characteristics of B are derivable from certain propositions expressing characteristics of A in regard to representations that concern characteristics of A. “A determines B wholly or completely” means then that all propositions concerning characteristics of B are derivable from certain propositions concerning characteristics of A (for example, centre and radius determine circle completely, because all characteristics of the circle can be derived from their characteristics).

“Either-or” -structure, Bolzano says, means in its strongest sense that only one of the options presented is true, but it can also mean that at least one and possibly several of the alternatives are true. Similarly, he adds, it can refer to options present in the sentence or use some variables. Furthermore, Bolzano points out, “all A are either B or C” does not mean that either all A are B or that all A are C, but that each individual A is always either B or C.

Bolzano goes into great detail with the so-called modal terms or necessity, impossibility, possibility and contingency. In their strongest sense, he thinks, these concepts refer to actual existence of objects – “must” means “must exist” and “can” means “can exist”. More precisely, Bolzano explains, if something (say, God) is necessary = there is a conceptual truth of the form “A exists”, where A comprehends the object in question. Something is then impossible if its existence is a conceptual falsity. Possible thing is then defined as not impossible and contingent thing as existing, but not necessarily. Bolzano's definitions imply that any object that can be represented with a pure concept referring only to it is either necessary or impossible, thus, contingent is something that we can represent only through mixed representations or intuitions.

This strong sense of the modalities can be relativised, so that e.g. something is necessary relative to another thing (Bolzano speaks also of external necessity) if neither is by itself necessary, but the existence of one is derivable from the existence of the other (like punishment is necessary relative to a sin). Furthermore, Bolzano adds, in an improper sense these concepts can be extended to truths in themselves. Thus, b belongs necessarily to A, if “A has b” is a conceptual truth; b is impossible for A = “A has no b” is a conceptual truth.

Bolzano points out a final way to use especially the concept of possibility. In this sense, something is possible or something could be, if we do not know it to be impossible, that is, if we know of no conceptual truth that would express the opposite. Obviously this does not mean that it might not really be impossible.

If a proposition speaks of an actual object and the characteristics ascribed to it in the proposition do not always belong to it, Bolzano thinks, the proposition must determine a time, when the characteristics belong to the object. Thus, we have verb tenses for present, past and future times, and sometimes the determination of time is even the object dealt in a proposition (Bolzano’s example is “cherries bloom earlier than grapes”, which should mean that the relation of the season of cherries blooming to the season of the grapes blooming is the relation of an earlier to a later season).

Bolzano is especially interested in propositions about the beginning, the enduring and the end of a state. A state endures through some period of time, he says, when the proposition that the state is actual at the moment x remains true, whatever moment included in the period of time is put in the place of x. A state enduring now means then that it endures through a period of time including the present. Bolzano finally defines a beginning or an end of a state at a certain moment in such a way that the proposition that the state is actual at the moment x is true, as long as x is replaced with a moment that is a bit later than the moment of beginning or a bit earlier than the moment of ending, but false, if the x is replaced with a moment earlier than the moment of beginning or later than the moment of ending.

Bolzano also emphasises the importance or propositions speaking of becoming. Such becoming, he says, involves actual existence in a future time, but mere existence in future time is not always becoming. The difference lies not just in how far in the future the event is, Bolzano thinks, because we do speak of events centuries in the future as already becoming. Instead, he suggests, becoming requires change, and the future event becoming must be caused by this change. Then again, Bolzano continues, change requires something that is changed: an internal characteristic of the changing thing does not remain the same even for the smallest time, when the change is going on. Given this definition of change, becoming of M can then be defined as an on-going change causing M to exist in the future. If we say that M becomes from A, this means that the object A is undergoing the change, which causes the future existence of M. Finally, Bolzano concludes, if M becomes through or because of P, the object P is a total or partial cause for the change that causes the future existence of M.

Bolzano concludes the section on propositions by suggesting a few examples, where one and the same sentence actually combines several propositions. Thus, differentiating proposition “not X, but Y does have b” is a combination of proposition “X does not have b” and “Y has b”, compared to one another, while extensive expression “all A, including those that are X, have b” says that all A have b but that the listeners probably won't believe this and think that X are an exception.

