13.1. Undoubtedly

(13.i)       how can we prove the truth of our cognitions?

is the basic question of every gnosiology. The well known sceptic answer is that if we try to prove the truth of our cognitions we meet with

- either a vicious circle (diallelus)

- or a regressum ad infinitum

- or some apodictically postulated truth.

(Münchhausen trilemma)

and that therefore such a proof is unattainable.

I think that a more constructive position is possible.

13.1.1. The coherence of the reality we live in, thence the coherence of the cognitions constituting any actual statute kact, is an irremovable presupposition of our knowledge. Yet a cognition about a world does not entail the objective existence of the same world. The powerful faculties of the human mind allow the creation of fictitious worlds and therefore the possibility of referring alethic predicates to the respective fictitious statutes (as, say, kmyt for the Greek mythology). This notwithstanding, a peculiar rank must be recognized to kact: in fact a fundamental asymmetry distinguishes it from any fictitious statute. I mean that, for instance, just as

(13.ii)       Socrates was one-headed

is kact-true because unquestionable evidence tells us that really there was a one-headed man named "Socrates",

(13.iii)      Cerberus was three-headed

is kmyt-true because the Greek mythology tells us that the portals of Hades were guarded by a three-headed dog named "Cerberus". Analogously, just as

(13.iv)       Socrates was three-headed

is kact-false,

(13.v)       Cerberus was one-headed

is kmyt-false. The acceptation of "true" (or "false") is the same in both cases: exactly as stating that (13.ii) is kact-true (that (13.iv) is kact-false) is stating that the piece of information under scrutiny is pro-collated (anti-collated) in the respective statute kact, stating that (13.iii) is kmyt-true (that (13.v) is kmyt-false) is stating that the piece of information under scrutiny is pro-collated (anti-collated) in the respective statute kmyt. The mentioned fundamental asymmetry is that fictitious statutes are, so to say, son-statutes of the actual mother-statute; in fact, for instance,

(13.iii)* in the Greek mythology Cerberus was three-headed


in the Greek mythology Cerberus was one-headed

are respectively kact-true and kact-false because unquestionable evidence tells us that the Greek mythology tells us that et cetera..

13.2. In spite of the peculiar rank we assign to kact, the real existence of the respective world is not absolutely sure: the ontological dimension of such a world can only be postulated. Personally I agree with this postulate, nevertheless a margin of arbitrariness survives. The super-ascertained coherence of the huge flow of information we are immersed in from the date of our birth renders probabilistically absurd thinking that the same coherence follows from a merely casual knowledge of a universal chaos. Yet none of us can a priori refuse the hypothesis that such a coherence is born by the coherent but deceiving intervention of a Superior Will. How could I peremptorily exclude that, one day or another, I should awake in an ultra-galactic laboratory, thus realizing I am only a sphere of intelligent matter stimulated by a lot of electrodes? All my life was only a virtual experience, and a smiling God in a bright uniform asks me: And so? What about the objective existence of the ‘real' world you believed to live in?

This non-absolute sureness, nevertheless, does not legitimate the scepticism in its canonical or modern version (Albert 1985, Chapter I, section 2), because (13.i) can be proposed even under such a non-absolute sureness.

Anyhow, for the sake of expressive simplicity, I will reason in compliance with the stone-age metaphysics represented by the most common realism (no ultra-galactic laboratory, no smiling God et cetera); and just in this restrained sense I will speak of an absolute truth.

13.3. In the current practice, the statute of reference is often omitted even where it is not the actual one. This elliptic habit may find a justification in the remark that, usually, the omitted reference is specified by the context. For instance we read (13.iii) as an elliptical form of (13.iii)* because the reference to kmyt is contextually clear. Some ambiguities may arise where actual and fictitious statutes overlap. Let me suppose that, according to the official historical report,

(       Vae conspiratoribus!

were the first Marco Antonio's words after Caesar was stabbed to death. Therefore ( is the right answer for the student questioned by his history teacher about those first words. But since, according to Shakespeare's tragedy,

(13.vii)       Friends, Romans, countrymen

were the words under scrutiny, (13.vii) is the right answer for the same student questioned by his English literature teacher. No contradiction affects these two incompatible answers to the same question because the two teachers refer to different (and locally incompatible) statutes as kact and kShakespeare are. Here too, obviously, any ambiguity disappears as soon as the existence of two overlapping statutes is focused on: in fact

According to Shakespeare's tragedy "Friends, Romans, countrymen" were

the first Marco Antonio's words after Caesar was stabbed to death

is an unquestionable kact-truth.

