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.
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