Bernard Bolzano: Study of science – Propositions about propositions

Bolzano has already mentioned the possibility of propositions about propositions, but he then left a fuller account of that notion to a time after the discussion of relations between propositions, since many propositions about propositions deal with just such relations. As he is now finally in a position to deal with the topic, Bolzano is especially interested in showing how to present such propositions in his official form of all propositions: “subject has a predicate”. Thus, for instance, the notion of compatibility of propositions would be fully expressed as “representation of a collection of certain representations, which put in place of i, j… in the propositions A, B, C… makes them all true, refers to some object” (the notion of incompatibility is, of course, expressed by a negation of this proposition). As he is well aware, we express such complex propositions in a much simpler manner and especially the notion of variable representations in the propositions is usually left implicit (for instance, we just say that the propositions can be true at the same time).

Bolzano goes quickly through all the various forms of compatibility and incompatibility, but we really need to look at just the notin of derivability, as it is the only one where he feels the need to give the full official form. It is really no wonder he refrains from doing it in other cases, since even this one is a mouthful: “representation of certain parts in A, B, C… M, N, O…, which are such parts that every arbitrary collection of representations, which put in their place makes the A, B, C… true, makes also the M, N, O… true, refers to objects”.

From the species of compatibility and incompatibility of propositions, Bolzano moves on to discuss the relation of complementarity, which is obviously important to disjunctions. We may quickly go through Bolzano's official structure for the basic notions in this regard, that is, the material complementarity – “representation of a true proposition among M, N, O… refers to some object” – and formal complementarity – “representation of a true proposition among a collection of propositions, whatever else is put in place of representations i, j…, following rule R, refers to some object” (it is not clear what this rule R refers to). Bolzano is as quick in presenting his official structure of propositions on probability – “relation of the set of all such representations that make the propositions A, B, C… true to the set of those that in addition to A, B, C… make also M true is m” – although he does add that we commonly say something to be probable, when this m is more than ½.

Bolzano's official structures for material and formal consequence are by now probably not surprising, former being “relation of truths M, N, O… to truths A, B, C… has the characteristic of a relation of consequence to its ground”, and the latter “the relation of propositions M, N, O… to the propositions A, B, C… has the characteristic that each collection of representations that put in the place of i, j… makes the A, B, C… true, makes also the M, N, O… into such truths that are the consequence of the former”. What is more interesting is his remark that we often speak of grounds and consequences, when we should actually speak of partial grounds and consequences. Furthermore, Bolzano defines cause as an existing object, the existence of which is a partial ground of some truth, and effect as an existing object, the existence of which is one consequence of an existence of a cause. He also points out that the existence of a cause implies the existence of an effect, but not necessarily the other way around, since an effect could have many possible causes. Thus, he defines condition as a full or partial cause, the existence of which is derivable from the existence of the effect.

Bernard Bolzano: Study of science – Relations between propositions

Moving on to relations of propositions, Bolzano begins by noting, just like with representations, that there are no two completely equal propositions in themselves. Again like with representations, he continues, propositions in themselves can still be similar in the sense they have so many commonalities that they can be easily confused with one another. Bolzano points out that propositions with similar representations are not necessarily similar and that similar propositions need not have very similar representations. For instance, “some As are B” is similar to “representation of an A that is also B has objectivity”, although the representations in them are not, and “Sun lights the Earth” and “Earth lights the Sun” have similar representations, but are not similar propositions.

Just like with representations, Bolzano states, we can compare the content of propositions, that is, their constituents. He points out that all propositions share at least the copula “have”. Thus, to make the definition sensible, Bolzano takes affinity of propositions to mean that they share at least one constituent beyond “have”. Clearly, he points out, such affinity can have degrees, and propositions with affine representations are affine, and affine propositions have some affine or same representations.

Bolzano goes on to define some types of affine propositions. Some of them, he says, can have the same subject or predicate or a contradictory subject or predicate. Bolzano particularly defines converted propositions as such that are otherwise completely same, but two representations have switched their places. Such switches can occur in different places of the propositions, but Bolzano defines as perfectly converted the propositions “what has a has b” and “what has b has a”; technically “a has b” and “b has a” are even more converted, he admits but it is so rare to find two characteristics that characterise one another. Bolzano defines as conceptually converted two propositions “representation of A that has the characteristic b has objectivity” and “representation of B that has the characteristic a has objectivity” – in other words, this is the familiar relation of “some As are Bs” and “some Bs are As”. He also mentions the traditional notion contraposition as the relation between contraposed “what has a has b” and contrapositing “what has not-b has not-a”.

Bolzano notes that the extensions of the propositions bring forth new relations. Still, he passes these quite quickly, since the extensions of the propositions were defined by the extension of the subjects of the propositions so that the relations really introduce nothing radically new. Bolzano also adds that similar relations could be defined in terms of extensions of predicates.

