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Document status: June 16, 2004


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


Kinds of Knowledge    |    Detailed descriptions    |    Map of Mind    |    On Language [See On Signs, below]    |    Box: On Signs    |    Tracing Back: Roots of Language, Varieties of Linguistic Expression    |    Animal Communication    |    Language    |    Box: Language as a Biological Phenomenon    |    Box: Solipsism and Experience    |    Tracing Forward: From Language to Knowledge by Description: A Very Brief Sketch    |    Box: Propositions    |    Generating Knowledge    |    The Possibility of Inference or Logic    |    Initial Comments on Unity Among the Kinds of Knowledge    |    Inference: From Perception to Description    |    Implications for Logic    |    Logic and Certainty    |    Logic and Language    |    The Propositional Calculus    |    Box: Formal Systems    |    Box: Formal Axiomatic Theory for the Propositional Calculus PC    |    Predicates    |    Categorical Propositions    |    Box: Recasting Arguments in Terms of Categorical Propositions    |    Immediate Inference    |    Syllogism    |    Box: Predicate Calculi    |    Logical Systems    |    Introduction to Concepts    |    Systems of Logic    |    Formal Logic    |    Metalogic    |    Applied Logic    |    Formal Logic and the Variety of Logics    |    Modal Logics    |    Box: Systems of Modal Logic    |    Other Deductive Systems    |    Mathematics    |    Inductive Logics    |    Mathematics, Deduction and Induction    |    Postscript: Plans for This Document


Kinds of Knowledge

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

Detailed descriptions

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

Map of Mind

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 “OM” is regarded as an intrinsic symbol in certain traditions from India

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

Directive: world to mind direction of fit

Commissive: world to mind direction of fit

Expressive: null direction of fit

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

Animal Communication

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

Generating Knowledge

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

The recent treatments of logic in Journey in Being and, especially, Foundation and ‘Whereof one cannot speak…’ go beyond the treatment here in understanding the nature, concept, scope, and possibilities of logic. The understanding of logic here is quite traditional in taking the scope of logic to be a tool for inference. The newer treatment embeds the traditional scope in one where LOGIC is at or close to the core of being. The newer treatment has affinities with Hegel’s concept of logic but avoids his lack of foundation; further the traditional scope is placed within the more fundamental concept. Additionally, the new treatment shows the way in which the variety of specialized logics (propositional calculus, predicate calculi, tensed logics and so on) relate to one another and to LOGIC. But because of the expanded scope, the concept of LOGIC allows us to see the affinities between all activities of being –at a primal level– and creative endeavors at the symbolic level without conflation or confusion. I do not presently plan to incorporate the new ideas in the present document. Rather, I may use the technical information – after enhancement– to round out the treatments in the foregoing documents

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?

Implications for Logic

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

Logic and Certainty

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

Logic and Language

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

The Propositional Calculus

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} [~ 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


[p Ú p] É p

q É [p Ú q]

[p Ú q] É [q Ú p]

[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:

The result of uniformly replacing any variable in a theorem by any wff is a theorem [rule of substitution]

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:

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

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

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

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 A and for each B Ì D, G Þ B, then G Þ A


Box: Formal Axiomatic Theory for the Propositional Calculus PC

The symbols of PC are the statement [proposition] letters Ai, i = 1, 2, … and the primitive connectives ~ and É

[a] All Ai are wfs

[b] if A and B are wfs then ~ A and A É B are wfs

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]]

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

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

Immediate Inference

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 {phi}[phi], {psi}[psi], {chi}[chi], . . . as predicate variables [replaceable by predicates], the formula {phi}x is used to express the form of the propositions in question. Here x is said to be the argument of {phi}; a predicate [or predicate variable] with only a single argument is said to be a monadic, or one-place, predicate [variable]. Predicates with two or more arguments stand not for properties of single individuals but for relations between individuals. Thus the proposition “Tom is a son of John” is analyzable into two names of individuals [“Tom” and “John”] and a dyadic or two-place predicate [“is a son of”], of which they are the arguments; and the proposition is thus of the form {phi}xy. Analogously, “. . . is between . . . and . . .” is a three-place predicate, requiring three arguments, and so on. In general, a predicate variable followed by any number of individual variables is a wff of the predicate calculus. Such a wff is known as an atomic formula, and the predicate variable in it is said to be of degree n, if n is the number of individual variables following it. The degree of a predicate variable is sometimes indicated by a superscript--e.g., {phi}xyz may be written as {phi}{sup 3}xyz; {phi}{sup 3}xy would then be regarded as not well formed. This practice is theoretically more accurate, but the superscripts are commonly omitted for ease of reading when no confusion is likely to arise

