1.The Singular Modalities.

2.The Plural Modalities.

3.Analogies and Contrasts.

4.Apodictic Knowledge.

1.The Singular Modalities.

I do not claim that my theory of logical modality as it stands solves all issues, but I think you will find it very productive, an impressive integrative force.

The concepts of ‘logical modality’ enable us to predict systematically all the ways credibility may arise in knowledge over the long-term. Credibility itself is not a type of modality, but the ground and outcome of logical modality. We shall immediately define the primary categories of logical modality, and thereafter discuss their development, their significance, and their justification:

Truthis the character of a proposition which seems more convincing than its negation, in a given context of knowledge. In the case of any proposition implied by its own negation, its credibility is extreme.

Falsehoodis the character of a proposition which seems less convincing than its negation, in a given context of knowledge. In the case of any proposition implying its own negation, its incredibility is extreme.

A proposition isproblematic, with regard to its truth or falsehood, if it seems to carry neither more nor less conviction than its negation, in the given context of knowledge. This is indicated by such expressions as ‘might or not be’ or ‘perhaps is and perhaps is not’.

In practical terms, thedegree of credibility, whether high, low, or median, of a proposition is a measure of the amount of evidence or counterevidence put forward on its behalf or against it. This refers to the weighting of information by confirmation or undermining, which topic will be dealt with more fully under the heading of adduction.

By (logical)contextis meant, the accumulated experiences and conceptual insights of the knower (a person or society) at the time concerned.

The context-specific concepts of logical modality are built on the awareness that: at every stage of knowledge, some things somehow seem ‘true’, other things somehow seem ‘false’, yet others seem ‘problematic’; and that these attributes often vary with the growth of experience and reasoning.

These observations suggest that, although every appearance is accompanied by some such characterization, the characterization is not in all cases firmly attached to the object, but is often a function of the experience and reasoning which have preceded them.

The concepts are thus formed, to begin with, only in recognition that such events occur, and that they are distinguishable by our consciousness, and that they each display such and such properties. Then we say: ‘Let us call this truth or falsehood or problemacy, as the case may be….’

It must be stressed that underlying the foregoing definitions of truth, falsehood, and problemacy, is the assumption that a sincere effort of awareness took place. It is difficult to insert such technical specifications in our definitions explicitly, without engaging in circularity, but there is no doubt that the definitions would lose all their value and significance without this tacit understanding.

A true or false proposition is called‘assertoric’, because it makes a definite claim. A problematic proposition is not assertoric: it presents an appearance with equal tendency in both directions, and therefore devoid of tendency; it calls upon us to consider a hypothesis.

Problemacy signifies a suspension of judgment. It does not signify the existence of ‘real’ indeterminacy, but only recognizes the appearance of indeterminacy in contexts less than complete. In reality, we believe, every issue is settled, once the event takes place; in omniscience, there would accordingly be no problemacy — it only arises in more limited viewpoints.

Problemacy has no equivalent outside logical modality; being freely open to change as knowledge evolves, there is no error in saying that any proposition we choose to formulate is at first encounter problematic.

Note that meaningful, precise, and clear, propositions may be true, false or problematic. Meaningless propositions are classified as false. Vague or obscure propositions, as at best problematic, if not false.

Factual assertorics of less than extreme credibility and problematics, give a semblance of co-presence or co-absence of opposites. The laws of contradiction and of the excluded middle are our reminders that that impression is transient; ultimately, everything is either totally credible or completely incredible. In other words, so long as we make no attempt to at once apply both truth and falsehood, or both untruth and unfalsehood, no law is broken; but as soon as we lay claim to more than the propositions suggest, we err.

For this reason, we can effectively discard nonextreme assertions and problems, and say of any proposition: it cannot be both true and false, and cannot be neither true nor false. There is ultimately no mixing or in-between of these attributes; our goal is to arrive to the extremes, not to linger on intermediate stages. There would be no point in constructing a logical system with reference to the finer gradations of credibility: it would be immobile.

2.The Plural Modalities.

Truth and falsehood are the categories of logical modalitywith a single, given contextas their frame of reference.