What Bolzano considers the most important relations of propositions come up with the use of variables, introduced with the notion of validity of propositions. What the notion of variables allows Bolzano to do is to introduce in modern logic very familiar analogies between propositional logic and algebra of sets (in Bolzano’s logic, relations of extensions of representations). For instance, he defines compatible propositions as such that, when taking the same representations in them as variable, some choice of these variables makes all the propositions true – similarly, incompatible propositions are then such that taking the same representations as variable, no choice of these variables makes all propositions true. Clearly the notions of compatibility and incompatibility of propositions are analogical to the compatibility and incompatibility of representations, and indeed, one could be defined in terms of the other – propositions are compatible, if the collections of representations making them true are compatible, and representations are compatible, if propositions saying that an object represented by one of these representations are compatible.

Yet, there is a catch in Bolzano’s definitions: the relations of compatibility and incompatibility depend on the choice of variables, so that propositions that are compatible with one choice could be incompatible with another choice. In practice, this dependence on the chosen variables does not usually affect that much of Bolzano’s discussion, just as long as the chosen variables are always the same.

Bolzano goes on to describe simple features of these new concepts, such as that if a proposition is compatible with another and this with a third, the first and the third might not be compatible (compatibility is not a transitive relation, we would say nowadays). What makes his discussion a bit more complex is that instead of propositions he is usually discussing collections of propositions, and as he notes a few times, what holds for individual propositions might not hold for collections of propositions (this is more evident with notions a bit more complex than just compatibility and incompatibility).

Bolzano shows special interest in propositions of the form “X has a” or “A has x”, where either the subject X or the predicate x is the only variable. Thus, he notes, propositions with predicates as variables – say, “A has x” and “B has y” – are compatible, just as long as their subjects refer to some objects: we just have to pick suitable predicates that fit with the kinds of objects. Similarly, propositions with the subjects as variables – “X has a” and “Y has b” – are compatible, just as long as their predicates are characteristics of something: we just have to pick suitable subjects that have these characteristics. Then again, Bolzano points out, the case is somewhat more complex, if the propositions had the same subject that was then taken as variable – for instance, with “X has a” and “X has b” – since such propositions are compatible if and only if the predicate representations are compatible or are characteristics of some shared objects.

So, what are the other similar notions discussed by Bolzano. Well, he defines derivability of a collection of propositions, in analogy with comprehension of representations, as a relation where the derivable collection of conclusions is always true, when the collection of compatible propositions or premisses from which it is derived is also. Here we find an example of the complexity involved with Bolzano speaking of collections of propositions, instead of just individual propositions. Thus, an individual proposition can never be derivable from both an individual proposition and its negation, Bolzano points out, unless it is formally true, that is, always true with this particular choice of variables. Then again, if a proposition is derivable from a collection of several propositions, it could be derivable, even if some or even all propositions would be replaced by their negations: for instance, conclusion “B and C are not equivalent” is derivable from premisses “every A is B” and “it is false that every A is C”, but also from their negations.

Bolzano also defines the stronger relation of exact or adequate derivability, where none of the premisses or their constituents could be dropped without removing the relation of derivability. In this case, he points out, neither the conclusion nor any of the premisses can be formally true and no premiss is derivable from other premisses. While derivability is a transitive relation, exact derivability is not, although it has particular instances of transitivity. Thus, Bolzano can define composed derivability as a relation where the premisses of an exactly derivable conclusion are again exactly derivable from further premisses and the final conclusion is then also exactly derivable from these more ultimate premisses. Then again, he thinks, there are also cases of simple or non-composed derivability such as a proposition “all A are C” being derivable from “all A are B” and “all B are C”.

Further relations of this kind are equivalence (collections of propositions being derivable from one another), subordination (one-sided derivability), independence (collections of propositions being compatible, but not derivable from each other), exclusion (when all propositions of a collection are true, propositions of another collection are false), mutual exclusion (two collections excluding one another), contradiction (if all propositions in one mutually exclusive collection are false, all propositions in the other are true) and contrariety (mutually exclusive collections are not contradictories). None of these have anything surprising in them, so we can skip Bolzano’s discussion of them.