13.3.1. Henceforth I put aside fictitious statutes (owing to their reducibility), therefore the index "act" can be omitted. Nevertheless, since different knowers or even the same knower at different moments may have a different knowledge of the actual world, wherever there is the risk of a potential ambiguity, the punctual reference will be specified through indexes like "g" and "t". This subordination, far from being a theoretical limit, is a weapon to refine our analyses wherever, say, a kg°t°-truth must be distinguished from a kg°t'-truth or from a kg't°-truth. Let me quote §8.8.4: contexts where one of the two different statutes is the speaker's one … are peculiarly insidious. Anyhow this theme will be re-taken in Appendix 16.

13.4. An alethic procedure is always organized through the institutive, the propositive and the collative stages. The institutive stage consists basically in the sensorial perception of the reality we live in. Once we banish hallucinations (mirages, deliriums et cetera), the trustworthiness of our sensorial perceptions is assured. This notwithstanding current statutes (even mine!) include false cognitions. But how can a non-hallucinatory sensorial cognition be false?

13.4.1. My answer runs as follows. Although the interpretative process is normally unrealized because of its naturalness and immediateness, our usual cognitions are not the brute sensorial data, but the pieces of information inferred by our interpretation of such data. Therefore banishing hallucinations et cetera assures the trustworthiness of the brute perceptions, but of course it cannot assure the trustworthiness of the inferred pieces of information. The little spot I saw squatted beneath the bush is not a hallucination, yet if subsequent and unquestionable acquirements should show that it was a hare I would be compelled to admit that my previous identification was wrong.

13.5. Once the coherence of our informational flow (of our statute in progress) is assumed as the fundamental gnosiologic milestone which must be always preserved, acknowledging that sometimes a new acquirement results incompatible with our previous statute is acknowledging that, in order to restore the coherence, some correction (that is the substitution of some piece of information with its opposite) is necessary. Yet, in general, the coherence can be restored through different corrections; I claim that our choice complies with a rule (let me call it "criterion of minimal charge") shortly and intuitively expressible by saying that we must choose the simplest (the cheapest?) correction.

The plain theorization of this rule follows.

13.5.1. A characteristic rustle awakes me; I bet it is raining. I open the shutters and my opinion is confirmed: pines are trickling under a dark sky and the surface of the swimming pool is stippled by thousand of drops.

Let me assume "(degree of) reliability" as a synonym of "(value of) probability" (Carnap would speak of the degree of confirmation) peculiarly intended to emphasize the eventual presence of non objectively quantifiable components in the respective assignation.

A momentous part of our gnosiologic life consists in assigning a degree of reliability to hypotheses. Given a certain h, its degree of reliability, obviously, depends on the statute k; In the example h is *it is raining* and the acquirement k' obtained by opening the shutters, improving my previous statute (the rustle), increases the reliability of h, that is its probability (P(h|k°&k') > P(h|k°)).

13.5.2. Let me start from the simplest possibility space, that is from a dilemma, in order to introduce a simplified version ®* of ®. If we represent the probability of h and of ~h (obviously linked by P(h|k) + P(~h|k)=1) on a unitary segment 0÷1, we get two symmetric points, whose absolute distance |Pb(h|k) - Pb(~h|k)| varies from 0 (the two opposite hypotheses are equiprobable, so both points are in the middle of the segment) to 1 (one of the opposite hypotheses is true and the other false, so the two points are in the extremes).

The quantity

ih = (ph-p~h)/ (1-| ph-p~h |)

(where "ph" is an abbreviation for "P(h|k)" et cetera) is called "import (of the h-correction)" or also, picturesquely, "cost (of the h-correction)". The import is a ratio, and since in its numerator the difference between the two probabilities does not occur in absolute value, the same import is a relative quantity (a negative cost is a gain). So

are some instances of the import function (whose asymptotic course is highly significant).

