consistency, truth and ontology

Studia Logica (2011) 97: 7–29DOI: 10.1007/s11225-010-9295-x © Springer 2010

Evandro Agazzi Consistency, Truth and

Ontology

Abstract. After a brief survey of the different meanings of consistency, the study is

restricted to consistency understood as non-contradiction of sets of sentences. The philo-

sophical reasons for this requirement are discussed, both in relation to the problem of sense

and the problem of truth (also with historical references). The issue of mathematical truth

is then addressed, and the different conceptions of it are put in relation with consistency.

The formal treatment of consistency and truth in mathematical logic is then considered,

with particular attention paid to the relation between syntactic and semantic properties

of sets and calculi. After the crisis of mathematical intuition and the dominance of the

formalistic view, it seemed that consistency could totally replace the requirement of truth

in mathematics, also in the sense that the existence of “objects” of axiomatic systems

could be granted by their consistency. A rejection of this claim is presented, whose central

point is a detailed analysis of the theorem that any consistent set S of sentences of first

order logic has a model. A critical scrutiny shows that this model is very peculiar, being

offered by the elements of the same language that is being interpreted, and the satisfia-

bility conditions for any sentence being constituted by the mere fact of belonging to S.

Though not being insignificant from a metatheoretical point of view, this theorem fails

to endow consistency (even in this privileged case) with an “ontological creativity”, that

is, with the capability of providing a model ontologically distinct from the language itself

(which is the precondition for the classical notion of truth that is also preserved in the

Tarskian semantics and model theory). A final discussion regarding the different “ontolog-

ical regions” and the referential nature of truth clarifies the different aspects of the whole

issue discussed.

Keywords: Consistency, non-contradiction, non-triviality, syntactic consistency, seman-

tic consistency, semantic completeness, analyticity, consistency and satisfiability, truth,

ontology, ontological regions.

1. What is consistency?

The terms “consistency” and “consistent” not only have a rather broad spec-trum of meanings when their literal translations are considered within differ-ent languages, but also have a not less significant variety of meanings whenthey are considered within one and the same language, depending on the par-ticular context where they are used. Limiting our attention to the context

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8 E. Agazzi

of philosophy, it seems correct to say that these terms do not belong to theset of the traditional or classical notions of the philosophical vocabulary,having been introduced in the metatheoretical discussions regarding formallogic and mathematics in the course of the twentieth century. Within thiscontext, “consistency” means something like “logical coherence” or “logicalcompatibility”, but these two notions remain still rather vague in them-selves, and their meaning can be made more precise by relating it with thenotion of non-contradiction. Following this path we can say that consistencyis the property of a set of propositions that are “mutually compatible”, inthe sense that none of them directly contradicts anyone of the others; butin addition it is usually required that no contradiction can be derived fromthis set of propositions, so that a “logically coherent” construction can beobtained by taking them as premises of correct logical deductions. Theclarification (or at least simplification) obtained through this characteriza-tion of consistency in terms of non-contradiction is useful, yet still in needof additional precisions. In fact, consistency is meant here simply as thesubstantive linguistically related with the adjective “consistent”, and in theforegoing definitions we have stated that this adjective properly applies tosets of propositions. One can ask, however, whether also a single proposition

could be consistent or inconsistent, and this is a not trivial issue (disputesabout “selfcontradictory statements” have occurred in the history of philos-ophy, and the “paradoxes” discussed by ancient and modern logicians arealso examples of this kind). Moreover, it seems legitimate to apply the no-tion of consistency also to concepts, and say, for instance, that the conceptsof a square circle or of a wooden stone are inconsistent or contradictory inthemselves. Discussions on such topics are far from being idle curiosities,and are parts of such serious philosophical problems as that of analyticity.

We shall not enter these issues for the moment, and single out insteada couple of points still in need of clarification. The first regards the verynotion of contradiction: when it is applied to sentences, it is usually un-derstood in the sense that the contradictory sentence of A is its “negation”not-A, but it is still not determined whether this “not-A” is the “simplenegation” or something like an “opposition”. For example, “p is red” and “pis not red” are contradictory sentences, and the same happens for “p is red”and “p is black”. But “p is red” and “p is sweet” are not contradictory, andthe difference of these situations cannot be spelled out without consideringthe “ontological reference” of p and of the predicates attributed to it in oursentences. This is an elementary example of the fact that the very notionof contradiction requires a logical analysis of the sentences and an explicitcharacterization of the logical operators in order to be properly defined, so

Consistency, Truth and Ontology 9

that two sentences may or may not be qualified as contradictory depend-ing on the definition accepted. The second point concerns the fact that a“consistent set of sentences” is usually understood as the one from whichno contradiction can be derived by means of a “logically correct deduction”.It is well known that the proof of such a consistency became the most im-portant concern of the scholars working in the domain of the foundations ofmathematics in the first decades of the twentieth century, particularly forthose who accepted the “modern” formalistic view of the axiomatic method,according to which a mathematical theory is the set of theorems that can beformally deduced from a set of primitive sentences whose only indispensablerequirement is consistency. In that perspective it was tacitly understoodthat the “correct logical deduction” was the one obtained by using some ofthe classical logical calculi which, in spite of admitting a variety of technicalformulations, were all united by the same commitment to satisfy the mini-mal requirement of soundness (i.e., to enable one to deduce from any set ofpremises only logical consequences of them) and hopefully also the maximalrequirement of semantic completeness (i.e., to enable one to deduce fromany set of premises all their logical consequences). This was the ideal thathad inspired logic from the times of its foundation by Aristotle on, until thecreation of modern mathematical logic, and which reflected the convictionthat logic, after all, aims at making explicit the “laws of thought”, and thata calculus deserves being called “logical” only if it is able to satisfy thisrequirement. This attitude prevented people from accepting the idea thatcontradiction might not be the absolute evil, and that, instead of pursuingthe desperate enterprise of totally eradicating it, it could be more advisableto try to live with it, by delimiting and reducing its sphere of action, but thiscould be done only by modifying the logic used for the deductions. In thissense it is possible to say that consistency becomes a property not of setsof sentences or of concepts, that can be defined in slightly different ways,but also a property of a logic that also admits of different formulations. Ifwe agree to call “strictly consistent” a logic that absolutely excludes contra-dictions, we can call “paraconsistent” a logic that operates with a limiteddomain of contradictions and keeps it, so to speak, under control. Whatthis precisely means can be summarized by saying that consistency has beenoften defined as “non triviality” in the following sense: a system of sentencesis consistent if not whatever sentence can be correctly deduced from it. Thisdefinition was in keeping with classical logic, in which a famous statement(sometimes called the “law of the Pseudo-Scotus”) affirmed “ex contradic-

tione sequitur quodlibet” (“anything can be deduced from a contradiction”).For this reason that statement is often called “principle of explosion” in