Bolzano moves on to discuss relations concerning not just what happens if propositions or their collections are true, but determining what propositions in a collection are true. Here, it can be a case of speaking about the truth and falsity of the propositions as such or as case of speaking about the truth and falsity of the propositions with some representations taken as variable – in the former case, the concepts are defined, Bolzano says, materially, and in the latter, formally, although most of the concepts make best sense in their formal meaning. The simplest example is when all propositions of a collection are true or all are false, but more interesting examples are discovered, when some, but not all propositions in a collection are true – Bolzano calls this a case of complementary propositions. Complementation, he points out, can be one-membered complementation or disjunction, so that only one proposition in a collection is true, and disjunction can be exact, when no proposition could be removed from the collection, without cancelling the feature that it has always at least one true proposition. Complementation might also be multimembered, if it has more than one true proposition, and in this case, it might even always have some exact number of true propositions. Finally, even if a collection of complementary propositions does not have any exact number of true propositions, its propositions might still be compatible, that is, all its propositions could be true in some cases. At the end of all this, Bolzano notes that all thus defined notions could also be conditional, that is, we could restrict our attention to such representations that make certain presuppositions also true.

Bolzano extends this notion of presuppositions to his previously defined idea of validity of propositions, calling it comparative validity and identifying it with the notion of probability. In principle, he suggests that we find out representations making the presuppositions true and then from these the particular representations making a proposition we are interested in true – the probability of this proposition is the number of the latter collection of representations divided with the number of the former collection. In practice, Bolzano admits, we have to make some limiting assumptions about the number of the cases, because the number of possible representations that could be used is infinite, due to all equivalent representations: basically, he restricts his whole discussion to rather simple cases of pulling balls out of a box. As always, Bolzano is interested to extend his discussion from individual proposition to collections of propositions, which he manages to do by proving that with suitable choice of variables and presuppositions, the probability of a collection of propositions is the product of the probabilities of the individual propositions

Bolzano discusses the relation of a proposition that is ground to a proposition that is its consequence. He notes that this relation does bear some affinity with the earlier notion of derivability, but is stronger: for instance, the proposition of weather getting warmer is derivable from the proposition of thermometer rising and vice versa, but only the former is the ground for the latter, not the other way around. Bolzano points out that the consequence relation can also be connected with the notion of probability. Thus, the presupposition of summer coming increases internally the probability of a proposition that thermometer rises, because the coming of summer is, together with other propositions, a ground for thermometer rising, while a weather report increases its probability only externally, because this report is no part of a ground of thermometer rising.

The final relation Bolzano considers is that of an answer to a question – or actually we have here many relations to consider. We may speak of an answer fitting a question, which is simply the truth desired in the question, if it is not absurd and has one. Furthermore, Bolzano notes, we can also define an answer given to a question as any proposition that someone suggests as the required truth. Such an answer can be correct or fit the question or incorrect, he adds, and while a correct answer must be a true proposition, incorrect can be true or false. A determined question, Bolzano points out, can have only one answer, or at least just many equivalent answers, while there are innumerably many incorrect answers, even for absurd questions. He also defines solution as an answer to a problem in the restricted sense: such a solution, at least if it is a correct one, will consist of rules, that is, of propositions expressing how a certain undertaking for a determined goal is to be effected.

Bernard Bolzano: Study of science – Different types of propositions

From general characteristics of all or most propositions, Bolzano moves to characteristics that could be used to classify different propositions. He begins by noting that no propositions are truly simple, because they all have subject, copula and predicate. Still, he points out, we can differentiate between simple propositions, where both the subject and the predicate are simple (copula is, of course, always simple “have”), and complex propositions, where at least one of the subject and the predicate is not simple.

Another of Bolzano's divisions concerns the question, whether some of the representations in a proposition are intuitions. Thus, he calls a proposition with nothing but pure concepts a conceptual proposition (conceptual truth, if it is a true proposition) and a proposition with at least one intuition an empirical proposition.

Bolzano notes that in his suggested form of propositions, the predicate is always abstract. Then again, depending on whether the subject is abstract or concrete, we could speak of abstract and concrete propositions. Bolzano points out that some representations are not abstract nor concrete (e.g. pure intuitions), so if such happens to be the subject of a proposition, this proposition is then not abstract nor concrete.

Bolzano lists types of propositions involving notions of collections, which due to their simplicity we need not go through in detail. Thus, he mentions e.g. collective and distributive propositions, with collective and distributive representations as their subjects (“ A, B and C together…” and “each of A, B and C…” being exemplary forms) which differ from propositions with collective and distributive predicates (both have the form “A has b and c”, but in one the predicate is understood collectively, in the other distributively). We might still mention Bolzano’s assertion of equality (“characteristic m belongs to objects A, B, C… together”), assertion of difference (“relation of A, B, C… to F, G, H… is such that the first have a characteristic m, the latter don't”) and determination (“characteristic m belongs exclusively to A, B, C…).

