Consider the two historically important kinds of logic, deduction and induction. Inductive logic is nowhere near as emphasized as deduction in the 19^th, 20^th and early 21^st centuries but, to begin, let us consider both kinds. We want as many “data points” as possible; even the consideration of spurious data may be revealing.

In deduction, the process of deduction leads from premises to conclusions. If the premises are true, the conclusions are true. If the premises are known or agreed upon to be true, so must be the conclusions. Thus, we get new knowledge from given knowledge. Of course, the conclusions are in a sense contained in the premises – deductive logic is one big system of tautology. But, there is more to deduction than that. Even within a given system, although the conclusions are contained in the premises, that may be difficult or impossible to see and therefore deduction is necessary; and, deduction is not a linear process from premise to conclusion for not only are the conclusions not known in advance and must be initially guessed, the proof itself must be found by trial and error. Additionally, premises, too, may be called into question, are tentative and thus entire axiomatic systems are, to some extent, in evolution.

On rigor

Conclusions follow from premises. The first point to rigor is to make sure that conclusions do indeed follow from premises. The history of logic from before Euclid, to Aristotle and to today is this: [1] expanding the scope of the kinds of assertion permitted as premise or conclusion – from simple to compound to function [predicate] statements, modal statements, an apparent incompleteness of truth and falsity as encompassing the range of truth values; and, [2] enhancing rigor. There is, however, no absolute rigor – the story begins with the logical paradoxes starting from Bertrand Russell, the doubts cast on the classical forms by LEJ Brouwer, and the completeness theorems of Gödel and continues to today; and, today, mathematicians are willing to accept paradox so as to enrich content.

It is important to know that error is eliminated from each step. Therefore, rigor is important even when the overall validity is of a lower grade. But, conclusions do not simply follow from premises. Where do premises come from, what is their source of power – obviousness and common agreement… and, hidden, from success. The last point is a hint. An entire system of premises, deduction and conclusions is tentative; it is empirical and when no longer valid across the domain of application, premises must be inductively modified.

“Rigor” is incomplete. Further, practice has it that we focus on the domain of success – this is the institutional blindness to the domain of failure. This is also true of biology, of biological constitution. In the end, we still die for our ideas – even though Popper popularized the idea that our ideas now die for us… he was talking of science; he also talked of the searchlight metaphor for science

Induction

Induction is, typically, a process of generalization. From a number of data points we “induce” a pattern. Or, from a set of scientific data, we conclude a law or a theory; the reality is more complex than just that. An example is Newton’s system of mechanics; it involved, at least, three main components: the idea that constant momentum is the natural state of a body so that force equals rate of change of momentum, the idea of force as mutual interaction between bodies – the concept of action and reaction… and the important special case of gravitation being given by the inverse square law, and the development of a calculus that permitted the analysis of complex forces and motions. Clearly, this was not a one step process and Newton’s efforts were continuous with those his predecessors and successors. Finally Newton’s theories, after centuries during which its conceptual and empirical domain expanded, met against some of its limits and gave way to the radical new theories of the 20^th century: the new theories of space, time, gravitation and quantum mechanics. Thus, induction, is more like a conversation between data and law or between world and theory than an instruction from world to theory. But, we saw that deduction is also a “dialog”. There was a concept of the necessity of induction which exalted its status; but, the reality is less exalted but more real. Let us consider a simple example of induction to make some points. Consider the sequence 1, 2, 4 – what is the next number? That requires to discover the pattern that the sequence follows. It is easy to see that each number is double the previous and so the next number is eight. But that is just one possibility. For each number is 2 raised to the power of the previous number and, according to this pattern, the next number is 16. In this example, we found two simple patterns:

However, generally, it is not necessary to find two patterns or two simple formulas. If a law covers a set of data points, then clearly a second law that is identical to the first only at the data points also covers the data points. In practice, it should count as something that one pattern makes understanding clearer, prediction simpler just as the Copernican system was simpler than the Ptolemaic. This would figure into a “logic of deduction” and does but as an intuitive and practical point and not in any formally necessary way.

What is being done in induction? Generalization allows us to come up with a pattern or theory; then, we think, the theory tells us, at least approximately, about the nature of the universe. Practically, it permits prediction. In either case, we get new knowledge from old.