10 E. Agazzi

modern literature. Therefore, if it can be shown that from a given set ofsentences (e.g. mathematical axioms) even one single elementary sentencecannot be deduced, this set of sentences is proved to be consistent. Thisfact explains why equating consistency with non triviality was very com-mon at the beginning of the twentieth century, when such a strategy washoped to be successful in the consistency proof of formal systems. What ischallenging, however, is the following: to create a logic in which the law ofthe Pseudo-Scotus does not hold and, therefore, certain kinds of contradic-tions can be admitted without the undesirable consequence that anythingbecomes provable. Paraconsistent logics are of this kind and, in order toappreciate their sense and importance, it is useful to make a survey of thereflections on the nature of mathematical knowledge that eventually led togive to consistency a paramount importance and then to interpret it in away that prepared the emergence of paraconsistent logics.

2. The principle of non-contradiction

During the whole history of Western thought the principle of non-contradic-tion (PNC) has been considered as the soundest and indispensable principleof human knowledge. Its simplest classical formulation is the following: “it isimpossible to affirm and deny something at the same time and under thesame respect” (note that “affirming” is meant here as something more thansimply “uttering”, since it entails the expression of a “judgment” and, there-fore, the requirement of meaningfulness). In this formulation the principleis clearly a metalinguistic rule, indeed the highest metalinguistic rule sinceit states the basic condition for the existence of a declarative discourse, thecondition for “saying” something at all. In this sense it could be called “theprinciple of diction”, and Aristotle stresses this point when he shows (in thefourth book of Metaphysics) that even the one who wanted to deny this prin-ciple would make use of it, if he simply wants to state something, to declarehis position. This is often called the “logical form” of the PNC and it appearsas a condition of sense for any discourse. Its indispensable role, however,is reinforced when we consider the aim of a declarative discourse, that is,to state how things are, and here we find the “ontological formulation” ofthe PNC, “nothing can at the same time and under the same respect, beand not be”, from which the logical form of the PNC follows, since affirmingthat something is (the case) excludes affirming that it is not (the case), i.e.denying it. Let us now examine the classical definition of truth, consideredas a property of a declarative discourse: “true is the discourse that aboutwhat is (the case) says that it is (the case), and about what is not (the case)


says that it is not (the case)” while “false is the discourse that about whatis (the case) says that it is not (the case) and about what is not (the case)says that it is (the case)”. We have as an immediate consequence the rule“no (declarative) discourse can at the same time and under the same respectbe true and false”.

According to classical philosophy, knowledge in its proper and fullestsense had to be expressed in a (declarative) discourse (logos) that had tobe not only true, but also equipped with the reasons for its truth, thatwe usually call “justifications” today. These justifications were soon con-ceived as correct inferences performed according to truth-preserving proce-dures whose explicitation gave rise to formal logic already in the work ofAristotle. Though already in antiquity and in the Middle Ages logic had de-veloped through the proposal of a respectable amount of “laws” and “rules”,the PNC was considered the fountainhead of all of them, it was qualified asthe “first principle” (primum principium), and a perfect logical proof wasusually qualified as a reductio in primum principium, a “reduction to thefirst principle”. As a matter of fact, it is hardly possible to find in the writ-ings of the ancient logicians a formal derivation of the single valid logicalrules from the PNC; it is undeniable, however, that when they present a jus-

tification of such rules, reference to truth, and to the logical and ontologicalversions of the PNC are at work, at least implicitly.

We must consider, in addition, why the requirement of consistency inone of the above mentioned modern senses was essential also from the pointof view of classical knowledge, that is, the requirement that a set of proposi-tions accepted as the premises of a cognitive discourse (typically of a science)be such that no contradiction (i.e. “A and not-A”) could be correctly de-duced from it. The reason is that, since logical deduction consists in theadoption of truth-preserving rules, and since a contradiction is always false,it follows that if a contradiction could be deduced from a set of premises, atleast some of them must be false. But now we can also understand why thetraditional mathematicians and philosophers never engaged themselves insomething like a consistency proof of their axiomatic systems (a fact that issometimes presented as an evidence of the superior critical awareness of mod-ern mathematicians and philosophers of mathematics with respect to theirpredecessors). We must not overlook the fact that, during the history ofWestern culture until the middle of the nineteenth century, mathematicianswere convinced that the axioms of their theories were evident in the sense ofbeing true in themselves and absolutely such. Therefore, their consistencywas granted by the fact that no false consequences (and a fortiori no contra-diction) could be correctly derived from them. It was the change in the way

12 E. Agazzi

of conceiving the mathematical knowledge that gave to the requirement ofconsistency a different sense.

3. On the conceptions of mathematical truth

It belongs to the most widespread convictions of cultivated humans, withinthe most diverse past and present cultures, that those propositions that havebeen recognized as true in mathematics are absolutely true and endowed withcertainty. “2+2 = 4” has been always presented as the paradigmatic exampleof a “truth” that holds independently of any spatio-temporal situation, ofany psychological or sociological conditioning, of any cultural influence, thatis, therefore, universal and necessary. When we take into consideration,however, the sense attributed to this universality and necessity, and evenmore, the reasons that are advanced as a justification of such a conviction,differences and divergences become soon visible and they bring to light, inparticular, deep issues regarding the relation between thought and realityor, if we prefer, between thought and the existence of the objects towardwhich it addresses itself.