Moving on to propositions with negative representations, Bolzano thinks that since completely negative representations represent no object and thus no characteristics, they cannot appear as subject or predicate in any propositions. On the other hand, he continues, “something that is no A” can, so that we could at least have propositions with negative subjects.

Bolzano notes that often negation is said to be attached to copula, but he thinks it is actually the predicate that is actually then negative. True, he admits, language does often attach negation to the verb, but so it does with probability or necessity, which, according to him, do not characterise the copula, but the whole proposition. Thus, Bolzano concludes, negation seemingly attached to copula is actually attached to the whole proposition and indicates that the proposition is false. He justifies this statement with the example of a proposition with many objects in the subject: “all A do not have b” does not mean that for each A, they would not have b, but only that it is not true that all A have b. Even if the subject is singular object, we can say “A has no b”, which indicates a lack of characteristic b, that is, a characteristic of not-b, which Bolzano then takes as the defining moment of negative propositions, in difference from affirmative propositions.

Bolzano notes an interesting connection between negative and collective propositions. While propositions with collective and distributive predicates are equivalent when affirmative, they are not so when negative: “A has not the collection of characteristics b, c …” means not the same as “A has not any of characteristics b, c…”, because in the former case it still can have some of the characteristics in the collection. Generally all types of propositions have their affirmative and negative version, and if there’s nothing remarkable about their difference, I shall not mention them, even if Bolzano does.

From the standpoint of logic itself, Bolzano notes, it is remarkable that some propositions handle either representations or other propositions. If a proposition deals with a representation, he points out, its subject must then be a representation of a representation. A simple example is a proposition stating that a representation has – or has not – objects. Indeed, all the notions dealt in Bolzano's discussion of representations have their own corresponding propositions – for instance, we might assert a representation to be general or to be comprehend by another representation – and I will also skip this part of Bolzano's account as rather simple repetition.

Moving on to propositions about propositions, Bolzano notes that at this point he can deal only with propositions about characteristics of propositions, since their relations haven't been considered yet. In fact, he at this point mentions only the affirmation of A, saying that a proposition A has truth, and the corresponding negation of A, which says that proposition A has no truth.

Previous types of propositions appear in all sciences, Bolzano continues, but there are some that are useful, although they appear only in some sciences. His first example is the existential proposition that asserts or denies the existence of something.

An important type of propositions for Bolzano is formed by those describing mental phenomena, that is, effects caused by the soul, either within the soul itself or outside it. Two of such phenomena are already familiar to us, namely, subjective representations and. judgements. In addition to these, Bolzano mentions sensations of convenience or inconvenience, accompanying representations, wishes or desires, caused by judgements that certain objects would cause a sensation of convenience, volitions, which differ from desires by concerning what we should do, no matter if it is inconvenient, and actions, which are changes caused by volition on our soul or on certain other substances, primarily our organs and through them surrounding objects. All of these phenomena, Bolzano suggests, come with their own propositions, which may or may not assert the person e.g. sensing something.

An important subtype of propositions concerning mental phenomena is formed by propositions with the concept described by the German word Sollen (what ought to or should be done). Bolzano notes that the concept of Sollen properly applies only to actions or actually to decisions of will: each decision has a characteristic it should have. Thus, he defines an ethically good decision as such that is as it should be, whether it is a duty or just commendable decision, and the corresponding proposition he then calls an ethical proposition. On the other hand, Bolzano notes, propositions like “it should rain” use the concept improperly, expressing merely the uncertainty of what is asserted. A related notion is that what we may or are allowed to do, which can be defined as not something we should not do, and the corresponding proposition Bolzano calls an assertion of permission.

Another important subtype of propositions involving mental phenomena Bolzano considers are propositions about desires or problems (something is desired to be done). He is especially interested in problems that assert a wish for truth with certain characteristics, that is, in questions. More precisely, Bolzano explains, questions do not ask for truths in themselves (as he has said many times, these do not exist), but their appearance in mind as a thought or their linguistic expressions. He lists some subspecies of questions, such as questions for truth or falsity of a given proposition (e.g. is it true that God exists), questions about predicates for given subjects (e.g. what are the characteristics of triangles) and questions about subject for given predicates (e.g. who is the tallest person on Earth). An important subtype for sciences is formed of practical or technical questions, which Bolzano also calls problems in a stricter sense and which require truths describing how to reach a certain goal. Some questions, he notes, are determined in the sense that they correspond to only one or several equivalent truths, some are undetermined or correspond to many non-equivalent truths, while others are impossible or imaginary in the sense that they demand characteristics that are not to be found in any truth.

