ANIL MITRA PHD, COPYRIGHT ©
2001 AND REVISED July 2003
Document status: July 20, 2003
Maintain
Essential content absorbed to Journey in Being
May develop certain topics or details of topics such as the origin and kinds of knowledge, symbol, language and logic; non-symbolic varieties
May absorb the above e.g. to Journey in Being
On Language [See On
Signs, below]
Tracing Back: Roots of Language, Varieties of Linguistic
Expression
Box: Language as a Biological Phenomenon
Tracing Forward: From Language to Knowledge by
Description: A Very Brief Sketch
The Possibility of Inference or Logic
Initial Comments on Unity Among the Kinds of Knowledge
Inference: From Perception to Description
Box:
Formal Axiomatic Theory for the Propositional Calculus PC
Box: Recasting Arguments in Terms of
Categorical Propositions
Formal Logic
and the Variety of Logics
Mathematics, Deduction and Induction
Postscript: Plans for This Document
There is a traditional distinction[1] of knowledge by acquaintance and knowledge by description. The difference is based on the degree to which the knowledge is direct. Knowledge by acquaintance is direct, sensory. Knowledge by description is inferred, described in symbols e.g. language. Knowledge by acquaintance is depicted rather than described: one may draw a picture of a scene, evoke a mood[2] with music, act out an event
These traditionally identified kinds of knowledge show the individual as separated from the world. Identity requires individuation but not absolute individuation. I look to enlarge the scope of knowledge. I will do this by adding a new kind of knowledge – one that includes the traditional kinds. The origins of the new concept are as follows. I sought, first, to understand knowledge as a relationship between organisms and the world. As far as the traditional kinds and definitions “from the inside” such as justified–true–belief are concerned the purpose was not to displace but to give context and understand them. I sought a “Copernican”[3] move away from the “Ptolemaic” system of understanding for knowledge. A purpose is to provide foundation or grounding of the traditional modes – except, perhaps, origins in nothingness there is no self–foundation. As a result of this grounding knowledge is closer to being understood as a universal rather than a local phenomenon. I asked the following question. What element in an abstract and general specification of being such as being–meaning–action or being–relationship–process corresponds to what is commonly taken for knowledge? That is, what element is a knowledge–primitive in the sense that knowledge as we conceive it appears as its elaboration. An objective and a result is to show the nature of knowledge as an essential part of the nature of ultimate being. Additionally, a broader conception of knowledge is a switch that opens up vistas of knowledge and possibilities of being
I want to add a third kind, knowledge by participation. Knowledge by participation is even more direct than knowledge by acquaintance. In empiricism, perception is thought to found all knowledge; if I were of an empirical bent I would find that knowledge by participation, which does not require special sensory apparati, might found the other kinds. Knowledge by participation is structure and processing of an organism that enables it to live in its environment. These structures and processes must in some, at least rough way, mirror the environment and its processes and therefore they may be called knowledge. Knowledge by participation is organismic, possessed by the organism as such without reference to any special system for knowledge. A possible difficulty in accepting knowledge by participation as a kind of knowledge is the natural focus on knowledge explicitly obtained by or held in specialized organs
The answer to this difficulty is as follows. The word “knowledge” stands for a concept and refers to something. The following is in the nature of concepts and referents in general. We can say that the referent is well specified when a system of explanation including the concept achieves precise description and prediction. I assert, and the objective of this narrative is to show, that introduction of a third “kind” of knowledge does introduce a broader framework of description and explanation for knowledge even in its restriction to the two traditional kinds. Further, knowledge by participation is not merely an abstract concept. Finally, the other kinds of knowledge are not displaced and they do not lose their significance. Rather, their place in nature –as parts of a more comprehensive concept of knowledge– is better understood. They are understood as elaborations of knowledge by participation. The concept of knowledge by participation relates the more advanced kinds of knowledge to the organism, its origins and to its pre–vital [physical] nature
Knowledge by participation is a move in the direction of providing ultimate grounding for knowledge. Knowledge and being are not essentially distinct. A second purpose of this narrative is to understand and move forward to the ultimate possibility of knowledge – at least in the human realm. This is the realm of perception, thought –in consciousness, language, logic, mathematics and their application– and being. A third purpose is to show that –in contrast to regarding one of the modes as fundamental– the different “modes” together provide a complete picture of knowledge
Knowledge by acquaintance. Sensation, perception, immediate–direct, not intrinsically expressible in symbols or language but depictable, vague, bound to the nature of the object that is known and to the physiology of the organism
Knowledge by description. Conceptual knowledge, derived or inferred, expressed in language, precise and distinct, free. It is the freedom of symbolic expression that distances knowledge by description from what is known, but also makes it more precise. Knowledge by description also makes possible hypotheses that may be tested… and, consequently, makes possible knowledge of domains beyond the environment to which the organism is adapted. Alternatively, knowledge by description may be seen as a kind of adaptability of the organism. Knowledge by description is an evolutionary elaboration of perception
Knowledge by participation. The adapted physiology of the organism; especially those elements that involve response to the environment. The sequence in evolution is knowledge by participation, by acquaintance, and by description. An earlier kind does not stop when a later kind starts. Knowledge by description can be seen as a variation of knowledge by acquaintance which can be seen as being built upon knowledge by participation. Knowledge by participation occurs prior to the evolution of distinct and discrete sense organs and occurs at a stage when afference–efference or perception–locomotion are bound together. A fourth kind “knowledge by mutual adaptation” could be added but this can be seen as a kind of knowledge by participation. Knowledge by participation can be acquired in evolution or in the development of the individual. Adaptation is acquired in evolution. Knowledge by participation is the structure–process of an organism or a group of organisms that makes living – function and procreation possible
Knowledge by participation covers the range from evolutionary adaptation to innate knowledge to acquired knowledge. It does so without elaborating as the later modes. It also appears in certain elaborations as the later modes. The range includes development of perception and so the foundation of the later modes. The range also includes a foundation for Dasein’s mode of being as approximately described by Heidegger
The distinctions among the kinds of knowledge are not absolute. Knowledge by acquaintance is involves concepts as in the recognition of a table as a table. Knowledge by description is, to some extent, built from elements and relations that are perceptual
Can knowledge by participation also include or show a connection to the following levels? 1. Primal modes of being such as being–relationship–process. 2. A coupling of the organism to the environment that appears in awareness as a connection of immediate consciousness to the depth of consciousness and the unconscious modes, and so as the identity of Atman and Brahman. I begin to affirm the connections as follows. At the evolutionary–developmental stage when all knowledge is by participation we see that knowledge = organism and so all knowledge is a variety and elaboration of being
See Journey in Being
On Language
[See On Signs,
below]
The following comments are continuous with the development above. The comments approach the question of the “foundation” knowledge from its most recent human elaboration – language. Kinds of knowledge approached this question from roots in the evolution of life and earlier
Knowledge by description is expressed in language. A somewhat but not significantly generalized concept of language is required to make this assertion correct. “It is raining” is an assertion, a proposition, here expressed as a sentence in English; its assertion is the assertion of a fact. The assertion is the assertion of a fact. This fact can be traced both backward and forward.