I claim that wherever we have to correct our statute, the rule to follow is: minimize the total import of the corrections (hence "criterion of minimal charge"). The rule, for instance, states that a correction concerning a surely true piece of information (ph = 1) entails an infinite cost (is a logical contradiction) exactly as a correction concerning a surely false piece of information (ph = 0) entails an infinite gain (is a logical necessity). Analogously, if two opposite hypotheses have the same reliability (ph = 1/2 = p~h) the import of the respective correction is null. All these instances comply with our intuition, Exactly as in ® a confirmation value is represented by the (relative) areal difference between the sector representing h under and the sector representing h under k°&k' (that is under k°&e), in ®* a confirmation value is represented by the (relative) distance between the point representing h under and under k°&k' (under k°&e). Of course any e validating (invalidating) h invalidates (validates) ~h. Before abandoning this topic I wish to remark that also an n-ary possibility space can be represented through ®*. In this case we have to deal with a unitary segment and n points whose distances from the origin have a unitary sum because they represent the probability of the respective piece of information. Obviously any distribution is dictated by the statute of reference; therefore a new acquirement increasing some distances will decrease the others et cetera.

13.6. Let me re-approach the matter through some easy examples.

Example 1. I am walking along a lonely and narrow road. Over yonder a white car is daringly overtaken by a very fast red coupé, whose driver waves at me when he passes by. I am pleased with the acuteness of my sight because the quick reflections in his windshield did not forbid me from recognizing my old friend Bob. In the meanwhile the white car arrives, its driver stops and gets out: he too is my old friend Bob! I feel dismayed: are there two Bobs? A hypothesis I reject: if Bob were a pair of persons, after sixty years of frequentation, sooner or later my proverbial acumen would have realized so unusual a peculiarity. A hallucination? I am not drunk, anyhow I pinch myself and I force him to pinch me. No hallucination. A mistaken identity? I control his features scrutinize his identity card and ask him some particulars that only Bob can know; at the end I must reject also the hypothesis of a mistaken identity. Was then the red coupé a hallucination? I remember the roar of its engine, the Doppler effect, the quake of the ground at its passage, the birds flying away and so on. Another hypothesis I must reject: too punctual and well organized was the chain of poly-sensorial perceptions. Indeed I am facing an incoherence born by some mistaken inference from absolutely trustworthy sensorial data. And I think that all of us would overcome such an incoherence by the same correction: the driver of the coupé was not Bob. A conclusion dictated just by the criterion of minimal charge. In fact

(13.x)       a red coupé passed me by


(13.xi)       the driver of the white car is not the driver of the coupé


(13.xii)       the driver of the white car is Bob

have so high a degree of reliability that to replace one of them with its respective opposite would be too expensive a correction. On the other hand the low reliability of

(13.xiii)       the driver of the coupé is Bob

makes its substitution with

(13.xiv)       the driver of the coupé is not Bob

a reasonable correction. In fact a tanned and moustached person resembling Bob might have deceived even the acuteness of my sight, while his casual gesture to chase a fly or to encourage so energetic an old walker might have deceived even the acuteness of my mind, inducing me to read it as the wave of an old friend. Briefly: as (13.xiv) is perfectly compatible with (13.x), (13.xi) and (13.xii), its replacement to (13.xiii) is the cheapest way out toward a restored coherence. And just because such a replacement is not only the cheapest way out, but even the only reasonable one, all of us would accept it spontaneously.

13.6.1. Anyhow the same procedure rules also less univocal contexts.

Example 2. The driver of the white car too passes by and waves at me without stopping, he too resembles Bob. As (13.xii) is no longer a sure cognition, I find myself in perplexity facing a possibility space partitioned in the three incompatible alternatives (13.xii), (13.xiii) and

(13.xv)       Bob is neither the first nor the second driver

each of them with its degree of reliability (the absurdity of the fourth alternative according to which Bob ought to be both drivers allows me to neglect it directly). The respective representation is then a circle partitioned in three sectors whose measures may be modified by any subsequent acquirement or consideration. For instance, as soon as I ponder that Bob is a prudent driver the reliability of (13.xiii) decreases, and it continues decreasing as soon as I ponder that hitherto Bob's reservedness (and parsimoniousness?) kept him far from ostentatious cars. And as soon as I remember that few days ago Bob sprained his ankle, the highly increased reliability of (13.xv) makes it a practically irreplaceable piece of information (that is, in ®: the highly increased area of the sector representing (13.xv) makes it the only partition of the circle).