The most ancient point of view (and in a way the most spontaneous one)that supported this conviction was the admission of a more or less explicitontology of the mathematical entities. According to this view (that is oftencalled “Platonism” because Plato was its explicit founder, but has countedwith many followers until our days) the universality and necessity of themathematical truths depend on the fact that they concern certain simple,immutable, immaterial entities endowed with an autonomous existence “inthemselves”, that can be grasped in a kind of intellectual intuition throughwhich we are able to single out and describe in an immediate way theirmost elementary properties, and then proceed to discover more complexand concealed properties by means of the infallible tools of the logical rigor.This conviction was the ground for the axiomatic structure imposed on thedifferent branches of mathematics from the times of Euclid on, and whichhas dominated Western culture until the modern age.

But precisely in modern age a strong empiricist philosophical approachgained influence, within which the ontological presuppositions just men-tioned seemed very questionable, and a new ground for explaining the uni-versality and necessity of the mathematical truths was advocated. This newground was the thesis that mathematics are simply human constructions

and, as such, derive from something like a general tacit human convention:to the extent that we “pose” certain definitions and primitive propositions,we feel also compelled to admit with a value of universality and necessity,


everything that can be logically derived from these starting points. Butthis does not correspond to any “truth content” of mathematics, since themathematical propositions present themselves as true in virtue of a kindof convention and not because they express a true knowledge of some realentity. Not having their own objects, mathematics cannot be conceived asa system of “true” propositions in the proper sense of this term.

An intermediate position is represented by the conception according towhich mathematical propositions are purely analytic. In this sense, theyare considered genuinely “true”, however not because they “tell the truth”about any particular entities, but as a consequence of their logical form.

It lies outside the aims of this paper to make a historical reconstructionof these positions (and of their multifaceted variants) and we shall rathertry to analyze the distinguishing features of this traditional partition inorder to point out the different role that logical rigor plays for establishingmathematical truth in each of these approaches. More precisely, this “logicalrigor” will be conceived as the instrument that assures to mathematics theirexceptional internal consistency (i.e., their being free of contradictions). Insuch a way we are brought back to our main concern (the investigation ofconsistency) and our problem receives the more precise formulation of seeingwhether mathematical truth can be reduced or not to being identical withconsistency or non-contradiction.

4. Consistency and truth of the mathematical propositions

The three positions mentioned above agree that consistency is a necessarycondition for truth because, first, any contradiction is necessarily false andas such cannot be part of a system of true propositions and, second, becauseone of the warranties of mathematical truth is rigor, that “imposes” to ac-cept the logical consequences of accepted premises and this is tantamountto being “consistent” with our premises. These positions, however, disagreeon the fact that consistency is also a sufficient condition for truth: only theposition that we have called “analytic” (that within modern foundationalresearch on mathematics may be recognized, at least to a certain extent, inthe basic conception of the so-called “logicism”) maintains that consistencyis both a necessary and sufficient condition for truth in mathematics. Asto the other two positions, the first (Platonism) maintains that the sourceof any mathematical knowledge is a particular kind of intellectual evidenceor intuition and, therefore, consistency does not exhaust the conditions formathematical truth. The second (empiricism, in the particular sense ex-plained above) does not ascribe to mathematics a real “truth content” and,

14 E. Agazzi

as a consequence, admits that consistency, after all, is the only kind of truththat can be required for mathematics; it is a rather poor truth (a kind oftruth by convention) but we do not have other sources for crediting mathe-matics with a more substantial truth.

The last remark enables us to understand in what sense we can consider“formalists”, for instance, as representatives of such an “empiricist” attituderegarding mathematics. It could sound a little strange to classify them thisway and, in addition, one could also point out that the most importantrepresentative of the formalistic school (i.e. Hilbert) did not strictly identifymathematical existence with non-contradiction. Still one must recognizethat, though formalists admitted that some intuition is needed as a practicaland heuristic source of mathematical knowledge, all that really counts is thatthis intuition be rigorously formulated by means of exact axiomatic systems,and consistency was, in the last analysis, the only condition for acceptingsuch systems. Therefore, consistency was the radical condition for everymathematical truth and nothing else seemed worthy of being looked forbeyond it.

The distinction that we have sketched between the advocates of the om-nipotence of consistency and those who required additional sources for math-ematical truth seems still rather vague, since it makes use of such insuffi-ciently precise concepts as that of a particular ontology of the mathematicalentities and that of a special intuition on which mathematical knowledge isgrounded. A possibility of better clarifying this traditional discussion in thephilosophy of mathematics can be offered by modern mathematical logic thathas provided us with certain precise notions useful for analyzing our issue.Indeed we find in it some precise definitions for the notions of consistency(non-contradiction) and of truth for a set of mathematical propositions and,in addition, certain results regarding the mutual relationships between theserequirements.

5. The definitions of consistency and truth in mathematical

logic

The notion of consistency, as we have already seen, is typically syntactic, inthe sense that it concerns the logical form of sentences and the proceduresof derivation or deduction. In fact, the fundamental sense of consistency isthat of the absence of contradictions, and a contradiction is defined as thesimultaneous affirmation of A and not-A, independently of the meaning of A.This immediate application of the PNC occurs in the “indirect proofs”, or


proofs by reductio ad absurdum, in which in order to prove a propositionP it is shown that from the negation of P follows P itself, or the negationof an axiom or of an already proved proposition. In short, consistency wasconsidered, for reasons already explained, the necessary requirement of anyset of scientific propositions, in the sense that no contradiction could bederived from them. How can one be certain, however, that a prima facie

consistent set of propositions is really consistent, i.e., that a contradictioncould never be correctly derived from it? The fact of having deduced verymany propositions without encountering a contradiction is in itself no war-ranty that a contradiction will never appear. This issue became importantafter the construction of the Non-Euclidean geometries that were intuitivelymutually incompatible, but at the same time such that no contradictionhad been discovered within any of them. The strategy of “discharging” theconsistency of a theory T on the consistency of another theory T’ was ob-viously provisional since at the bottom some primitive theory had to befound whose consistency could be proved directly. At this point the law ofthe Pseudo-Scotus offered a precious help: if whatever sentence is derivablefrom a contradiction, it is sufficient to show that at least one single sen-tence (even artificially created) cannot be deduced from a set of sentences,in order to be sure that this set is consistent. This is why the conditionof “non-triviality” or “non-explosiveness” became the most usual way ofcharacterizing consistency syntactically:

Cons M =def ∃α [Not (M � α)]

We find in textbooks also another definition of consistency, that is calledsemantic, according to which a set M of sentences is consistent if it admitsa model, or is satisfiable (i.e. if it can be interpreted on an arbitrary set ω

of objects in such a way that its sentences become true propositions aboutthose objects).