Box: On Signs
Sign: a pointer Community sign: a shared sign Individual sign: private and community Mark: a sign that is constructed as such Elementary mark: a mark used to build composite marks – e.g. letters of an alphabet Artifactual sign: something constructed that later becomes or is chosen as a sign Symbol: a sign that embodies or evokes at least some of the essence of the object Embodiment: actual or in perception – in the mind Therefore the association: symbol and object Intrinsic symbol: a symbol that intrinsically contains the essence of the object. This does not mean there are any… but many symbols are treated as such because of the psychological association of word and object The sacred sound “ Signs that because of common heritage or genetics, evoke common feelings/thoughts/objects would seem to be intrinsic Are there objects whose existence require and imply the existence of a universe? Universal symbol: common to humankind, to primates… |
Tracing Back[4]: Roots of Language, Varieties of Linguistic Expression
Not every sentence or meaning is the expression of a fact. This is related to the concept of illocutionary point which is different from propositional content of which there is an infinite variety. The question is how many kinds of speech act are there from among promises, commands, assertions, questions…? Based on a development of the theory of intensionality and upon the concept of direction of fit of a speech act –mind or word to world, world to mind and so on– there are essentially four kinds. The appearance of five kinds below is due to the world to mind fit being associated with two kinds[5]
Assertive: mind to world direction of fit. This is the form of the proposition. The assertive form is capable of being true or false. The other forms are not associated with truth or falsity due to their direction of fit not being from mind to world. Note that terminology is not uniform and what is called assertive here is elsewhere labeled declarative
1. Directive: world to mind direction of fit
2. Commissive: world to mind direction of fit
3. Expressive: null direction of fit
4. Declarative: two way direction of fit
The point is: these forms of expression come with certain mental attitudes; their expression comes with and the interpretation depends upon non–verbal performance and context
In Tractacus Logico–Philosophicus, Wittgenstein provided an elaborate account of a foundation of language in meaning. His analysis amounted to the following. The world is made up of facts. Compound facts are made up, ultimately, of atomic facts that have no further structure. Similarly, language has an atomic form that is in and mirrors the world. Thus every valid linguistic unit has meaning; and every fact has expression. The “later” Wittgenstein rejected this view; an account of reasoning is in Philosophical Investigations published posthumously in 1953. The starting point for the rejection was the observation that not every linguistic unit has meaning… rather, language functions in a context and so we should look to use and the nature of use for understanding
Now consider a herd of antelope grazing over a range of open grassland. At one edge of the herd a leopard approaches and is sensed by one antelope that makes a response. The response is a sudden tension, a tremor to a muscle in readiness for action, that is at once in association with a nervous excitement, a heightening of alertness, and sensed –perhaps there is a wave– by the other antelope. The others, recognizing the tension–excitement, immediately assume the same attitude. The response of each individual may involve a number of evident and internal changes and there may be a sequence of events, interactions and mutual galvanization into action and flight
Consider next, a similar situation – a group of baboon in a migration on the African savanna. They are confronted by lion. The response is similar to that of the antelope but more elaborate and organized. There is an explicit rapid alert, organized defense and possible confusion of the predator. There are similarities and differences among groups showing both inherited and learned behavior
Note the following points:
Interplay of explicit sign, internal mental –cognitive, emotional– state and context
The sign is simpler, of lower dimension, than the mental state and context but serves as a communication and induction to action through mutual participation of like animals in a common context
The lower the dimensionality of the sign, the higher its functionality. Calls are vocal and have pitch and volume… but, in certain limits, become one dimensional
The multifunction of communication: alert, confusion
The basic communication does not follow a particular kind or direction of fit –assertive, directive and so on– but has a number of kinds
Imagine, now, a band of chimpanzee at ease; there is no danger. It is easy to imagine in this situation the elaboration of functions of signals: expression, assertion, induction or command…
The elaboration continues until modern Homo sapiens[6] and includes the following elements:
Elaboration of kinds of expression
Socialization of the expressive function
Brain development in certain directions that enhance language ability
Various levels of separation of language from context and mental state – with advantages and problems
As human beings, it is natural to think of human language as Language. That is partly due to self–interest. Most of human communication in language is with other human beings. However, other animals possess some elements of language as we understand it. Whether language is limited to life as known to humankind, it may be useful to consider language in the abstract – unless the limitation is necessary. One step away from an anthropocentric concept of language is to distance the concept from the view from the inside. That may be achieved by setting aside the concept of intensionality – not altogether but for the purpose of the abstract concept. An example would be machine languages and information. Whatever the limits on such alternatives, the two approaches can be used to inform each other and so, perhaps, avoid pitfalls of both.