Truth is a category of logical modality lying between logical necessity and possibility. Falsehood is the exact contradictory of truth, lying between logical impossibility and unnecessity. Truth is fact and falsehood is fiction, ideally. So we may call them the‘factual’level of logical modality; in analogy to the actual level of natural or temporal modality, or the singular level of extensional modality; but this is only an analogy, not an equation.

The categories of logical modality referring toa plurality of unspecified contexts:

Logicalnecessitycharacterizes a proposition which is true in every context, and in that sense is true irrespective of any given context.

Logicalimpossibilitycharacterizes a proposition which is false in every context, and in that sense is false irrespective of any given context.

Logicalcontingencycharacterizes a proposition which has neither the attribute of necessity nor that of impossibility, as they are above defined, so that it is true in some contexts and false in others.

Logicalincontingencyis the negation of contingency, the common attribute of necessary and impossible propositions. Logicalpossibilityis the negation of impossibility, the common attribute of necessary and contingent propositions: truth in some contexts. Logicalunnecessityis the negation of necessity, the common attribute of impossible and contingent propositions: falsehood in some contexts.

With regard to corresponding concepts of logicalprobabilityorimprobability.

We can say that, in this system, truth or falsehood correspond to mere incidence or nonincidence; necessity or impossibility signify the extremes (100%) of probability or improbability, and contingency concerns intermediate degrees (less than 100%) of these. Thus, to be consistent, we must define the logically probable as what would be true in most contexts (or false in a minority of contexts), and the logically improbable as what would be true in few contexts (or false in a majority of contexts).

These concepts would then enable us to specify our breadth of vision — effectively, how many eventual changes of context we have taken into consideration in making a prediction. The practical feasibility of this, with some precision, and the relation of logical probability and credibility, will be explored when we deal with adduction.

Thus, in summary,logical modalitymay be defined as a qualification of propositions as such, informing us as to whether each is true or false, in this (i.e. a given) context, only some (unspecified) contexts, or all contexts, or somewhere in between these main categories.

Here again, it must be emphasized that ‘is true’ (meaning, seems more convincing than not) and ‘is false’ (seems less convincing than its contradictory), depend for their plausibility on our having sought out and scrutinized the available information with integrity. This issue is discussed in more detail in the next chapter.

I want to emphasize here thatthe concepts of logical modality, as here defined, are prior to concepts of logical relation, like implication, which (as we shall see) they are used to define.

The former are built on the vague, notion of a proposition being variously credible‘in’some context(s). Although this ‘in’ suggests that a kind of causality is taking place, it is not yet at the stage where specific relations like implication may be discussed. There is only a mental image of items ‘pushing’ others into existence; a very sensory notion.

Likewise, our first encounter with ‘credibility’ is very intuitive, something intrinsic to our every consciousness. The later systematic understanding of credibility, with reference to adduction, is merely a report on when it occurs, not a substitute for that primitive, inner notion.

It is interesting that, in Hebrew, the word for ‘with’ is‘im’(spelt ayin-mem), and that for ‘if’ is‘im’(spelt aleph-mem). In that language, if I am not mistaken, when verbal roots are that close, it signifies that the thoughts underlying them are also close. I wonder if the English words ‘in’ and ‘if’ have similar origins, rather than those most philologists assume.

Incidentally, also similar in Hebrew, are the words‘az’(spelt alef-zayin), meaning ‘then’ in time or logic, and‘oz’(spelt ayin-zayin), meaning ‘strength’. This confirms what I said above, that the notion of logical causality is rooted in an intuitive analogy to physical force.

3.Analogies and Contrasts.

Various analogies and contrasts between the singular and plural modalities are worthy of note. The former measure credibilities in any one context. The latter take a broader perspective, and compare credibilities in a variety of contexts. Thus, true, false, and problematic are comparable to necessary, impossible, and contingent — but they are not identical.

Contingent truth and falsehood are contextual, whereas necessity and impossibility (incontingent truth and falsehood) effectively transcend context. What holds in every context, holds no matter what the context, whereas the contextual is tied to context and in principle liable to revision (though that may never happen).

Note that it is therealizationof contingency as truth or falsehood, which is relative to context, but the contingency in itself is no less absolute (with respect to context) than necessity or impossibility.