Cons M =def Sat M

This definition is semantic because it essentially relies on the notion oftruth (via the more sophisticated notion of satisfaction), and simply reflectsthe old thesis that no false consequences can be correctly derived from truepremises; therefore, if a set of sentences can become “true of something”,even in a rather artificial way, no contradiction will be derivable from it,because a contradiction is never true. Traditional mathematicians thoughtthat their axiom systems were true in virtue of an intellectual intuition capa-ble of grasping the properties of the mathematical objects, and were thereforefree of the concern for consistency, but when Non-Euclidean geometries and

16 E. Agazzi

abstract algebras did not allow for such an intuitive support, the searchfor some more modest accessible model appeared as a good replacement,especially when the efforts of proving syntactically the consistency of math-ematical theories turned out to be little successful.

The details of this story need not be considered here, while it is worthnoting that the most interesting difference between the syntactic and seman-tic definition of consistency is that the second calls into play the existenceof a set ω of objects while the first does not and, in addition, can dispensewith the notion of truth. One could wonder why such features should beappreciated, especially considering that in the whole of our cultural tradi-tion mathematics have been considered as the domain of the uncontroversialtruth and as the sciences capable of leading humans to the knowledge ofthe most abstract entities, but one must consider, on the other hand, howmany difficulties had been encountered by mathematicians of the nineteenthcentury precisely in trying to determine the nature of such entities and thesense of the mathematical truths. Therefore, it seemed hopeful to find inconsistency the adequate tool for dispensing with these difficulties, not nec-essarily in the sense of “eliminating” truth and ontology from mathematics,but in the sense that consistency could be an adequate substitute for them,at least under certain conditions.

The traditional view that consistency is a necessary (and tacitly presup-posed) condition for the mathematical truth can be easily translated in theclaim that every set M of propositions that admits a model (in the aboveexplained sense) is consistent: this is an easy well-known theorem of math-ematical logic. The most challenging enterprise is to prove the reciprocalthesis, that is, that consistency is also a sufficient condition for mathemat-ical truth. Such a claim could sound rather innocent and almost a way forhinting at the “abstract” nature of mathematics, but it takes up a moreengaging sense if we translate it in the claim that every consistent set ofsentences possesses a model. In fact this formulation endows consistencywith something like an “ontological creativity” that is far from self-evident.There is no more the possibility of escaping the difficulty by saying thatmathematical truth is “abstract”, because it should be in any case a truthconcerning a certain structure of “objects” (possibly of abstract objects, butalways of objects). How can consistency be endowed with an ontologicalcreativity? How can a system of arbitrary, though consistent, conditionsensure that there exists in the world a structure of objects about which itturns out to be true?


6. Consistency and analyticity

The traditional view according to which mathematics constitute an “ana-lytic” science had found an interesting way for avoiding the difficulty of on-tological creativity. Traditional analyticity was in fact bound to consistencyin a peculiar way, that made consistency a necessary and sufficient conditionfor truth, in the following sense: a proposition was qualified as analytic if,first, it was non-contradictory (i.e. consistent) and, secondly, if it could notbe denied without contradiction (i.e., if its negation was inconsistent). Insuch a way analyticity was equated with logical necessity and, hence, ananalytic proposition was endowed with necessary truth. It is interesting tonote that such a truth was still vaguely conceived as having a relation withobjects and structures, but without being bound to the concrete indicationof anyone of them. A logically true proposition was considered to be “true inall possible worlds” and this, though having the appearance of linking truthwith a reference to certain structures (the “worlds”) exempted it practicallyfrom mentioning any world at all. This is why consistency, in its “strong”form equating it with analyticity was relieved from the burden of ontolog-ical creativity. The conception of analyticity just outlined was explicitlyadopted by philosophers such as Wolff, Leibniz and Kant, but it is inter-esting to consider the reason adduced by those thinkers in order to justifythat an analytical “judgment” could not be denied without contradiction:this reason was that in an analytic judgment the predicate is already (im-plicitly) included in the subject, such that it would be contradictory to saythat the subject does not contain the predicate. Indeed this justificationwas often presented as another equivalent definition of analyticity. Preciselythis justification, however, led Kant to maintain that analytic judgments,in spite of being endowed with universal and necessary truth, were unableto express genuine knowledge, because nothing new was said in them aboutthe subject since the concept of the predicate was already contained in theconcept of the subject. In particular, no reference to experience was neededin order to affirm them, because they expressed a purely logical relationbetween concepts that was valid a priori, that is, independently of any em-pirical confirmation. Therefore, said Kant, fullfledged knowledge must beexpressed through synthetic a priori judgments, in which the universalityand necessity of analyticity (a priori) is joined with the effective noveltyadduced by the “synthetic” contribution of a sensory intuition. Even in thecase of mathematics Kant maintained that this requirement was satisfied.

If we want to express this position using contemporary terminology, wecould say that we are in the presence of two different concepts of truth:

18 E. Agazzi

the notion of analytic truth is circumscribed to the sphere of sense (the con-

cept of the predicate is already included in the concept of the subject) andholds without relation to a referent, but only in virtue of the logical propertyof non-contradiction (consistency). The notion of synthetic truth is veryclose to the traditional view and requires that the sense of the judgmentconforms to the status of the “objects” that are in fact the referents of thediscourse. The difference with respect to the traditional definition of truthis that, for Kant, the objects are not the “things in themselves” but con-structions obtained by structuring the contents of sensory intuitions by thecategories of the intellect. This distinction was central to Kant’s philosophyand backed the distinction between thinking and knowing: not whatever thatcan be “thought” can also be “known”, or, to put it differently, “thoughts”are not “objects” and since a claim of existence usually regards the domainof objects, it follows that simple thoughts are not sufficient for stating theexistence of anything. Even in the most favorable situations (such as thatof consistency) simple thought is not endowed with ontological creativity.