Box:
Language as a Biological Phenomenon
There is a description of the origin or constitution of language as a biological phenomenon The Tree of Knowledge[7] by Maturana and Varela. The following are some essentials: Social phenomena: coupled behavior made possible by structural coupling among organisms. Complex behavior is made possible by nervous systems and is necessary for continuity of the lineage in organisms with sexual reproduction for gametes to meet Communication: coordination of behavior in social phenomena Cultural behavior: stable trans-generational patterns of ontogenetically acquired communicative behavior Linguistic domains: acquired communicative behaviors that depend on the particular ontogeny of the particular organism contingent on its specific history of social interactions. Linguistic domains are so called because the behaviors constitute the basis for language but are not yet language Semantic description: the claim that the meaning of the different communicative behaviors arises as above Language: when actions are coordinated about actions that pertain to the linguistic domain itself Consciousness – according to Maturana and Varela in The Tree of Knowledge: the reflexive
aspect of language makes for consciousness. The idea is that awareness can be
described and becomes a topic or point of focus. Maturana and Varela refer to
self–description as the key feature that makes consciousness possible |
Box:
Solipsism and Experience
In the example of the antelope herd it is assumed that the experience of one animal is essentially the same as that of any other – the differences for normal adults are hues and overtones but not in the main features. Here “experience” refers to the subjective state Solipsism is the view that the individual is “trapped” in his or her own experience – that it cannot be known whether the experience corresponds to what is real. Specifically, granted the existence of other individuals, I cannot know whether their experience in a given situation is the same as mine. In the case of the antelope, according to solipsism, it cannot be known whether the outer behaviors correspond to the same inner mental states. This is a problem for the account of communication given above An objection to solipsism is that, given its premise, I cannot know what “the same experience” means. I cannot know, due to faulty memory, whether my own experience is the same from moment to moment… or whether the phrase “my experience” has meaning. What is the basis for connecting “this experience” with “this body”? The way I know the practical identity of my experience with all non–pathological human experience is through shared development and use. The way I know the practical identity of experience among antelope is [1] there is sufficient similarity among vertebrates to assert the existence of experiential states in all vertebrates[8], and [2] from the similarity in physiology and behavior among antelope Here is a practical objection to solipsism. I know that
I did not create the world of my experience – not only is it largely
autonomous, it is far too detailed, too complex. The argument is practical of
course because I may be deceived as to the existence and identity of a
creator. In as much as it is a good argument it is an argument against
solipsism, and its variants such as the idea that I may be a brain stimulated
by a scientist to give me my experience. A practical objection to the
brain-in-a-vat argument for solipsism is that the thoughts of the scientist
would have to be far more complex than my thinking as the brain-in-a-vat.
But, for this to be a good argument for solipsism, the scientist would, in
turn, have to be a brain-in-a-vat. There would be an infinite regress of
successively more complex brains-in-vats. Here is another practical objection
to solipsism. I can compare my own behavior [my third person experience of
myself] with the behavior of others [my third person experience of others] and
draw conclusions about the identity of the nature of the first person
experiences; it is curious that some people insist on finding a divide by
insisting on a comparison of the essential difference between their first
person experience of themselves with their third person experience of others |
Tracing Forward: From Language to Knowledge
by Description:
A Very Brief Sketch
The various kinds of speech act though not prima facie knowledge constitute a phase of knowledge by participation. Alternatively, view the effects –or correlates– of speech acts upon the community as performances that reflect knowledge possessed by the group as a unit
Here, however, concern is with assertive or declarative[9] knowledge. In language an assertion is a claim about a fact. An assertion is about the world and, as such, can be true or false. An assertion is a claim to knowledge and, if true, is descriptive knowledge
What is the origin of the various kinds of speech act –specifically of assertion– from the phase of social evolution where communication is integral with action? There are two factors, [1] an elaboration of the kinds of communication and [2] development of communication as an activity that can stand alone as an activity in itself – through reflexion. Thus a speech act may be about something and may thus be true or false
Some linguistic characteristics of knowledge are:
Assertion: true or false
Abstraction: what is common to a group of facts
Compounding: “and”, “or”
Box:
Propositions
Two related definitions: A proposition is what is capable of being the meaning of a declarative sentence – in some language actual or possible. This concept of proposition agrees with most current philosophical use Alternatively, a proposition is meaningful declarative sentence A proposition is thought to be the primary bearer of truth and modal properties, and the object of knowledge, belief, hope, desire and other intensional states. The concept: proposition is whatever is the bearer of truth Two doubts: 1. The subject-predicate form of expression, and 2. Language as an adequate expression of propositions |
New knowledge is generated by new experience and by new or novel propositional combinations of what is known – including simplification. It may be questioned whether such new combinations constitute truly new knowledge. When the process of combination is difficult to see then the resulting knowledge is effectively new for the fact was not known even if contained. From a pragmatic point of view it is new for it enables actions that were not possible before
We may say that the new combination was “implied” but not known. The generation of new knowledge from given knowledge is, generally, inference or logic. “Logic” is sometimes reserved for formalized inference or for established systems of inference
The Possibility of Inference or Logic
Inference is usually done in words – the medium of logic is language. The purpose of this section is to show how inference is possible and necessary without language, i.e. in terms of knowledge by acquaintance. This is a form of foundation or grounding and makes for a fuller understanding of logic in terms of language and of the logical calculi
Initial Comments on Unity Among the Kinds of Knowledge
In a sense the only knowledge is direct knowledge – by acquaintance. Knowledge by participation is too diffuse while knowledge by description is too indirect to count as knowledge. Knowledge by acquaintance, however, is not certain – the senses may mislead. Practically, however, we accept degrees of certainty
If knowledge is regarded as a function rather than an object –something that is in and of itself– then it is natural to accept an idea such as degrees of certainty. When I say that knowledge is a function I mean nothing more than knowledge is an element in a complex system of interactions. Viewed that way, the different “kinds of knowledge” constitute Knowledge, condition, reinforce and have continuities with one another
Inference: From Perception to Description
It remains true, at least, that knowledge by acquaintance can be a paradigm for all knowledge. Is there a kind of inference within this realm and what is the nature of that inference? Knowledge by acquaintance is not merely data or experience but includes perception –or experience– of patterns, e.g., in time and space. Given experience, and therefore memory, of patterns it is possible from exposure to a limited “scene” to “predict” beyond that scene, e.g., in time and space or to fill in the elements of a scene. For perception to be part of an adaptation that includes action there would seem to have to be some predictive or selective ability
Knowledge by description follows this prescription. In this case the pattern of propositions in a prediction bears some relation to the pattern of signs – in the corresponding sentences. The pattern of signs that enable “derivation” of knowledge from given knowledge is a calculus or logic
Deduction in science vs. logic
In science, “natural” laws enable prediction or deduction. True logical deduction is by tautology. What is the iconic analog? One answer: the iconic analogs of logic include whatever can be concluded by abstraction
Is there a “boundary” place where Logic = Law?