Bolzano goes on to mention some quite basic divisions of propositions: they can have a subject that represents no object or a subject that represents at least one object, and if latter, they can have only one object or many and maybe even infinite objects. He also points out the division that has already been mentioned many times, at least covertly, namely, that of true and false propositions.

Furthermore, Bolzano again reminds the reader that truth or falsity is an unchanging feature of a proposition, and if we appear to speak of propositions changing truth value, we are actually speaking of linguistic expressions that can change the proposition they signify, if they include words like now or this. He develops this idea by suggesting that we could think of some parts of propositions as variables, where the change of the variable part could generate propositions with different features – proposition referring to no object could become a proposition referring to some object or a true proposition could become false. If we determined the possible results of the variable, Bolzano notes, we could then measure how many of these variations are true and how many false. Thus, he defines validity of a proposition as a relation of the number of its true variations to the number of all variations: of course, this proportion is dependent on what parts are chosen as being variable. Bolzabo then defines a universally valid or formally true proposition as having the validity of 1, while a universally invalid or formally false proposition has then the validity of 0.

Related to this notion of validity, Bolzano points out that no proposition is formally true or false, if all its representations are taken as variables. Still, he thinks, it is of interest if there is at least one representation, such that taking it as variable, the proposition is formally true or false (for instance, “a human that is evil deserves no praise”, when the representation “human” is taken as variable). Bolzano decides to call such propositions analytical, while a synthetical proposition is then such that it has no representation that could be taken as variable so that the proposition would then be formally true or false. He then goes on to list some examples of very general analytical truths: identical or tautological proposition “A is A” – or as Bolzano prefers to say, “A has a” – “A that is B is A”, “A that is B is B” and “every object is either B or not-B”. He points out that all the examples he just listed were such that they are formally true, if all but logical parts (whatever that means – Bolzano admits that the notion is hazy) are taken as variables and decides to call them logically analytical propositions.

Another feature related to the notion of variables in propositions is what Bolzano calls a conversion of a proposition, where two representations within the same proposition change their place. If such replacing does not change the truth value of the proposition (for instance, if it is true both that Titus loves Cajus and that Cajus loves Titus), he calls the proposition convertible or reciprocable proposition. Bolzano then defines analytically reciprocable proposition as such where the changed representations could be anything without changing truth or falsehood, like in the proposition “A that is B is A”.

Bernard Bolzano: Study of science – What are propositions made of?

Having described various characteristics and relations of representations in great detail, Bolzano turns next to certain combinations of representations or propositions. He reminds the reader that we are again talking about propositions in themselves, not our thoughts of propositions or judgements. In Bolzano’s opinion, he has sufficiently justified the notion of a proposition in itself in his fundamental theory, so he can right away move to discussing their further characteristics. Just like with representations, he will begin with general characteristics of all or most of propositions, then move on to speak about different types of internal characteristics propositions and finally deal with different relations propositions have to one another. Somewhat differently from representations, Bolzano will then consider propositions involving such relations and in the last section propositions that have a linguistic expression which makes it hard to analyse their constituents.

Starting with the characteristics of all propositions, Bolzano begins with the most obvious one that propositions in themselves do not exist unlike our thoughts or assertions about propositions. He adds another quite familiar characteristic that a proposition is composed of representations, which makes it then sensible to define a content of a proposition as the sum of all its parts. What is perhaps not as evident is that propositions can be used to make indefinitely more new propositions, Bolzano notes: we can make representations out of propositions or of collections of them, and when something is said of this representation, we formulate a new proposition with the original ones as its constituents.

Bolzano is dedicated to the idea that every proposition is either true or false. True, he admits, a verbal expression might be true in one sense and false in another or its truth could be completely indeterminate, but this is not true of propositions in themselves. Furthermore, as Bolzano has pointed out earlier, every proposition is fixed to a particular time, so that it is either always true or always false, that is, incapable of changing its truth or falsity.

Bolzano states that at least all true propositions, and if not all, at least many of the others also, concern some object. The representation of this object, he continues, is then a constituent of the proposition, namely, its subject. Furthermore, Bolzano adds, if not all, at least many propositions assert some characteristic about this object, and the representation of this characteristic is the constituent of the proposition called the predicate. In addition to the subject and the predicate, he concludes, a proposition must still contain a third representation connecting the predicate to the subject or the copula, which Bolzano always takes to be the representation of having.