Let us now briefly see how the above considerations apply to the dif-ferent trends in the foundational research on mathematics that we havealready sketched. Logical empiricists have essentially advocated the “ana-lytic” position regarding logic and mathematics and have even coined theword “tautology” to denote the sentences of these disciplines, that “do notsay anything new” and do not transmit any real knowledge. On the otherhand, thinkers such as Frege, who wanted at the same time to attribute tomathematics an analytic nature, but also the status of a genuine cognitivescience, were obliged to introduce a kind of additional ontology, according towhich logical entities have a “real” existence (which is a form of Platonism).We are not interested here in showing the shortcomings of the said positions,since we want to go back to our specific topic and investigate what mathe-matical logic has offered us in order to evaluate the “ontological creativity”of simple consistency, that is, not of the “strong” one of analyticity (in whichthe negation of a consistent set of sentences is inconsistent). In fact, thiscorresponds much better to what a “working mathematician” attributes tohis science: he normally does not believe that the axioms of a mathematicaltheory are “logical truths”, but simply that they constitute a consistent setof sentences. Therefore, they are not meant to hold in “all possible worlds”,but only in some possible world. Now our problem can be made precise inthe following form: once we have given a consistent set of propositions, canwe be sure that there exists at least one world in which they are true? Manymathematicians have been of this opinion, that is, that every consistent setof axioms of a mathematical theory has a model, but is this really the case?


7. Consistency and existence of models

7.1. Semantic completeness as a consequence of the satisfiability

of every consistent set of sentences

A fundamental theorem of mathematical logic states that if, for a givenformal language L, every consistent set M of sentences expressed in L hasa model (is “satisfiable”), then it is possible to construct within such alanguage a classical logical calculus that is “semantically complete”. Theproof of this theorem is simple. Let us write “Cons M” as an abbreviationfor “M is consistent”, and “Sat M” for “M is satisfiable”, and expresssemantic completeness in the usual form “M |= α ⇒ M � α”. The theoremthen says:

[For all M (Cons M ⇒ Sat M)] ⇒ [For all M (M |= α ⇒ M � α)]

Proof. Let us assume that M |= α. This means that every model ofM is also a model of α and, therefore, cannot be a model of ¬α. HenceM ∪{¬α} has no model, or is not satisfiable. Our hypothesis was thatevery consistent set of sentences is satisfiable; therefore M ∪{¬α}, not beingsatisfiable, is inconsistent. From an inconsistent set of premises any sentence(in particular α) is classically derivable; therefore M ∪{¬α} � α, from which,by elementary sentential rules, follows M � α.

This result already shows the weakness of the conception just mentionedabove, according to which every consistent set of mathematical propositionspossesses a model: if this were true we should conclude that all logicalcalculi are semantically complete, but we know that this is the case onlyfor the first order predicate calculi (including sentential calculi). Therefore,we should conclude that the ontological creativity of consistency can at bestbe envisaged in the case of systems of propositions formulated in first orderlanguages, whereas it could not hold for systems of propositions for whoseformulation a more powerful language is needed.

This conclusion is already sufficient for our problem: we were tryingto see whether the existence of mathematical entities could be consideredsuperfluous and replaced by the simple consistency of the systems of math-ematical propositions, though accepting the traditional notion of truth thatentails reference to “objects”. This could be possible if consistency were able,either to “generate” itself certain objects or to offer a warranty concerningthe existence of objects capable of “making true” the consistent propositions.We see now that this condition could obtain, at best, for particular systemsof propositions, depending on the language used, and this leaves us puzzled,

20 E. Agazzi

for one does not see how the existence of the mathematical objects could de-pend on the structure and richness of the language that speaks about them.But we shall see more in depth this issue by passing to the analysis of theprivileged case in which mathematical logic seems to indicate that consis-tency is endowed with an ontological power, that is, the case of first orderlogic, for which it has been proved that every consistent set of sentences hasa model. By examining this theorem we shall try to see whether it actuallyentails the ontological creativity that it prima facie entails.

7.2. Every consistent set of sentences of first order logic is satis-

fiable

This proof, proposed by Henkin in 1947, consists essentially in embedding anoriginal set S of sentences (supposed consistent), in a succession of larger andlarger sets obtained by adding new sentences and new individual constantsin ways that always preserve consistency. One obtains in such a way a setS∗ which is consistent and enjoys several properties, among which the factthat if it has a model, this will also automatically be a model of S. Thena model is found for S∗, that is also a model of S. We shall pass over thetechnicalities of this well known proof and shall simply recall a few moresalient points.

The final goal is to show that, for every S,

Cons S ⇒ Sat S

and this requires that a certain domain of objects ω and an interpretation j

on ω be found such that S has a model in ω, or is satisfied in ω, according tothis interpretation. This amounts to saying that, for any sentence α of ourlanguage, j satisfies α on ω if and only if α ∈ S. The original idea of Henkinwas to take as ω the language itself, but two conditions had to be fulfilled:

a) For any sentence αi, either αi ∈ S or ¬αi ∈ S. Indeed, if S were “toosmall” such that both α and ¬α were not included in it, the interpretationj would make “false” both of them, and this is incompatible with thesemantic definition of an interpretation. Therefore, before proposing aninterpretation, it is necessary to “increase” the size of the original S byembedding it in a much larger set S∗ that fulfils the said condition.

b) Any existential sentence ∃xα(x) belonging to S should be satisfiableeven if an infinite sequence of sentences ¬α1,¬α2, . . .¬αi . . . were alsopresent in S. In order this to happen the “linguistic pool” must be en-riched by adding, for every existential sentence of the mentioned kind,


an “exemplification” α(y) of it, in which a “not yet occurring” individualvariable y appears.