It is not clear that there can be any universal calculus or logic. Rather, examples of logical calculi can be set down from an examination of patterns of thought. There is no proof of such calculi but they can be studied as objects in themselves. The logic of propositions is found to be a framework for the development of various logical machinery. Concepts of consistency and completeness give meaning and significance to logic. Assumption of a metaphysics or elements of a metaphysics may result in a universal logic or universal elements to logic
From the nature of the concept of logic and its place in the relationship among knowledge items it follows that proposition, truth and falsity and inference or implication are fundamental terms or concepts
When conclusions are contained in premises [even if difficult to see] inference is deductive or necessary. When inference contains essentially new information it is inductive. It is not clear whether there is any necessary induction
The functions described above are linguistic – assertion, abstraction, compounding; and inferential or logical
Some inference is possible from the form of the premise. From “it is raining and it is 10 in the morning” it follows that “it is raining.” The conclusion is necessarily true from the fact that it is “included” in the premise. Given the premise, the conclusion certainly follows. This is an example of deduction. In deduction or deductive logic the conclusions follow necessarily from the truth of the premises. The example given is trivial but deductive logic is useful when the conclusions do not follow obviously from the premises and ingenuity is required to develop the chain of inference. As an example, deductive logic is a large part of the basis of proof in mathematics
There are other kinds of inference in which the conclusion does not necessarily follow from the premise. An example is generalization from experience to truth. If all ravens seen are black we may be tempted to conclude that all ravens are black even though there may be, somewhere, a brown raven. We make such practical generalizations every day; life would be difficult or impossible without them. Such inference is called “induction”: the generalization is induced from what is given. Much more interesting than this kind of example is the situation where the result does not follow in any obvious way from what is given. Much of science is like that. From observation of phenomena, and from experience a cause is hypothesized. The hypothesis results in predictions. A large class of predictions is correct and this makes the science useful. Some predictions are not correct and this requires tinkering with and generalizing the concepts underlying causes. Further predictions follow… In this way large parts of nature become understood and predictable from a few basic theories or principles. It is clear from this description that science is not a one–step inference of laws and theories –conclusions– from phenomena or sets of observations –premises. However, science has sometimes been portrayed that way and scientists and philosophers have sought a logic of induction or scientific inference. Intuition and creativity are a large part of the formulation of the causes in terms of which the fundamental theories are expressed; and, once theories and laws are given expression, they may be tested in a number of ways – experimentally by testing the predictions and logically by examining the theories for consistency… What then is a logic of induction for science? One part of scientific inference is the testing of the theories or laws. A second part is the formulation of new or modified theories. As we have seen this involves intuition. However, a reading of the history of science indicates that there are some useful principles that guide in the intuiting and criticism of theories. These principles which include simplicity, beauty, symmetry are useful rather than definitive. Also to be considered are questions such as utility, likelihood, probability. These topics are what might be considered in an outline of inductive logic
It is clear that inductive logic is not as well defined an activity as deductive logic; that the process of “induction” is always open to intuition and creativity that cannot or have not been formulated as a principle or method. It is also clear that inductive reasoning does not guarantee the certainty of the conclusions. This is good at least in that it reflects the nature of the world. Further, even though the conclusions of deductive logic follow necessarily from the premises it does not follow that all true conclusions can be imagined or deduced by human beings or that there can be a method for such deduction. Further, deduction from premises [e.g., primitive concepts and axioms] to conclusions [e.g. theorems] is not the whole story to logic or one of its major uses – mathematics. In the progress of logic mathematics there is always the question of the origin, formulation and adequacy of a system of axioms; and when advancing or generalizing a branch of mathematics, looking at what are the theorems that should be in the field and in what way may the axioms be modified or abstracted to achieve the, as yet, only partially defined results
An interesting question is as follows. Given a deduction from premises to conclusions and truth of premises, the truth of the conclusions follows. We have seen an example of deduction in which the conclusion was included in the premise. Such a deduction or implication is called a tautology. We also saw that tautological conclusion was not obvious or trivial in general. The question is whether there is any deduction in situations where the conclusion is not in any way contained in the conclusion. The importance of the question is diminished by the lack of complete distinction between deductive and inductive phases of thought
The purpose is to show how new knowledge can be derived from given knowledge with words – i.e. language. We saw that knowledge expressed in language –knowledge by description– loses direct connection with the object of knowledge but may thereby gain in universality. When logic or inference is carried out in words it gains the same advantage. Logic is an instrument in the extension of knowledge beyond the direct realm – an instrument of progress toward universality. Additionally, when expressed in words and symbols, logic can be formalized, patterns or methods discerned and logic itself can be studied
The form of language, grammar, corresponds, in some way, to how the world is. Thus grammar is the expression of a kind of knowledge – the form of knowledge. Thus grammar and logic are related. Logic may be thought to be part of grammar. This requires an extended meaning of “grammar.” This is [related to] Wittgenstein’s view of logic as grammar
Similar remarks may be made about mathematics. In the view of mathematics called logicism, primarily associated with Bertrand Russell, mathematics is logic or a part of logic. In formalism, mathematics requires its own concepts and axioms in addition to those of logic. Formalism is associated with David Hilbert
The “unit” of knowledge is the proposition – or, in language, the declarative sentence. An example is “It is raining.” The function of logic, then, is to show what true propositions follow from given sets of true propositions
Suppose rain is defined “drops of water falling and striking earth.” Accept the definition even if it is arguable. It is sufficient to the point to be made. Then from “It is raining” it follows that “drops of water are striking the earth.” In other words “it is raining” implies “drops of water are striking earth.” In this example the implication is due to the conclusion being contained in the premise
Now consider the statement “it is raining implies that the ground is wet.” This would usually but not necessarily be true – the ground might be molten lava and cause evaporation without becoming wet. Still we may insist that for practical purposes “it is raining implies that the ground is wet.”