Only some absolutely sure new acquirement contradicting some absolutely sure previous cognition could throw me into a deeply bewildering puzzle. But just the fact that after trillions and trillions of acquirements all the (relatively rare) situations affected by an apparent incoherence have been solved through a severe re-examination of the whole context and the detection of a mistake affecting the acquirement or the cognition (therefore their only presumed irreplaceableness), just this fact, then, induces me to accept as an incontrovertible datum the coherence of the informational flow constituting my mental life (quite independently of the ontological dimension of its source). In this sense the falsity of a hypothesis results from its incoherence with respect to the informational flow (§13.8 below).

13.6.2. I recall §6.13. Making reference to Example 1, let me represent the possibility space concerning the four t°-alternatives determined by applying the opposition between being Bob (=b) and not being Bob (b) to the two drivers (d1 for the driver of the coupé and d2 for the driver of the white car). Once excluded the possibility that both drivers are Bob (alternative 4) the t°-circle is partitioned in three only sectors whose areas, before the coupé passes me by, are determined, say, by some previous statistical pieces of information about the identity of the drivers running that road; so the measure of the alternative 1 (d1b & d2b) will be rather high et cetera.

When at t' the coupé passes me by, the sight of its driver modifies my cognitive situation increasing the probability (the measure) of the alternative 2 (d1=b & d2b). If I were absolutely sure that d1=b, I could represent my (new) t'-cognitive situation by shading the sectors 1 and 3. but if I am not sure, an appeal to a (new) t' diagram is necessary in order to represent through the re-partitioning technique the new assignation concerning the three mentioned alternative. In other words. Either we introduce a scale of shading intensities, or we must acknowledge that shadings can only represent the elimination of an alternative. In this sense shadings are an abbreviative technique only somewhere applicable, since the most powerful and universal representation is realized by a sequence of diachronic diagrams each of them ruled by the re-partitioning technique. Of course a hybridization of the two techniques is realizable in order to reduce the number of diagrams.

13.7. A paradigm where the opposition between cognitions and propositions is combined with the opposition between alinguistic (factual) and linguistic informational sources, comprehends four points. Opposing *factual* to *linguistic*, strictly, is an approximation, since the hearing of a voice or the reading of a text too are factual perceptions. This notwithstanding the opposition is right because a linguistic source, besides adducing the non conventional pieces of information concerning its material nature, adduces also further and conventional pieces of information, that is because the essential step does not regard the inference from the listening of modulated sounds or the sight of ink arabesques to their classification as meaningful expressions, but from the text resulting from such a classification to its meaning. Therefore, although the above examples privilege alinguistic sources, the analysis is valid even where the pieces of information are linguistically mediated, that is where they result from a semantic interpretation.

Example 3. Like Example 1, except that the driver who gets out from the white car is Tom, who tells me: "Did you see? Since he fell in love and bought a Ferrari, Bob has become a risky driver!". In a situation like this, not only is (13.xiii) no longer an incompatible assumption, but its reliability is increased by Tom's comment, that is by a completely separate phonic utterance.

Example 4. Like Example 1, except that my wife is walking with me (better: I am walking with my wife). When the red coupé passes us by, I do not recognize its driver but I hear her exclaiming "It is Bob!". The piece of information (13.xiii) is now inferred exclusively from the semantic interpretation of sounds modulated by my wife's voice; the reliability of (13.xiii) depends on the confidence I have in her sight (and her sincerity). Of course if I believe her, when the white car stops and Bob himself gets out, the criterion of minimal charge compels me to conclude that sometimes we confide excessively in our wives.

Example 5. My wife's exclamation (Example 4) is verified by Tom's comment (Example 3). I only deal with uttered sounds, but in the end I believe (13.xiii) et cetera.