“A has b”, where A represents an object and b its abstracted characteristic, is for Bolzano a general formula of propositions. He admits that all propositions do not appear to fit into this formula, but thinks that on a closer analysis they can be seen to follow it. Thus, Bolzano argues, every proposition contains a verb, which always contains, according to him, the representation of having. For instance, “A works” can be analysed into “A is working” and this again further into “A has the characteristic of working”. Generally, Bolzano says, every proposition of the form “A is B”, where B means a concrete object with some characteristic b, can be turned into a form “A has b”. This is true, he insists, even of propositions with seemingly simple verbs like “A should”, “A acts”, “A wants” and “A senses”, which he turns into propositions “A has an obligation”, “A has activity”, “A has a want” and “A has a sensation”. A peculiar case is that of the propositions of the form “A is”, but Bolzano thinks they could be transformed into “A has existence”.

There are still quite difficult looking propositions, such as the traditional hypothetical and disjunctive propositions, and Bolzano is willing to ignore them for now and to deal with them later. Instead, he notes that the supposed inflections of the verb according to its person, number or gender are just a feature of natural language, which has the habit of repeating information, just in case the listener or the reader misses something, and do not concern the copula. Similarly, Bolzano says, the temporal determination of verbs – even when dealing with timeless entities, like numbers – is just another feature of the natural language, and in a proposition it is actually the subject that is determined by time (in other words, it always represented an object at some time – or then a timeless entity). Similar questions concern such additional determinations as “often”, “rarely” and “probably”, which he understands to relate the whole proposition to our capacity of knowledge or to other propositions. Then again, negation Bolzano considers to be a determination of the predicate (in other words, “that picture is not beautiful” means actually “that picture has a lack of beauty”).

Bolzano has now managed to define the concept of proposition through the concept of representation: it is a combination of two arbitrary representations through the concept of having, where having is the copula, that which has is the subject and that which is had is the predicate. Yet, he adds, there is nothing to guarantee that a representation is a simpler concept than proposition, since we could also define representations as constituents of propositions.

Bolzano has already mentioned that most and at least all true propositions speak of some objects. This makes it sensible to define, like with representations, the extension of proposition and to distinguish it from the mere quantity of this extension. In fact, Bolzano states, the extension of a proposition is simply the same as the extension of its subject, and indeed, the subject points out all the objects the proposition is about. On the contrary, extension of the predicate, he adds, is not the same as the extension of the proposition and it is quite indeterminate, which portion of this extension is handled by the proposition: for instance, “Caius has understanding” does not mean that Caius has every sort of understanding, but only that he has the kind of understanding he has, and which kind, is uncertain.

Bernard Bolzano: Study of science – Kinds of characteristics

The final section of Bolzano’s study of representations concerns relations representations have to other things, although Bolzano notes many of them have to be left later, when these other things are treated specifically. Still, he begins by defining a correct representation of an object simply as a representation that truly is a representation of that object and similarly an incorrect representation of an object as one that is not. Bolzano points out that correctness and incorrectness are only external characteristics or relations of representation to a certain object, so that we cannot speak of correct or incorrect representations in isolation, although he does make the exception that imaginary representations can be always called incorrect, because they are not representations of any object.

Bolzano continues by noting that a correct representation of an object can be complete in the sense that any characteristics of the object can be derived from the proposition stating that the object is represented by the representation and truths recounting the characteristics of the representation – other representations are then incomplete. He explains further that a complete representation does not need to contain all the characteristics of the object as its constituents – indeed, he adds, there would be an infinity of such characteristics. Bolzano points out that a representation of an individual object is always a complete representation: otherwise, some characteristics of the object could not be derived from this representation, so that we could have another object without these characteristics fitting the representation, which would then not be a representation of just this individual object. This means, he argues, that all intuitions are complete representations.

Furthermore, Bolzano notes, if there are no two actually existing objects with the exact same characteristics, all complete representations must be singular representations, because a representation with multiple objects would have to leave some of their individual characteristics undetermined. He also adds that just like with the notion of correctness, we cannot say from a representation by itself whether it is complete, but this requires comparison with the object: the same representation can be complete for one and incomplete for another object. Bolzano will go on to note the same thing for nearly all notions studied in this chapter, and indeed, it is rather obvious, since he is dealing with notions inherently dependent on relations of representations to other things.