It is an interesting question whether such “causal” or “material” implication is ever certain in the way that tautological implication is certain. The answer is not clear. However, if we develop a deductive logic, we can then apply it whenever implication is certain regardless of the reason for the certainty
Symbolic logic begins when propositions and other elements of logic are denoted by symbols. The letters A, B, C… may stand for propositions. Symbols are also introduced for logical relations and operations. Thus É is the symbol for implication. “A É B” is read “A implies B”. “A É B” is equivalent to “if A is true then B is true,” or “if A is true and if A É B then B is true.”
Propositions can be combined. “A . B” is read “A and B”. As an example consider “it is morning and the sky is blue”; that is true precisely when both “it is morning” and “the sky is blue”. Thus “A and B” is true precisely when both A and B are true. This specifies the meaning of the combination “A and B” of the propositions “A” and “B”. It is the purpose here to motivate the ideas rather than the details of logic. It is possible to develop a logical calculus of propositions without respect to the structure of the propositions. The foregoing shows the motivation for such a calculus. Additionally, the basic concepts and results of such a calculus would apply to any logical calculus that is formulated for propositions that have a restricted kind of structure. The primary example that we consider later is propositions of the subject-predicate form
Primitive symbols: ~, Ú, [,], and an infinite set of variables, p, q, r, . . . [with or without numerical subscripts]
Definitions of ., É, º
[a . b] = {sub Df} ~[~a Ú ~b ]
[a É b] = {sub Df} [~aÚ b]
[a º b] = {sub Df} [[a É b] . [b É a]]
Rules of formation
FR1. A variable standing alone is a wff
FR2. If a is a wff, so is ~ a
FR3. If a and b are wffs, [a . b] is a wff
Axioms:
1. [p Ú p] É p
2. q É [p Ú q]
3. [p Ú q] É [q Ú p]
4. [q É r] É [[p Ú q] É [p Ú r]]
Axiom 4 can be read, “If q implies r, then, if either p or q, either p or r.”
Transformation rules:
1. The result of uniformly replacing any variable in a theorem by any wff is a theorem [rule of substitution]
2. If a and [aÉ b] are theorems, then b is a theorem [rule of detachment, or modus ponens].
Box:
Formal Systems
A formal theory T: 1. A countable set of symbols is the given set of symbols of T. A finite sequence of symbols of T is an expression of T 2. There is a subset of expressions of T called the set of well formed formulas [wfs] - there is usually an effective procedure to determine whether a given expression is a wf 3. A set of wfs is identified as the axioms of T - if one can effectively decide whether a given wf is an axiom, T is an axiomatic theory 4. There is a finite set of relations R1, R2, …, Rn, among wfs called rules of inference. For each Ri there is a unique positive integer j such that for every set of j wfs and each wf A one can effectively decide whether the given j wfs are in the relation Ri to A, and if so, A is called a direct consequence of the given wfs by virtue of Ri A proof in T is a sequence A1, … An, of wfs such that, for each i, either Ai is an axiom of T or Ai, is a direct consequence of some previous wfs by virtue of one of the rules of inference A theorem of T is a wf of A such that there is a proof the last wf of which is A If there is an effective procedure -algorithm or mechanical method- for determining for any given wf whether it is a theorem, then the theory is called decidable; otherwise it is said to be undecidable A wf A is said to be a consequence in T of a set G of wfs if and only if A is theorem when the axioms are extended by G, i.e., there is effectively a new axiom set = the actual axiom set combined with G. Such a proof is called a proof of A from G. G Þ A is read “A is a consequence of G” - or if dealing with more than one theory we may avoid confusion by writing GTT Þ A to indicate the theory in question If j is the empty set then j Þ A, abbreviated Þ A, iff A is a theorem. Thus Þ A iff A is a theorem Note that [1] If G Í
D
and if G Þ A then D Þ
A [2] G Þ A iff there is a finite subset D G such that D Þ A [3] If DÞ A and for each B Ì D,
G Þ B, then G Þ
A |
Box: Formal Axiomatic Theory
for the Propositional Calculus PC
1. The symbols of PC are the statement [proposition] letters Ai, i = 1, 2, … and the primitive connectives ~ and É 2. [a] All Ai are wfs [b] if A and B are wfs then ~ A and A É B are wfs 3. If A, B and C are wfs of PC then the following are axioms [A1] [A É [ B É A]] [A2] [[A É [B É C]] É [[A É B] É [A É C]]] [A3] [[~ B É ~ A] É [[~B É A] É B]] 4. The only rule of inference is modus ponens: B is a direct consequence of A and A É B Definitions of the other connectives [D1] [A Ù B] for ~ [A É ~ B] [D2] [A Ú B] for [~ A] É B [D3] [A º B] for [A É B] Ù [B É A] |
Sentences –and therefore propositions– are frequently in a form in which something is “said” or predicated about a noun or subject. This is the subject-predicate form. It is sometimes held that this form is the form of all propositions. Regardless, it is the form of many propositions. The form may be written f[x] where f is the predicate and x is the subject. In the sentence “The boy is young”, “The boy” is the subject, x, and “is young” is the predicate. In this example f is a one place predicate. A two place predicate is a relation between two individual variables or subjects and may be written y[x, y]; as an example consider “x is greater than y” – here x and y may be numbers and y is “is greater than”
Categorical propositions are examples of the subject-predicate form. In many arguments the premises and conclusions that are stated or can be restated as categorical propositions
These are characterized by their quality - affirmative or negative and, in the second place, by their quantity - either universal or particular. There are four combinations:
A: universal affirmative All A’s are B’s
E: universal negative No A’s are B’s
I: particular affirmative Some A’s are B’s
O: particular negative Some A’s are not B’s
To be strictly categorical A and B are classes of objects. Many propositions are not of this form but can be reformulated as such - or, occasionally as more than one categorical proposition. In all cases the propositional calculus is applicable.