Obviously it must be shown that by performing these operations the initialconsistency of S is “inherited” by the newly constructed sets, and holds alsofor the maximal set S∗. This is granted by the familiar procedure throughwhich maximally consistent sets of sentences are obtained:

a) A first order language contains a denumerable set of sentences. It ispresupposed that a fixed enumeration of the sentences as well as of theindividual variables be established.

b) A succession of sets S0, S1, . . . , Si, . . ., is then inductively defined suchthat S0 ⊆ S1 ⊆ . . . Si ⊆ . . .

c) The maximal set S∗ is finally defined as the infinite union of the sets Si.

Here is the inductive definition:

S0 = S

Sj+1 = (i) Sj ∪ {αj}, if Sj ∪ {αj} is consistent(ii) Sj if Sj ∪ {αj} is inconsistent and αj is not a generalization(iii) Sj ∪ {¬α′

j(y)}, if Sj ∪ {αj} is inconsistent and αj is a generaliza-tion, where ¬α′

j is simply the expression of a “counterexample” of thegeneralized sentence αj, in which the “not yet used” individual variabley occurs.

S∗ =⋃

j Sj with 0 ≤ j ≤ ∞.

What we have to show is that S∗ is consistent.It is very easy to prove by induction that every set Sj is consistent. Let usnow suppose that S∗ is inconsistent; then for any α it would be S∗ � (α∧¬α)and this means that there is a finite sequence of sentences γ1, . . . , γn ∈ S∗

such that γ1 . . . γn � (α ∧ ¬α). Since every sentence γj is present in S∗

because it was present in a set Sj+1 and since the series of such successivesets increased monotonically, it is sufficient to consider the sentence withhighest index among our premises (let us call it γk) in order to determine acertain Sk that contained all these premises. Hence it would be Sk � (α∧¬α)contrary to the fact that all the sets Sj are consistent.

Moreover, S∗ is maximally consistent.We mean by this that it is impossible to add to S∗ any new sentence with-out producing an inconsistent set. This is a very peculiar property becausenot only our original set S, but in general all the sets of sentences we can

22 E. Agazzi

concretely consider are such that new sentences can be added to them with-out producing contradictions. Let us now consider an arbitrary sentenceα that does not belong to S∗. This sentence must necessarily be a certainαj according to our fixed enumeration and, if it does not occur in S∗, it isbecause it was excluded from Sj+1 since it would produce a contradiction ifadded to Sj .Therefore it would produce a fortiori a contradiction if added to S∗.

Other properties can be easily proved for S∗:

S∗ is “maximal” also in the sense that for any sentence α of the language,either α ∈ S∗ ¬α ∈ S∗;

no sentence can be deduced from S∗ that is not already contained in S∗;

all logical truths are contained in S∗.

For such an exceptionally gifted set S∗ a model is then offered, that will bealso a model of the original set S. The standard procedures for obtaininga model are: (i) that a “domain of objects” ω be provided; (ii) that aninterpretation j of the language of S∗ on ω be defined such that; (iii) amodel of S∗ can result, i.e. that the sentences of S∗ are satisfied on ω. Thisshould rigorously express, at least according to Tarski’s original proposals,the idea that the formal expressions of S∗ became true about ω under thisinterpretation.This domain is, in the simplest case, the set V of the individual variablesof L (or the set T of the equivalence classes of the closed terms of L in thecase of a first order logic with identity), and the subsequent steps of theinterpretation are articulated in such a way that functions and relations aredeclared to hold of their arguments if and only if the corresponding sentencebelongs to S∗.In the simplest case the interpretation j is defined as follows:

(a) For every individual variable xi ∈ V

j(xi) = xi

This means that every individual variable becomes, under this interpreta-tion, the name of itself. We shall, however, graphically stress the role of“object” played by the variable xi after the interpretation by writing it inbold: xi.

(b) For every n-place predicate variable P

j(P ) holds of (x1, . . ., xn) iff Px1, . . . , xn ∈ S∗

In order to show that the interpretation j so defined satisfies S∗ it mustbe shown that for every α ∈ S∗, j satisfies α. This can be easily proven by


induction on the construction of the sentences of S∗: for all atomic sentencesof the form P1, . . . , xn satisfaction immediately follows from the definitionof j. For composed sentences obtained by applying logical connectives andoperators to already satisfied sentences, satisfaction follows from the factthat such composed sentences also belong to S∗ (we omit such an elementaryproof). Since our original consistent set S is a subset of S∗, the interpretationj that satisfies S∗ (or, to say it differently, provides a model of S∗), alsosatisfies S (or provides a model of S).

The (undoubtedly genial) idea lying at the ground of this proof is totake as the structure on which the sentences of S∗ are interpreted the verystructure of the language of S∗. A spontaneous question, however, surfaces:which kind of “model” is this? Is it not too “artificial” and “selfreferential”?One can note that the procedure adopted here is similar to that not uncom-mon in abstract algebra when, for instance, a representation of an abstractgroup is obtained through permutations on the elements of the group itself,that are taken, so to speak, as representatives of themselves. In addition onecan also note that this semantic completeness, even if it is a little strange,does not hold (in general) for logical calculi beyond the first order and thisis certainly a not negligible information.

Still we cannot overlook that in the semantics of mathematical logicand model theory, as they have been started by Tarski and then remainedstandard, the domain of objects on which the interpretation of a formallanguage is made was considered as a structure different from the languagethat is going to be interpreted. This corresponds to the intention (explicitlyexpressed by Tarski) of respecting the fundamental characteristics of the no-tion of truth, which is the property of a discourse saying something about itsreferents and not about itself. The selfreferential features of the interpreta-tion of S∗, on the contrary, fail to respect this requirement since the domainon which S* is interpreted does not possess any structure of its own, butis structured according to the prescriptions contained in S∗ itself. There-fore, the situation obtained could be described, perhaps a little ironically, asfollows. Our theorem proves that any consistent set of propositions of firstorder logic describes a “possible world”. We ask “Which world”? and theanswer is “Of course, the world described by these propositions”! In moreserious terms we must recognize that even in the case of first order logic theconsistency of a set of propositions cannot secure the existence of a domainof objects “ontologically given” in an autonomous way, “about which” thesepropositions “tell the truth”. Hence, even in this privileged case consistencyis not endowed with ontological creativity.