Bolzano moves on to discuss especially notions related to characteristics. Characteristics as such are, of course, not representations, but features of objects, thus, he underlines in many places that we are here actually speaking of representations of characteristics: despite the importance of difference for explaining the inclusion of these notions in a discussion of representations, we can usually just ignore it and just speak of characteristics directly.

Bolzano begins simply by defining a correct characteristic as such that belongs to an object corresponding to a certain representation (other characteristics are, of course, incorrect). A more interesting notion is that of an essential characteristic, which he defines as characteristic that is correct for an object of a pure concept (other characteristics are then inessential). Evidently, the less objects there are in the concept chosen, the more the object has essential characteristics. Thus, Bolzano argues, for a singular and therefore complete concept, all the characteristics of the object are essential. He also suggests that if a characteristic can be expressed by a pure concept, it will be essential for some pure concept (for instance, the concept of something that has this characteristic).

Bolzano defines as absolutely proper or exclusive characteristic as one that belongs to this object and no other and contextually proper or exclusive characteristic as one that belongs to this object and no other in some kind, to which the object belongs. Characteristics that do not fit even the definition of contextually exclusive characteristic are then common or shared characteristics. Bolzano notes that for an exclusive characteristic, the representation “something that has this characteristic” is a complete representation of the object, since it represents only this object.

Bolzano goes on to point out that absolutely or contextually exclusive characteristics can be used for recognising the object that has them and could then be called its distinguishing feature (Kennzeichen). Such a distinguishing sign can then consist of several characteristics, which he calls marks, defining them as characteristics that at leas in connection with others are suitable for recognising an object. If a collection of such marks are enough for recognising the object, Bolzano states, they should be called sufficient marks, while otherwise they are object: if a single mark is sufficient by itself, it is then also a distinguishing sign.

Bolzano also divides marks into immediate and mediate marks, where a mediate mark is essentially a mark of a mark: for instance, speaking is an immediate mark of reason and thus a mediate mark of humanity, he explains. Furthermore, Bolzano defines affirmative mark as a characteristic that does not belong to all things of a kind, but to some of them, and negative mark or condition as a characteristic that belongs to all things of a kind, but not exclusively to them. The meaning of these definitions lies in that a presence of positive mark can always reveal that an object belongs to the kind, but its absence does not guarantee that an object does not belong to it, while conversely the absence of negative mark shows that an object does not belong to a kind, while its presence does not guarantee that an object belongs to the kind. In this sense, distinguishing signs of some kind can then be called both affirmative and negative.

Bolzano defines original or constitutive characteristic as such that its representation is already a constituent in the representation describing the object – other characteristics of the object are then called derived or consecutive. He notes that if the representation in question is a pure concept, an original characteristic is essential to that concept. Then again, Bolzano assures the reader, an original characteristic need not be internal, because some representation of an object can concern only its relations: Bolzan’s example is the representation of the middlemost rose on his window.

Bolzano considers the question whether an original characteristic can be used as a distinguishing sign of the object. He notes that if we are dealing with an original characteristic connected to a general representation, the answer is negative, since then the characteristic cannot be exclusive to the object in question. Then again, if we are dealing with a singular representation, Bolzano ponders, the question hinges on whether any of the original characteristics could be used to derive the others. He ends the discussion of original characteristics by pointing out that although representation of each characteristic of an object belongs to constituents composing the representation of the object, we cannot say that each constituent of its representation is original or even expresses characteristic of the object (a simple example is a representation of not-red that has red as its constituent, which still is not characteristic of the object).

Bolzano ends the discussion of representations with the notion of a difference between two objects by defining it as a characteristic that belongs to one, but not to the other object. Like all characteristics, he notes, differences can be divided into internal and external, essential and inessential, shared and proper, and original and derived. Furthermore, Bolzano differentiates a numeric difference between individual things from a specific or generic difference between kinds and quantitative differences involving difference of quantities from all other, which he calls qualitative differences.

Bolzano also considers the question whether all two objects (that is, not just two representations of the same object) differ by some characteristic. He notes the obvious answer that the characteristic of being this object and not the other does the trick, but then specifies the question to ask for internal differences and not mere relations. Bolzano himself believes, like Leibniz, that this is true of actually existing objects. He argues for this by asking the reader to picture two completely identical objects: in order to be identical, they would have to be surrounded by exactly identical things and experience the exact same events – otherwise, these things and events could change their features from one another – which all is extremely improbable. In any case, Bolzano concludes, the Leibnizian principle seems to be true about at least most things.