Box:
Recasting Arguments in Terms of Categorical Propositions[10]
Most arguments are not expressed as categorical propositions Propositions must be
expressed using two noun phrases joined by the appropriate copula, a form of
the verb to be Original: Some boys are singing Rewritten: Some boys are persons who are singing Languages contain
many more verbs than the standard copula; sentences with alternative verbs
must be rewritten: Original: All men think Rewritten: All men are persons that think Reference to one
individual The sentence “Ptolemy is a man” is considered to be a singular proposition. Some logicians allow such sentences in arguments and treat them as universal categorical propositions. It is usually better, however, to rewrite such sentences as explicit categorical propositions: All persons identical to Ptolemy are men Non-categorical
quantifiers: Languages usually have various devices for expressing quantifiers --occasionally the quantifier is not explicitly expressed: “A duck is a bird” instead of “All ducks are birds.” Further examples of non-categorical quantifiers recast in categorical form: “A few scientists are brilliant.” is rewritten “Some scientists are brilliant.” Conditional
sentences have the form “If . . ., then .” If the antecedent [“if” clause] and the consequent [“then” clause] refer to the same class of objects, the conditional can be rewritten in categorical form. Otherwise, it must be dealt with differently [see Other argument forms] If an animal is a deer, it’s four footed Rewritten: All deer are four footed When the antecedent and consequent refer to different classes, such rewriting is not possible [e.g., “If the president is reelected, then I shall never vote again”] The forms “Only”,
“The Only”, and “All Except” Finally consider quantifiers such as “Only”. Only A’s are B’s means that, if anything is a B, then it is also an A. The quantifier “The only” is different. “The only runners are Mexicans” means “All runners are Mexicans.” “All except” introduces an exceptive proposition which require two categorical propositions to state their content. “All except teachers teach” means: No teachers are persons who teach All non-teachers are persons who teach |
The simplest arguments from categorical propositions, called immediate inferences, have one premise and one conclusion
A: All A’s are B’s
E: No A’s are B’s
I: Some A’s are B’s
O: Some A’s are not B’s
There are three common transformations of these basic forms of categorical proposition: the converse, the obverse and the contrapositive of the original
Original Converse
The subject and predicate terms are interchanged. Thus A becomes “All B’s are A’s”. The argument from original to converse is valid only in the cases E and I
The Obverse
Affirmative / negative are interchanged, and the predicate term is replaced by its negation [frequently formed by prefixing “non-”]. Thus, “All A’s are B’s” becomes “No A’s are non-B’s”. Conclusion of the obverse from the original is valid in all cases - in fact the original and the obverse are logically equivalent
The contrapositive
Subject and predicate are interchanged and both are negated. Only for A and O do the contrapositives follow from the original
An argument with two categorical propositions as premise and one as conclusion is next in complexity after the immediate inference. When such an argument has three terms, the predicate of the conclusion occurs in the first premise and the subject of the conclusion occurs in the second premise, the argument is a categorical syllogism
The kinds of categorical syllogism can be enumerated. There are 64 moods AAA, AAE… and four figures - the patterns in which the terms S, M and P [subject, middle and predicate] are arranged:
|
Figure 1 |
Figure 2 |
Figure 3 |
Figure 4 |
|
M - P |
P - M |
M - P |
P - M |
|
S - M |
S - M |
M - S |
M - S |
|
\ S - P |
\ S - P |
\ S - P |
\ S - P |
There are 256 syllogisms. The task of syllogistic logic is to show which syllogisms are valid.