24 E. Agazzi

If we want to speak of truth also in the present case, we could qualify it as“truth by simple consistency” or “weak analytic truth”. In fact, if one askedhow it is possible to “falsify” the propositions of S∗ we were obliged to answerthat such a falsification could never occur since they do not speak about astructure of independent objects and, on the other hand, they are protectedagainst the only possibility of linguistic falsification, that is inconsistency.Therefore, consistency is the only warranty for affirming the truth of ourpropositions.

8. Truth and ontology

In the foregoing considerations we have admitted two different meanings forthe truth of a declarative proposition (or a set of propositions). The firstis the classical one, going back to ancient Greek philosophy, according towhich a proposition is true if and only if what it says conforms with the wayof being of the entities it is intended to describe. In modern terminologywe call these the referents of the proposition and for this reason we cancall referential this conception of truth (that is also called, with a certainambiguity, the “correspondence theory” of truth). Tarski’s definition of truthfor the formalized languages (as he explicitly declared) wanted to be a wayof expressing this conception free of any risk of inconsistency. The second isthe notion of analytic truth that can be qualified as “truth by consistency”and which apparently dispenses us with the condition of referentiality. Wehave seen, however, that there is a kind of secret ambition or hope, thatanalytic truth is also endowed with referentiality, since consistency of a set ofpropositions should grant the existence of a model for them. In such a way wecould say that consistency plays the role of a criterion of truth, that is, of asufficient condition for truth, considered in its classical traditional referentialsense. We have critically evaluated this expectation and concluded that itis not really satisfied. Our reasoning has followed the traditional paths ofanalytic philosophy of logic and mathematics that, in particular, consider alanguage as a system of signs and not as something that expresses thoughts

(so that the interpretation of a language is conceived as an association ofreferents to the signs without the mediation of any sense). Our conclusions,however, will appear more significant if we translate them in that discourseregarding human knowledge that, as we have seen, has been the field inwhich the concepts of truth, non-contradiction, logical consequence havebeen elaborated and developed.

In this perspective, knowledge is the result of the thinking activity ofa subject S. Thinking is a bipolar activity, one pole being the thinking


subject’s mind, and the other pole being that which is thought, or the “con-tent” of this activity, that is, thought. Since one cannot think of nothing,the content of thinking (that is, thought) cannot be thought of nothing, andtherefore is thought of being or reality. This is possible because the subject S

is “open” to a reality largely distinct from it and to which it refers, i.e., it di-rects its attention. This capability of the subject is called intentionality andin virtue of it the “intended” objects are present to the subject in form ofrepresentations or thoughts, though remaining ontologically distinct from it.

Having qualified thinking as an activity of the mind, it follows thatthought has the characteristics of a product of this activity, a product that,as such, remains within the mind and, in this sense, may be said to be inter-

nal to it. However, this must be understood correctly, that is, not in a spatial

sense, but in a way similar to that according to which we say, for example,that a certain theorem is proven within geometry (or in geometry), or thata given issue is treated within ethics (or in ethics), that Hector is a Trojanwarrior in the Iliades. An easy way to avoid the pictorial-spatial flavour ofsuch expressions could be to say that thoughts have a mental nature, thata theorem has a geometric nature, that Hector has a literary nature, andso on. With the same caveat we can also use the word “content” (which iscommonly associated with the notion of being “in” or “within” something),and say that thoughts are contents of the mind, theorems are contents ofgeometry, characters are contents of literary works. We must pay attention,however, to a possible ambiguity: it may seem spontaneous to say that thefeatures of reality are the “contents” of our thoughts, but this would producethe confusion of attributing a mental nature to the whole of reality. There-fore it is advisable to use two terms and say, for example, that thoughtsare contents of thinking, while reality is the object of thought. One can findin this distinction a kind of generalization of Frege’s well-known distinctionbetween sense (“Sinn”) and reference (“Bedeutung”), and this would havethe advantage of making us recognize that thought, being a “content” ofthinking, lies “within”, or is “internal” to thinking, while reality, being an“object” of thinking, lies “outside”, or is “external” to it. But is this reallyan happy solution? If external and internal are not meant according to anaive pictorial-spatial characterization (and we have seen that they mustnot be intended this way), this would oblige us to say that, while thinkingand thought are distinct but not different (since they share the same mentalnature), reality and thinking (or thought) are different (they do not havethe same nature). This conclusion, again, seems obvious at first sight, butit may give rise to serious difficulties, and in particular to the absurd thesisthat thoughts are not real.

26 E. Agazzi

In order to avoid this consequences we must require that the meaningof reality not be unduly biased by some hidden limiting presupposition, inorder to leave it the most general latitude. Such a meaning may be equatedwith the general notion of being, such that reality encompasses everythingwhich is different from nothing. From this point of view, one must acceptour statement above: thought “cannot be thought of nothing, and thereforeis thought of being, of reality”. But at the same time one must recognizethat thought itself is different from nothing, hence it is a form of being, andcannot be “alien” to being, or reality.