13.8. Let me insist (§13.4.1). The eventual occurrence of linguistic sources (then of semantic components in the interpretative stage) is a primary factor for emphasizing the distinction between the institutive and the propositive stage; nevertheless it is a secondary factor with respect to the alethic procedure because this procedure deals with a collation between pieces of information, quite independently of the way in which such pieces of information are obtained (a conclusion confirmed also by the applicability of alethic predicates to fictitious statutes). The fact that the core of an alethic procedure is not influenced by the informational source strongly evidences the short-sightedness of a merely linguistic approach to logic. Although the usual objects of alethics are the pieces of information drawn by interpreting linguistic texts, the linguistic component is not essential for alethics.

13.8.1. Where the informational source is non-linguistic, a clean distinction between brute sensorial data and inferred pieces of information is often hampered by the naturalness and immediateness of the same inferential stage. For instance what I wrote in §13.6 ("over yonder a white car is daringly overtaken ...") does not describe my sensorial experience; what I wrote describes my interpretation of a red and a white spots moving on a grey line. I firmly believe in that interpretation because all of my subsequent perceptions confirmed it: in fact the red spot over yonder became larger in conformity with its presumed speed and with the laws of perspective, the noise I heard corresponded to the roar of a speedily approaching powerful engine, and so on. The initial interpretation is progressively legitimated by the immense quantity of subsequent sensorial data which can be coherently inserted into the same interpretation, that is by the lack of a credible alternative interpretation. What should I have thought? that the red spot was a coupè-shaped horse whose whinnies sounded like a powerful engine and so on?

The vineyard in the fog (§5.6.1) is a perhaps more punctual example showing what I mean by "immediateness of an interpretative stage". The sight of a near and moving shadow is the sensorial datum, the awareness of my perfect clearness of mind (everything is relative) legitimates my convincement that such a datum is not a hallucination, its interpretation as a bad giant is the inferred piece of information. Hence the sensation of impending danger and my reaction. Yet subsequent and absolutely reliable acquirements incompatible with

a gigantic, anthropomorphic and hostile figure is threatening us

make it an untenable hypothesis; therefore its substitution with

(13.xvi)       no gigantic, anthropomorphic and hostile figure is threatening us

becomes a necessary correction. Analogously subsequent and absolutely reliable acquirements make

(13.xvii)       a bee-master overalls hung on a stake are tossed by a gust of wind

a further and absolutely reliable hypothesis whose perfect compatibility with (13.xvi) restores the perfect coherence of my whole experience (indeed (13.xvi) does not imply (13.xvii), the former is achieved before recognizing that the tossed cloth is a bee-master overalls). As for the ridiculousness of my instinctive reaction, it follows from the roughness of mistaking overalls and bad giants, two rather unlike categories of individuals, particularly with respect to their faculty of threatening.

13.8.2. A specification about *alinguistic*. The alinguisticity I spoke of in §5.6.1 concerns the practical impossibility of a linguistic mediation for my initial inference from the sensorial datum. The alinguisticity of Example 1 and Example 2 concerns the (factual) way in which the various pieces of information are obtained without conditioning eventual linguistic factors in the inferential procedure.

13.8.3. The assumption according to which hallucinations et cetera are banished is not theoretically reductive, since not banishing them would only mean admitting a subordinate order of possible mistakes. In fact and by far, the usual mistakes we (at least: we sober) fall into concern the interpretative stage, not the perceptive one (as for instance the fading out of the red spot I interpreted as a coupé over yonder).

Example 6. Like Example 1, but I am under a heavy LSD-effect. The red coupé, instead of passing me by, takes off and fades out in a flash. Yet as soon as my clearness revives, the recollection of my previous LSD-assumptions supplies a coherent explanation enlightening the pleasant hallucinatory phenomenon too.

13.9. Strictly, since everyone can likewise utter fallacious and veracious sentences, the intrinsic reliability of the piece of information inferred from a linguistic source (that is the reliability of the proposition we get from the interpretation of the sentence before collating it with our statute) ought to be null. Yet this is not the current case because, usually, the context supplies a meta-information suggesting us an a priori assignation.