Box:
Predicate Calculi
The Predicate Calculus Propositions may also be built up, not out of other
propositions but out of elements that are not themselves propositions. The
simplest kind to be considered here are propositions in which a certain
object or individual (in a wide sense) is said to possess a certain property
or characteristic; e.g., "Socrates is wise" and "The
number 7 is prime." Such a proposition contains two distinguishable
parts: (1) an expression that names or designates an individual and (2) an
expression, called a predicate, that stands for the property that that
individual is said to possess. If x, y, z, . . . are
used as individual variables (replaceable by names of individuals) and the
symbols Atomic formulas may be combined with truth-functional
operators to give formulas such as Intuitively, the notions expressed by the words
"some" and "every" are connected in the following way: to
assert that something has a certain property amounts to denying that
everything lacks that property (for example, to say that something is white
is to say that not everything is nonwhite); and, similarly, to assert that
everything has a certain property amounts to denying that there is something
that lacks it. These intuitive connections are reflected in the usual practice
of taking one of the quantifiers as primitive and defining the other in terms
of it. Thus ( in which a is any variable and ( Lower Predicate Calculus A predicate calculus in which the only variables that
occur in quantifiers are individual variables is known as a lower (or
first-order) predicate calculus. Various lower predicate calculi have been
constructed. In the most straightforward of these, to which the most
attention will be devoted in this discussion and which subsequently will be
referred to simply as LPC, the wffs can be specified as follows: Let the
primitive symbols be (1) x, y, . . . (individual variables), (2) 1. An expression consisting of a predicate variable of degree n followed by n individual variables is a wff. 2. If 3. If 4. If If a is any individual variable and For example, in Higher Order Predicate
Calculi A feature shared by LPC and all its extensions so far mentioned
is that the only variables that occur in quantifiers are individual
variables. It is in virtue of this feature that they are called lower (or
first-order) calculi. Various predicate calculi of higher order can be
formed, however, in which quantifiers may contain other variables as well,
hence binding all free occurrences of these that lie within their scope. In
particular, in the second-order predicate calculus, quantification is
permitted over both individual and predicate variables; hence wffs such as ( |
(phi), 
(psi), 
(chi), . . . as predicate variables (replaceable by predicates),
the formula 
xyz; 
)" or often simply as "()," where the blank
is filled by a variable, which may be read "For all "; and
existential quantifiers, written as "(
)," which may be read "For some " or
"There is a such that." ("Some" is to be understood as
meaning "at least one.") Thus (
={sub Df} 
, 
, . . . , each of some specified degree (predicate variables),
and (3) the symbols 
, 
, 
, (, and ). An infinite number of each type of variable can
now be secured as before by the use of numerical subscripts. The symbols 
, 
, and 
are defined as in PC, and 
as explained above. The 
is a wff, so is 
are wffs, so is (
, . . . ) is free. A wff containing no free individual variables
is said to be a closed wff of LPC. If a wff of LPC is considered as a
proposition form, instances of it are obtained by replacing all free
variables in it by predicates or by names of individuals, as appropriate. A
bound variable, on the other hand, indicates not a point in the wff where a
replacement is needed but a point (so to speak) at which the relevant
quantifier applies. 
)(
x )
Introduction to Concepts
Categorical propositions; immediate inference ; categorical syllogisms; other argument forms ; symbolic logic ; inductive logic
Systems of Logic
Formal Logic
Propositional and Predicate Calculi – lower and higher; Modal Logic; Set Theory
Metalogic
Its Nature – syntax and semantics; the axiomatic method; logic and metalogic; semiotics – the study of signs and sign using behavior
Formal Systems and [their] Formal Language[s]
Formal Mathematical Systems – theory: Gödel’s two incompleteness theorems; decidability and undecidability; consistency proofs
Logical Calculi – the calculi of formal logic; theory:
Propositional calculus; the first-order predicate calculus; Löwenheim-Skolem theorem – any system which can be formalized in the first order predicate calculus, if it has a model, will have an enumerable model; the completeness theorem; the undecidability theorem and reduction classes
Model Theory – in which the interpretation of theories formalized in the framework of formal logic are studied
Background and typical problems. Satisfaction of a theory by a structure: finite and infinite models; elementary logic; non-elementary logic and future developments
Characterizations of the first-order logic; generalizations and extensions of the Löwenheim-Skolem theorem; ultrafilters, ultraproducts, and ultrapowers – constructs that are useful in studying models
Applied Logic
Critique of Forms of Reasoning – correct and defective argument forms; kinds of fallacies - material, verbal and formal fallacies
Epistemic Logic – the logic of belief; theory of belief. The logic of knowing; the logic of questions
Practical Logic – theory
of reasoning with concepts of practice, of analyzing relations among statements
about actions and their accompaniments in choosing, planning, commanding,
permitting…
Logic of preference – or the logic of choice, also known as proairetic logic
Logic of commands; deontic logic – the permitted, the obligatory, the forbidden, or the meritorious are the deontic modalities; systematization and relation to alethic modal logic – what is commonly called modal logic, the modal logic or logics of truth [and falsehood]; alternative deontic systems
Logics of Physical Application
Temporal logic – classic historical treatments; fundamental concepts and relations of temporal logic; systematization of temporal reasoning
Mereology – founded by Stanislaw Lesniewski, mereology clarifies class expressions and axiomatizes the relation between parts and wholes; axiomatization of mereology
Computer Design and Programming
There is a clear connection between two valued logics and the 0 / 1 binary elementary computer states. However the details of computer design depend more on lattice theory than on logical theory. There is, however, a strong connection between computer programming and logical theory. Additionally, there is a connection between machine states and the possible worlds used in the semantics of modal logic. This connection, extended to include dynamic logic – the logic of dynamic or non-static descriptions of the world, temporal logic and process logic – the logic or argumentation that is applicable to all kinds of processes, has been used to study the properties and behavior of computer programs – e.g., does a program stop after a finite number of steps?
Hypothetical Reasoning and Counterfactual Conditionals
These have to do with what would obtain if something that is not known to be true or known to be not true were true. I have read that developments of the theory have various practical applications, including science and history
Formal Logic and the
Variety of Logics
One of the purposes of formalization is to abstract and make precise the kinds of informal argumentation. In so doing, we get systems that are specialized, even compartmentalized. The price of precision is that some of the variety and interconnections are lost. At the same time formalization leads, in addition to greater precision, to the study and extension of formal systems. Obviously, care is needed in the selection of one or another formal system to a particular situation. While some logical systems have been constructed out of or as curiosities, the origin of formal logic is in the desire to make explicit and analyze the structures of inference
The purposes of study of logic and of the variety, here, are as follows. First, out of interest in the analysis of, the need for and the application of the various forms. Second, to consider what structural unities there may be among the various logics; is there a single logic from which all may be derived? Third, in an attempt to formulate a theory of logic: what is logic, what is its relationship to knowledge and the knowledge process, is there a single logic or concept of logic from which all the foregoing follow, and, finally, is there a Logic and, if so, what is its relation to logic?