But not being alien or separated from reality can mean that thought andreality are identical? This is notoriously the idealistic thesis, that one is notobliged to share provided that we proceed from the general notion of realityto that of ontology. This last derives from the consideration of the internalarticulations of reality and, in particular, from the awareness that there aredifferent kinds of reality (as Aristotle had already clearly recognized). Usinga terminology of traditional philosophy, one would say that the notion of“existing” or “being real” is not univocal, but analogical. For example, astone, an equation, a dream, a toothache, an hallucination, Hamlet, are allreal, since they are different from nothing and, moreover, we can think ofthem, speak of them, refer to them. Yet they do not obviously possess thesame kind of reality, and we significantly say that they are real, or thatthey exist, “within a certain world” (physical world, mathematical world,psychic world, perceptual world, fictional world, and so on). These different“worlds” can be most adequately called ontological regions, in the sensethat all of them are encompassed in the general sphere of reality, but havespecial connotations that determine the particular level, or kind of realitythey belong to. This is what the idealistic thesis has overlooked, havingunderestimated the ontological differences. In fact the “world of thoughts” issimply one among many ontological regions. Having overlooked this has ledidealists to the striking thesis that reality is the product of thinking. True,thinking produces thought, but one can maintain that thinking producesreality only if one equates thought and reality (and it makes no real differenceif, instead of individual thinking, we speak of a transcendental Thinking orSubject, or Reason).

The obvious question is: “how can we determine the different ontologicalregions”? The answer is: by means of the criteria of reference we must usefor attributing reality (their kind of reality) to objects, and this also amountsto attributing truth or falsity to the propositions that intend to speak aboutthese objects. If we must rely upon physical operations, we have to do with aphysical reality, if we must rely upon mathematical operations, we have to do


with a mathematical reality, if we must rely upon perceptual experiences, wehave to do with a perceptual reality, if we have to rely upon reading a poem,we have to do with a literary reality, and so on. Several of such operations orexperiences may be even of an intellectual character, but they are differentfrom pure thinking. For example, I can think that a golden mountain existsin the Alps, but I must find it through a geographical exploration that, inthe end, might show me that such a mountain does not exist in this part ofthe physical world. I can also think that Hamlet is a Trojan warrior in theIliades, but by carefully reading this poem I do not find this character in theIliades. In other words: whatever can be thought or conceived has a noematic

reality (it belongs to the “world of thoughts”), and the particular attributescontained in its noema indicate in which other ontological region it could

exist. But its actual (and not just possible) existence in that region must beestablished by means of referential tools different from pure thinking.

9. The ontology of thought and language

Thought, as we have seen, has an authentic reality, though it does not ex-haust the whole of reality. Let us note, incidentally, that the ontological real-ity of thought does not even coincide with the ontological reality of thinking,since thought is the product of thinking. Yet reality cannot be known outsidethought, and this simply because to know essentially means to make “inter-nal” to the subject things that are “external” to it (in the above explainednon-spatial, but ontological sense). This capital fact can be expressed bysaying that thoughts are primarily not the objects of thinking. Of course,they can be occasionally also the objects of thinking, when, through an actof reflection, we consider our thoughts, compare them, relate them, combinethem in a great deal of ways. This means that the realm of thought has(as any ontological region) its own features, structure and, perhaps, laws,and this vindicates a certain way of conceiving logic (in particular, Frege’sway). The study of thought (i.e., any investigation in which thought be-comes the object or the “intended referent” of the thinking activity) canbe done in different ways, one of which could be introspection, and anothervery useful one is the study of language considered as a tool by means ofwhich thought is expressed or rendered public. For this reason the questionwhether thought can “produce” external reality can be usefully translatedinto the question whether language can produce external reality and this isthe principal reason for which we have devoted a certain detailed attentionto the problem of the ontological creativity of sets of propositions that couldenjoy the highest privilege that is linguistically conceivable, that of being

28 E. Agazzi

consistent. What we have seen is that, even under the optimal conditions,language cannot generate an ontological model outside its own ontologicalregion, that is, a purely linguistic model.

How does the ontological region of thought come about? Till now we havesaid that thoughts are “produced” by thinking, but this is a really vague wayof speaking. The activity of thinking consists primarily in being “open” andoriented toward regions of reality that are different from thinking itself, thatis, towards certain objects. This openness, as we have already said, is calledintentionality in philosophical language, and it is thanks to this capabilityof the subject that things can be present to the subject (and in this senseinternal to it), though remaining ontologically distinct and independent ofit (and hence also external in a more radical sense).

10. Concluding remarks

A few statements can summarize the result of our investigation. There is asense in which it is possible to say that thought and reality constitute twodistinct realms, and therefore cannot be reduced to be the same thing, butpreserve a certain autonomy. Such a sense is expressed through statementslike these: “not everything that can be thought also exists”, and “not every-thing which exists is actually thought”. Common sense easily accepts thesestatements, but they must be carefully understood in order not to lead tomisunderstandings. They cannot mean that thinking and thought are notpart of reality. They are, but what is implicit in these statements is that“exists” is meant as to exist in those ontological regions of reality that donot coincide with thought. In this sense it is possible to say that thoughtand reality are distinct and not reducible to one another, provided that weare aware that this should be made more precise by speaking in terms ofontologies. As a consequence, the problem of the relation between thinkingand “reality” appears to be that of a conformity that cannot be taken forgranted. This is the problem of truth, that derives from the awareness thatthinking “produces” thoughts, but does not produce “reality” in its mostgeneral sense.

The way out of this difficulty (thoughts are real, but “reality” mightbe different) is offered by the analogical sense of “real” that we have out-lined, and which gives rise to an articulation of ontology. Thoughts arereal, but theirs is a noematic reality which (to use the Fregean terminologyalready mentioned) belongs to the ontological level of sense. Similarly, lan-guage is real, but its reality is different from that of thought (with which itcan be related through the operation of interpretation) and from any other


ontological region. Different is the level of referential reality (which includesseveral ontological regions), that is, of that reality which is the target of theintentionality characteristic of the acts of thinking. The difference and therelation between these two levels relies upon the availability of non-mental(or non-linguistic) tools or ways of access to the referents. These tools deter-mine those “regional ontologies” with regard to which the concept of truthis “relativized”, in the sense that the truth of a proposition is relative toits intended “referential domain”. In this way the relation between thinkingand being becomes the more precise relation between thoughts (the concretecontents of acts of thinking) and ontologies (that is, the delimited aspectsof reality that are actually intended by an act of thinking).

Evandro Agazzi

Department of PhilosophyUniversity of Genoa, Italyand

Department of HumanitiesUniversidad Autonoma MetropolitanaCuajimalpa, Mexico [email protected]

consistency, truth and ontology

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