Logic as the Process of Knowledge Acquisition and “Verification”

Verification in quotes because it is meant in the sense of justification; not in the sense of positive verification.

As a simplification, we saw that both deduction and induction involved coming up with new knowledge from given knowledge. Deduction as a straightforward process was rather lateral – premises and conclusions were rather on the same level and therefore deductive conclusions were “certain” but that was only within a given system and the certainty was no more secure than the system itself. On, the other hand, induction as generalization was not certain but that is the price to pay for generalization. However, when generalization lead to simplicity, greater predictive power; and when we find theories to transcend origins into a degree of universality we then think that the theory has captured something of the nature of things – even if approximately and in a limited domain.

But, is all logic acquisition / “verification” / the process of knowledge?

I see a cow in the field, I then know that there is a cow in the field. Is that logic? Not in the traditional view; traditionally that is perception. What if I wanted to be really sure? Or, what if I saw a cow in my bedroom when I woke and I doubted its presence. Or, I had a dream of a cow in my bedroom and on waking I felt confused and wanted to ascertain whether there really was a cow? I could do various things. If I wanted to be sure that there were a cow in the field I could touch it, I could prod it and see if it mooed like a cow, I could look for other cows, I could look for manure and see if were like the cow manure I have seen so many times before, I could ask others for corroboration. In the case of the cow in my bedroom or in a dream, I would probably search for evidence to invalidate rather than corroborate my first impression. But, there is a process of testing in all cases. In everyday situations, I bypass corroboration; or, rather, I assume the context is adequate corroboration. Thus, there is a kind of logic that, in the contextual case, is “null” logic.

Logic is the process of knowledge.

Then note:

Logos is the process of being.

But:

There is a dual between being and process;
[this is analogous to the duality, in quantum mechanics, of position and momentum.]

and, therefore:

Logos is the form of being.

And,

Logic is the form of knowledge.

There is no absolute reason that everything ever called “logic” needs to fit into this framework. That is, if for example we use “family resemblance” as the mark of all things of a kind, not everything thus far called “logic” need fall under the kind that we call “logic”. Some of the things called “logic” may fall under some other kind, others may fall under some other thing, others may be discarded as “error”. This discounts the possibility that there may be two kinds called “logic”. All this is not the result, only, of divergent / convergent / erroneous / metaphorical / analogical thinking; there is also historical accident. We can rid ourselves of the delusion that the system into which we are enculturated / educated is absolute, infallible, completely consistent, complete, perfect…

So, what I want to do is to focus on logic as the process of acquiring knowledge. I will not require that there be an algorithmic formulation. I will allow intuition and so on; but we will distinguish between formal and informal approaches and not confuse one for the other.

First, I want to enquire: have we covered the modes of knowledge acquisition? Seeing a cow in a field is knowledge by acquaintance: perception. For the usual inductive and deductive process, we use language and therefore require knowledge by description. However, we saw in Kinds of Knowledge, that we can think or infer in pictures. Thus, logic is used for both kinds. But, are there other kinds? We also saw another kind in Kinds of Knowledge – knowledge by immersion. A model is tacit knowledge. But, to be able to have knowledge, the being must have the capability for knowledge – this is contained in part in the genetic code which gives the individual capacities. Now, the genetic code contains information – does it not? Is that a form of knowledge? Not in the usual anthropic or biotropic way. But, as I noted in Evolution and Design, it may be regarded as phylogenetic knowledge: picked up, without intention, during evolution; this is in contrast to the usual ontogenetic knowledge. Of course, phylogenetic knowledge is not knowledge under the acquisitive process of individuals or societies. The observation is useful because it rounds out a conception of knowledge. Can we go further? The furthest is that the possibility of universal structure, as in the formation or beginning of a universe, is the Platonic form. The Platonic forms are the possibilities of structures of all universes. Thus, there is not a separate Platonic universe; there is not a Platonic universe at all except as a metaphor.

The Analysis of Logic

This is done preliminarily in Knowledge, Logic, Existence, in Kinds of Knowledge, and in as yet un-typed notes “On Inference”, and “Mathematics and its Foundations” from Journey 2001. This needs to be refined and elaborated.