For instance I did not witness Caesar's murder, and by hypothesis neither did I see any indisputable Hollywood movie on the topic, nor do I know how he died; therefore when I read in a celebrated encyclopaedia

(13.xviii)       Brutus stabbed Caesar

I cannot collate (the proposition adduced by) (13.xviii) with its homologous cognition in my statute. Nevertheless I assign a very high degree of reliability to (13.xviii) uniquely on the basis of the authoritativeness I recognize to the speaker as having. In fact its falsity would entail the correction of many nearly incorrigible ‘meta-cognitions' about the trustworthiness of the propositions warranted by the same celebrated encyclopaedia et cetera (or should I believe in a world-wide conspiracy aimed at deceiving my opinion about Caesar's death?). In contexts like this the standard procedure (reading the sentence, understanding the adduced proposition, collating it with its homologous cognition and on the basis of such a collation assigning the respective degree of reliability to the proposition under scrutiny) is inverted: the degree of reliability is derived from the authoritativeness of the speaker.

Of course these reciprocal procedures can be (and often are) hybridized, so that the reliability of a new proposition results from the combination of our previous cognitions and the authoritativeness of the speaker.

13.10. The millenary wisdom of natural languages grasps spontaneously the similarities among very heterogeneous referents of alethic predicates. For instance, there is a strong reason why *false*, besides its official acceptation concerning propositions, can be properly attributed to banknotes, to keels and so on. These attributions follow from the fact that in all cases something is false (fallacious) iff it adduces a piece of information incompatible with the statute (quite independently of its eventual apparent compatibility).

Therefore the falsity of a banknote follows from the accurate reproduction of the authentic ones; the banknotes of Monopoli are not false simply because, as they do not counterfeit anything, none of us is induced to think that a $100 Monopoli banknote is an actual one. Yet the deceptive component characterizing a ‘truly false' banknote is unessential. A false keel is a bar attached to the actual one (to the true one) in order to improve the nautical performances of the vessel, not in order to deceive the knower, its falsity follows from the fact that an incompetent person is induced to think that it is the spine of the hull, which it is not.

13.10.1. The criterion of minimal charge and the condition of confirmation (§12.2)

Wk', k°,h = P(h|k°&k') – P(h|k°)

are strictly related. In fact if k' validates (invalidates, is irrelevant to) h, by definition (§8.5.2) it increases (decreases, does not change) the import of the h-correction. In other words, the gt-reliability of h is increased by the admission in kgt of h-corroborating acquirements because as soon as ~h takes the place of h, such acquirements too must be corrected, thus incresing the cost of the correction. On the other hand the gt-reliability of h cannot be increased by the gt-lack of h-invalidating acquirements as this lack may follow from the gt-ignorance of such invalidating elements. And indeed our actual statutes are affected by many mistakes we shall never be aware of.

13.10.2. Incidentally. The more powerful the elaborative faculties of a mind, the more meaningful *false* is; in fact, once the coherence of the informational source and the non-hallucinatory character of the sensorial perceptions are postulated, the falsity can only concern the inferences drawn by the knower. And the more articulated is an inference, the more it is probable that some of its steps are misleading.

13.11. On the basis of the above mentioned homogeneity between the two terms involved in a collation (both cognitions and propositions, obviously, are pieces of information) the scholastic aphorism

(13.xix)       Veritas est adequatio intellectus et rei

finds in

(13.xx)       Veritas est adequatio intellectus et intellectus

an enlightening paraphrase.

13.12. It is easy to detect steady points of contact relating my approach with each of the three main alethic theories. In fact

- the intrinsically relational nature of alethic predicates presupposes a collation, which is to say a correspondence between two informational entities;

- the same opposition between *k-true* and *k-false* is the opposition between *k-coherent* and *k-incoherent*;

- the pragmatic functionality is the compatibility with the flow of information.

These points of contact do not follow from a pre-programmed syncretism. They are only the consequences of a completely autonomous approach (the informational one) which, moreover, is free from the difficulties affecting the other three. I mean that

- as soon as, in accordance with the substitution of (13.xx) to (13.xix), we recognize that the correlata linked by the correspondence are not heterogeneous (as facts and thoughts), but homogeneous (as cognitions and propositions) the nature of the same correspondence is no longer problematic, since it reduces itself to an implication;

- to say that a piece of information may be coherent (true) with respect to many statutes is to say that some lack of information affects the collation; yet as soon as such a gap is progressively reduced through further acquirements, the number of compatible statutes is also reduced, until singling out the right one

- since the truth of a proposition is its compatibility with the actual flow of information, the pragmatic functionality of every true proposition is necessary, and as such it can be assumed as an empiric criterion of truth.