A careful distinction must be made between the truth, falsehood, or problemacy, of a proposition whose logical necessity, contingency, or impossibility is unspecified — and the truth, falsehood, or problemacy, of any proposed modal specification for that proposition. Failure to distinguish between these perspectives can be very confusing.

A proposition may be problematic to the extent that, not only do we not know whether it is true or false, but we do not even know whether it is logically necessary, contingent, or impossible.

Less extremely, we may know the proposition to be true or false (and thus, possible or unnecessary), yet not know whether it is logically necessary, contingent (possibleandunnecessary), or impossible. In such case, the singular modality (the proposition per se) is assertoric, but the plural modality is still to some extent problematic.

If a proposition is known to be logically necessary or impossible, then it is assertoric with regard to both its plural modality (the incontingency) and to its singular modality (accordingly, true or false).

If a proposition is known to be logically contingent, it is assertoric with respect to its plural modality (the contingency). We may additionally know that the proposition per se is true or false, in which case it is also assertoric with respect to its singular modality. Or we may still be at a loss as to whether it is true or false, so that it is problematic with respect to its singular modality.

In any case, here again, problemacy does not signify real indeterminacy, but merely absence of sufficient knowledge, remember.

Our definitions make clear that problemacy should not be confused with logical contingency. A proposition may be definitely true or false, and so unproblematic, and still contingent; and a problematic proposition may after serious consideration be found to be necessary or impossible, whereas a properly contingent proposition should not thus change status.

Yet problemacy and contingency have marked technical analogies, which allow us to treat any problematic proposition (and therefore any proposition whatever, at first encounter) aseffectivelycontingent in logical properties. Logic repeatedly makes use of this valuable principle. As will be seen, if the proposition is not indeed contingent, it will be automatically revealed so eventually through dilemmatic argument, so that no permanent damage ensues from our assumption.

Note that the definitions of the logical modalities are very similar to those of extensional, natural and temporal modalities. There is a marked quantitative analogy (this, some, all), so that we can refer to them as ‘categories of modality’; and there is a broad qualitative analogy (inclusion or exclusion in a wider perspective), yet with enough difference that we can refer to them as distinct ‘types of modality’.

Logical modality puts more emphasis on epistemology than ontology, in comparison to the other types. It primarily qualifies knowledge, rather than the objects of knowledge. Whereas natural modality refers to the objective circumstantial environment of events, temporal modality to surrounding times, and extensional modality to cognate instances — logical modality looks at the informational setting.

With regard to technical properties, logical modality is often similar to the other types, but some notable differences also occur, as we shall see as we go along.

4.Apodictic Knowledge.

The many-contexts concepts of logical modality are formed by reference to the awareness that there are items of knowledge which somehow would seem to be true or false no matter what developments in knowledge may conceivably take shape, while others seem somehow more dependent on empirical evidence for their acceptance or rejection. The former are often called ‘a priori’ or‘apodictic’, and the latter ‘a posteriori’.

At first sight, apodictic statements present a difficulty. They seem inaccessible to anyone with less than total knowledge. Only the fully omniscient could know what is necessary or impossible in the widest context. A normally limited mind like ours cannot have foreknowledge of any final verities. Indeed, even if we ever reached omniscience, how could we be sure we have reached it?

However, these skeptical arguments can be rebutted on several grounds. To begin with, they are self-defeating in that they themselves claim knowledge about the capabilities of omniscience, and they do so in no uncertain terms: therefore, they are intrinsically conceptually flawed. Logically, then, it is conceivable for a limited mind to acquire apodictic knowledge, somehow.

Secondly, it is noteworthy that our minds, though admittedly less than omniscient, are not rigidly limited in their powers of imagination. We are able to construct innumerable hypotheses even with a limited amount of factual data to play with. Thus, we are never limited to one context, the present one, but can manipulate ideas which go beyond it. Of course, this does not mean that our imagination is able to foresee all contexts. The more factual data we have to feed on, the more our imagination can stretch out — but we never have all the seeds.