I will also use the following ideas. The idea of knowledge as a picture of the world as a picture; this implies, among other things, a regress without external foundation. Rather than being a criticism which it is from the “justified true belief” point of view, it is a freeing and a true anchoring or centering. We are in and of the real rather than alien critics; and, but for our neuroses, we need no alien concept of certainty or alien foundation; experiment is the nature of temporal being; we can go beyond the temporal by being ever in the moment or by knowing our true nature; and, our putative limits are just that – for if the certainty of knowledge is an illusion so must be the absolute and final nature of limits no matter how real and smack-in-the-face they seem from the immediate and pragmatic view.

Note the following point: the analysis of logic is the analysis of knowledge [and being.] Knowledge and logic are inseparably tied together. And therefore we need

Knowledge / process-logic / truth / cognition / language. What is the full field of analysis? When the system is compromised, the entire field of concepts may need simultaneous revision.

The Analysis of Knowledge

Presentation vs. representation. Representationalism: the mind perceives only mental images (representations) of material objects outside the mind, not the objects themselves.

Picture theory. How does picture theory fit in? Picture as presentation, representation, immersion in. Language as picture.

Immersion theory. Immersion and adaptation.

Phylogenetic vs. ontogenetic. Ontogenetic / acquired: dual interpretation as picture and as immersion. Phylogenetic as: body structure as adapted, as evolution incarnate, as the possibility of ontogenetic knowledge.

The Analysis of Truth

Truth and knowledge. Truth and logic. The concept of logic requires the concepts of truth and of inference. But if truth is compromised, so is logic. However, when the concept of truth seems compromised, it may simply be that we are using an incorrect or limited concept of truth.

“Definition” vs. criteria

The meaning of this is clear. It is something like “sense and reference.”

Concepts or theories of truth:

Aristotle: “to say of what is that it is not, or what is not that it is, is false; to say of what is that it is, or of what is not, that it is not, is true.

Aristotle directly influences the semantic view and has affinities with the redundancy and correspondence views

Redundancy (Ramsey)

This theory says that “true” is redundant because to say that “it is true that it is raining” is equivalent to saying that “it is raining”.

Variations are simple (Prior, Mackie, Williams), prosentential (Belnap, Camp, Grover), and performative (Strawson)

Semantic (Tarski who influenced Davidson, Kripke and Popper)

Correspondence (Russell, Wittgenstein who influenced Austin and Popper)

Pragmatist (Peirce, James, Dewey who influenced Wittgenstein and Dummett)

Coherence (Bradley who influenced Rescher). Notice that coherence theory is close to the idea that knowledge is the knowledge of patterns. But that makes it merge with the correspondence theory.

Relationship among and status of the theories

Independently complete and competing… or complementary?

There is an element of independence and competition but that is and should not be the whole picture. Pragmatism with its selectionist account may underlie other theories so that, for example, correspondence may explain selectional advantage. That is a “vertical” relation. There may also be horizontal relations: e.g. coherence being about internal or logical relations and correspondence about external or empirical relations. The redundancy theory can be seen as a clarification of Aristotle’s position; the semantic theory can be seen as providing a more precise version of the correspondence theories.

To see the various theories is a gross simplification; and ignores the multifaceted nature of knowledge and truth.

An example: the pragmatist theory

Roughly, the pragmatist theory is “what works.”

Nothing works; all we know is that something has worked in certain cases. This is Hume’s problem revisited and the answer must be the same.

“What works,” is refined as the practical or experimental consequences. But, the correspondence theory is one prescription of “what works,” or “what would work in all situations.” It is only in certain environments that we may determine what would work in a variety of situations. Consistency is one test; experiment is another. Thus pragmatism reduces to coherence and correspondence. What when determination of what would work is not possible, or not conceivable? We then have a broader concept of knowledge and truth: what has worked. But it remains necessary to find ways of identifying the bearer of knowledge and the criterion for knowledge.

The Analysis of Thinking… Cognition…

Thinking vs. imagination. Thinking as connected images for which the connection and the images correspond to the [structure and process of the] real.

The Analysis of Language

Elsewhere.

What I wrote at the lake.

Sunday 10.7.01

I think this is what I wanted to say.

There has long been concern (since before Euclid in the realm of History, but implicitly whenever any being wanted to transcend contingency) about the fundamental nature of axioms.

For, if the axioms are true, and if inference is that which from true inputs [premises] results in true results [conclusions] then the theorems would be true.