How will such a logic or Logic be generated? The bases will include:
The kinds of category: substance, space, time, cause, part-whole…
The kinds of knowledge: body, iconic, linguistic – these correspond roughly to immersion, acquaintance and description
The kinds of speech act or, more generally, in a theory of fit between mind and world in the knowledge function: assertive, directive, commissive, expressive and declarative… resulting in assertive or propositional, directive / deontic logics, commissive logics, expressive and declarative logics
Variety of practical application
Unifying and differentiating principles applied to the foregoing
The objective of the study of all logic is to recognize, analyze and classify various forms of inference. Modal logics deal with the concepts of “necessity”, “possibility” and “strict implication.” What is the motivation for these ideas?
Some true propositions are contingently true - they could be
false but happen to be true. An example is “There is a country called
A proposition is possible if its negation is not necessary. If
p, q, … are propositions then “p strictly implies q” means “it is necessarily
true that p implies q.”
Box:
Systems of Modal Logic
Alternative Systems of Modal Logic All the systems to
be considered here have the same wffs but differ in their axioms. The wffs
can be specified by adding to the symbols of PC a primitive monadic operator L
and to the formation rules of PC the rule that if The modal system
known as T has as axioms some set of axioms adequate for PC (such as those of
PM), and in addition 1. Lp
2. L(p
Axiom 1 expresses
the principle that whatever is necessarily true is true, and 2 the principle
that, if q logically follows from p, then, if p is a
necessary truth, so is q (i.e., that whatever follows from a
necessary truth is itself a necessary truth). These two principles seem to
have a high degree of intuitive plausibility, and 1 and 2 are theorems in
almost all modal systems. The transformation rules of T are uniform
substitution, modus ponens, and a rule to the effect that if Among the simpler
theorems of T are p L(p M(p (Lp M(p and LMp (p ( L(p There are many
modal formulas that are not theorems of T but that have a certain claim to
express truths about necessity and possibility. Among them are Lp The first of these
means that if a proposition is necessary, its being necessary is itself a
necessary truth; the second means that if a proposition is possible, its
being possible is a necessary truth; and the third means that if a
proposition is true, then not merely is it possible but its being possible is
a necessary truth. These are all various elements in the general thesis that
a proposition's having the modal characteristics it has (such as necessity,
possibility) is not a contingent matter but is determined by logical
considerations. Although this thesis may be philosophically controversial, it
is at least plausible, and its consequences are worth exploring. One way of
exploring them is to construct modal systems in which the formulas listed
above are theorems. None of these formulas, as was said, is a theorem of T;
but each could be consistently added to T as an extra axiom to produce a new
and more extensive system. The system obtained by adding Lp The relations
between these four systems are as follows: S4 is stronger than T--i.e.,
it contains all the theorems of T and others besides. B is also stronger than
T. S5 is stronger than S4 and also stronger than B. S4 and B, however, are
independent of each other in the sense that each contains some theorems that
the other does not have. It is of particular importance that if Mp Examples of
theorems of S4 that are not theorems of T are Mp Considerations of
space preclude an account of the many other axiomatic systems of modal logic
that have been investigated. Some of these are weaker than T; such systems
normally contain the axioms of T either as axioms or as theorems but have
only a restricted form of the rule of necessitation. Another group comprises
systems that are stronger than S4 but weaker than S5; some of these have
proved fruitful in developing a logic of temporal relations. Yet another
group includes systems that are stronger than S4 but independent of S5 in the
sense explained above. Modal predicate logics can be formed also by making
analogous additions to LPC instead of to PC. |
is a wff, so is L
(to be interpreted as "It is possible that"
and "strictly implies," respectively) can then be introduced by the
following definitions, which reflect in an obvious way the informal accounts
given above of the connections between necessity, possibility, and strict
implication: if 
L
are any wffs, then 
(Lp
q) Fuzzy logics
Multi-valued logics
Many logics, though useful, reduce to standard two valued
logics
1.
Logicism:
mathematics is branch of logic
Formalism:
mathematics is logic together with special mathematical terms and axioms. Thus
the terms and axioms of mathematics do not necessarily have meaning
Intuitionism:
mathematics is essentially independent of logic - it is an intellectual
activity that deals with intellectual constructions governed by self-evident
laws
2. The
equation
3. Mathematics
uses logic in deduction
In deduction the conclusions necessarily follow from
premises. In a sense the conclusions are contained in and determined by the
premises
In inductive logic the argument is by “generalization” e.g. interpolation
or extrapolation or by analogy. The development is not simple generalization
and may involve the use of conceptual understanding, laws and theories as part
of generalization. Seeing a pattern is an example of induction. Science is
inductive in its relation between observation and theory. There is a more
detailed discussion in the section Logic and
Certainty
Two questions arise. The first is whether there is a logic
of induction. A general approach to this question distinguishes between
creation and verification. The second question is in regard to necessity. Even
though the conclusions are not contained in the premises are there any
situations in which they are necessary? Generally, the theories of science are
regarded as revisable in light of new observations. The question is whether
there are any exceptions to this. The most general form of the question is
whether finite being may achieve or realize any absolutes
Mathematics, Deduction and Induction
Mathematics is commonly thought of as deductive. Intuitionism
questions that that defines the full nature of mathematics. Even if mathematics
is the discovery [or re-cognition] of a pre-existing Platonic world there is in
the interplay of formulation and reformulation of formal systems an inductive
process of matching that Platonic universe
Postscript: Plans for This Document
Metaphysical considerations: from the nature of being – implications
for the nature of knowledge. And, from the possibilities and ultimate nature of
knowledge – implications for the nature of Being and possibilities and limits
for individual being
Epistemological considerations: from the nature of knowledge
and relations among elements of knowledge – implications for the nature and
development of logic
From formal logic – completeness, consistency, decidability,
computability – implications for the nature and limits of mind in general, and
in particular as dependent on the varieties and kinds of knowledge
Add any details that will add to or support the arguments of
this narrative – from An Outline of Logic