Bernard Bolzano: Study of science – Inferences

No book of logic is not complete without some mention of inferences, and Bolzano’s is no exception. At worst, such an account devolves into a tedious listing of disparate specimens of inferences the author has deemed important. Bolzano is in danger of doing just that, but he does admit that such a listing can never be complete and tries at least to provide an inkling of systematicity to the proceedings.

Just to see more closely, what the method of Bolzano is in going through inferences, I shall give more details of the beginning of Bolzano’s account. He starts with what he calls the simplest kind of proposition: “A has b”. Note that by a lower case letter Bolzano always refers to a characteristic, while a corresponding upper case letter refers to all the objects that have such a characteristic. Thus, Bolzano’s simplest proposition means actually “all A have b”, and indeed, “all A are B” – the traditional form of universal proposition. Just like traditionally, the subject term has some objects to refer to, which Bolzano indeed takes as the first conclusion that can be drawn from this simple type of proposition.

After going through some equally simple conclusions that can be drawn from a single proposition of the simplest type, Bolzano moves on to discuss combinations of this type of propositions. To facilitate the discussion, he moves to an equivalent form “what has a, has b”. Bolzano points out that if we start with two propositions, they could share both representations a and b, only one of them or none.

Starting with the first case, Bolzano points out that the propositions could literally taken be only “what has a, has b” and “what has b, has a”, but he suggests adding the possibility of negating either a or b, leading to eight possible propositions:

1) what has a, has b,

2) what has b, has a,

3) what has a, has not b,

4) what has b, has not a,

5) what has not a, has b,

6) what has not b, has a,

7) what has not a, has not b and

8) what has not b, has not a.

The first proposition could always be taken as 1), because the important thing is the relation of the representations in the two propositions, and if needed, we can always say that not-not-a is equivalent to a. This leaves us seven other options for the second proposition, and even from these, we can take the options 3), 4) and 6), which cannot be connected with 1). Thus, we can infer from 1) and 2) that A and B are equivalent, from 1) and 5) that everything has b, and from 1) and 8) that B is not subordinate to A, but is also not equivalent to the representation of something in general (1) and 7) lead to various conclusions, which I’ll leave unmentioned for simplicity’s sake).

Bolzano goes on for quite a while with such a level of detail, although he eventually moves into a more summarised presentation. We do not need to follow him that far, but we can just note that next he goes on, as he promised, to discuss combination of propositions like “what has a, has b” and “what has a, has c”, sharing only one representation, which are, in effect, Aristotelian syllogisms using only universal premisses. He also considers the case where the propositions do not share any representations – “what has a, has b” and “what has c, has d” – the conclusion of which is that the whole combined of objects of A and B is a part of a whole combined of objects in B and D. Bolzano completes the study of inferences using only the simplest type of proposition with a consideration of cases with more than two premisses.

The next move Bolzano takes is to consider negations of type “it is false that A has b” (not to be confused with “A has not-b”, which would still imply that A refers to some objects, which the negation leaves open). Here we see the pattern Bolzano is making: he starts with inferences from a single negation, then proceeds to discuss inferences using both negations and affirmations of the first type and finally moves on to the case of several negations.

Bolzano follows the same systematic approach with propositions stating that a representation refers (or does not refer) to objects: he looks at individual propositions, then its combinations with propositions of the same or one of the previous types. The only interesting new feature is that in addition to propositions “representation A refers (or does not refer) to objects” he also considers propositions where the A has been analysed further, for instance, “representation of something that has characteristics a and b refers to an object” (these form the traditional particular premisses in Aristotelian syllogisms).

The next couple of types of propositions include such that indicate that a representation refers to one, several of a certain number of objects. What is interesting in these cases is that they imply further propositions about the sums of these sets of objects, at least if we know that they do not share any objects.

Bolzano moves on to discuss propositions about relations of representations, such as “representations A and C are compatible”. He notes that when we speak of mere individual representations, many of these propositions are actually equivalent to propositions considered earlier (thus, the just mentioned example says that some A are C), but his habit of speaking about collections of representations makes the discussion more complex. Thus, a proposition about comprehension between collections of representations (“every object that falls under one of the representations A, B, C… falls also under one of representations M, N, O…”) leads to a complex disjunction, where each individual object belonging to some representation of the first collection belongs to some representation of the second collection.

Bolzano has thus far discussed propositions about representations, but moves next to propositions about propositions, for example, the relations of derivability, equivalence and contradiction. What makes his account hard to follow is his use of variable representations to indicate that it is not individual propositions, but propositions of certain form he is discussing here. This feels an unnecessary complication at this point, when he is, for the most part, speaking only of inferences involving relations of propositions in general (e.g. variations of modus ponens and tollens), without indicating what the propositions are like (the only exception consists of such simple cases as premiss “if A is B, then A is D” leading to conclusion “every B is D”).

The final kind of inferences Bolzano considers involve probability, where the conclusion is not certainly true, but only of certain probability. Bolzano does not really dive deep into such probable inferences, but after some simple inferences (e.g. if probability of proposition is m, probability of its negation is 1 – m), he merely notes induction and analogy as important subtypes or probable inferences.

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.