Thirdly, the skeptical arguments misconstrue the issues. We defined the necessary as true, and the impossible as false — ‘in every context’. We did not say, the necessary is what is true, and the impossible is what is false — ‘to the omniscient’. Our definition does not exclude that the quality of necessity or impossibility begiven as such within any single context, as an inherent component of the appearance. It does not logically mean that we have to foretell what goes on in other contexts besides our own.

And indeed, we find within common knowledge many instances of manifest necessity or impossibility, without need of further investigations. Such events constitute the experiential basis for these concepts.

The primary examples of this are Aristotle’s laws of thought. They strike us as intrinsically overwhelming, as in themselves capable of overriding any other consideration of knowledge. We can only ever deny them reflectively, by obscuring their impact; but the moment we encounter them plainly, their practical force is felt. When we are face to face with a specific contradiction, we see that it is nonsense and that something, somewhere must be amiss. That is why the laws of identity, of contradiction, and the excluded middle are naturally adopted as the axioms of logical science.

But other examples abound. More generally, as we shall see, a proposition is self-evident, if it is implied by its own negation, or implied by any contradictories; and a proposition is self-contradictory, if its affirmation implies its own negation, or implies any contradictories. It will be shown that a self-evident proposition displays the consequent property of being implied by any conceivable proposition, and a self-contradictory proposition that of implying any conceivable proposition. ‘Any’ here means ‘every’ — so that these are cases of logical necessity or impossibility.

This may occurformally, for all propositions of a certain kind whatever values be assigned to their variables. Indeed, the science of logic itself may be viewed as a record of all such occurrences. Or it may occurcontentually(or ‘materially’), in the sense: not for all propositions of a certain kind, but only with certain specific contents. Note that this distinction is somewhat relative, depending on what we hold fixed and what we allow to vary.

Another way apodictic knowledge (or, for that matter, any knowledge) might conceivably be made available to a limited mind is throughrevelation, a communication from an omniscient mind. This is the logical premise of religion.Faithmight be defined as the conviction that the information does indeed come from an infallible source, G-d. This topic is too vast to be discussed in this treatise, but I merely wanted to indicate the entry point.

Now, if logical necessity or impossibility are somehow given as components of the appearance of things in any context of knowledge, what is their difference from (contingent) truth or falsehood, which are also given?

Theoretically, once a proposition has been seriously scrutinized and found not to be necessary or impossible, it henceforth remains permanently contingent — just as once a proposition is seen to be necessary or impossible, its status is thenceforth established. In practise a mistake might conceivably be made, but this does not affect the principle.

The essence of necessity or impossibility is their property of self-evidence or self-contradiction; it is not their permanence, which is only incidental. Contingent truths or falsehoods may also be permanent; a proposition may happen to remain true or false without change as knowledge evolves, and yet never lose its contingent status. That some contingent truths or falsehoods do change over time, is irrelevant. Even in a total knowledge context, truths or falsehoods may be characterized as contingent.

Thus, we do not regard an obvious empirical truth like ‘it is now raining’, or a well-established law of nature like ‘the amount of matter and energy in the universe are constant’, as logically necessary, even though we believe them to happen to be fixed truths (each in its own way), because they do not seem self-evident; they are both therefore intrinsically logically contingent. The raw, factual finality of the former or the natural necessity of the latter do not affect their common logical status.

On this basis, we can also say that logical contingency is conceptually distinct from problemacy. In omniscience, problemacy disappears, but not logical contingency. The latter remains as a further qualification of certain truths and falsehoods, distinguishing them from logical necessities and impossibilities, respectively. It follows that contingency as such is not a lower status than necessity or impossibility.

Lastly, note, a necessity or impossibility may be immediately apparent to anyone, or we may need to go though a long or complicated reasoning process to make it apparent. But in either case, the sense of obviousness is given within the appearance itself, so that the ease or difficulty with which we were brought to the insight are irrelevant to its finality.

It is hard to distinguish a priori and a posteriori knowledge by reference to the concepts of reason and experience. The former is indeed more purely analytical, but it cannot occur without the minimum of experience on the basis of which the concepts involved are meaningful and clear. Likewise, the latter is indeed more likely to be affected by changes in experience, but its conceptualization and logical evaluation involve a great deal of rational activity.

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