Atomic formulas may be combined with truth-functional operators to give formulas such as {phi}x {logical or}{psi}y [Example: “Either the customer [x] is friendly [{phi}] or else John [y] is disappointed [{psi}]”]; {phi}xy {contains}{similar to}{psi}x [Example: “If the road [x] is above [{phi} ] the flood line [y], then the road is not wet [{similar to} {psi}]”]; and so on. Formulas so formed, however, are valid when and only when they are substitution-instances of valid wffs of PC and hence in a sense do not transcend PC. More interesting formulas are formed by the use, in addition, of quantifiers. There are two kinds of quantifiers: universal quantifiers, written as “[{for all} ]” or often simply as “[],” where the blank is filled by a variable, which may be read “For all “; and existential quantifiers, written as “[{there exists}],” which may be read “For some “ or “There is a such that.” [“Some” is to be understood as meaning “at least one.”] Thus [{for all}x]{phi} x is to mean “For all x, x is {phi}“ or, more simply, “Everything is {phi}“; and [{there exists} x]{phi}x is to mean “For some x, x is {phi}“ or, more simply, “Something is {phi}“ or “There is a {phi}.” Slightly more complex examples are [{for all} x][{phi}x {contains}{psi}x] for “Whatever is {phi}is {psi},” [{there exists} x][{phi}x{dot} {psi}x] for “Something is both {phi}and {psi},” [{for all} x][{there exists} y]{phi}xy for “Everything bears the relation {phi}to at least one thing,” and [{there exists} x][{for all}y ]{phi}xy for “There is something that bears the relation {phi}to everything.” To take a concrete case, if {phi}xy means “x loves y” and the values of x and y are taken to be human beings, then the last two formulas mean, respectively, “Everybody loves somebody” and “Somebody loves everybody.”

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 {for all}may be taken as primitive, and {there exists}introduced by the definition

[{there exists}a]{alpha} ={sub Df} {similar to}[{for all}a]{similar to} {alpha}

in which a is any variable and {alpha}is any wff; or, alternatively, {there exists}may be taken as primitive, and {for all}introduced by the definition

[{for all}a]{alpha} ={sub Df} {similar to}[{there exists}a]{similar to} {alpha}

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] {phi}, {psi}, . . ., each of some specified degree [predicate variables], and [3] the symbols {similar to}, {logical or}, {for all}, [, and ]. An infinite number of each type of variable can now be secured as before by the use of numerical subscripts. The symbols {dot}, {contains}, and {congruence}are defined as in PC, and {there exists}as explained above. The formation rules are:

1. An expression consisting of a predicate variable of degree n followed by n individual variables is a wff

2. If {alpha}is a wff, so is {similar to}{alpha}

3. If {alpha}and {beta}are wffs, so is [{alpha} {logical or}{beta}]

4. If {alpha}is a wff and a is an individual variable, then [{for all} a]{alpha} is a wff. [In such a wff, {alpha}is said to be the scope of the quantifier]

If a is any individual variable and {alpha}is any wff, every occurrence of a in {alpha}is said to be bound [by the quantifiers] when occurring in the wffs [{for all}a] {alpha}and [{there exists} a]{alpha}. Any occurrence of a variable that is not bound is said to be free. Thus, in [{for all} x][{phi}x {logical or}{phi}y] the x in {phi}x is bound, since it occurs within the scope of a quantifier containing x, but y is free. In the wffs of a lower predicate calculus, every occurrence of a predicate variable [{phi}, {psi}, {chi}, . . . ] 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

For example, in {phi}x, in which both variables are free, each variable must be replaced appropriately if a proposition of the form in question [such as “Socrates is white”] is to be obtained; but in [{there exists} x]{phi}x, in which x is bound, it is necessary only to replace {phi}by a predicate in order to obtain a complete proposition [e. g., replacing {phi}by “is white” yields the proposition “Something is white”]

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 [{for all}{phi} ][{there exists}x ]{phi}x can be formed. This last formula, since it contains no free variables of any kind, expresses a determinate proposition--namely, the proposition that every property has at least one instance. One important feature of this system is that in it identity need not be taken as primitive but can be introduced by defining x = y as [{for all} {phi}][{phi} x {congruence}{phi}y]--i.e., “Every property possessed by x is also possessed by y and vice versa.” Whether such a definition is acceptable as a general account of identity is a question that raises philosophical issues too complex to be discussed here; they are substantially those raised by the principle of the identity of indiscernibles, best known for its exposition in the 17th century by Leibniz

Note: here and in Systems of Modal Logic  11, I have referred to Encyclopedia Britannica, 15th Edition

Logical Systems

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


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

Modal Logics

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 India.” Some true propositions could not be false. An example is “A ripe apple is an apple.” Such a proposition is necessarily true and is called a necessary proposition. The propositional calculus does not account for the difference between necessary propositions and propositions that are contingently true. That is the task of modal logic

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

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 {alpha}is a wff, so is L{alpha}. L is intended to be interpreted as "It is necessary that," so that Lp will be true if and only if p is a necessary proposition. The monadic operator M and the dyadic operator {dyadic operator}(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 {alpha}is any wff, then M{alpha} is to be an abbreviation of {similar to}L{similar to} {alpha}; and if {alpha}and {beta}are any wffs, then {alpha}{dyadic operator}{beta}is to be an abbreviation of L({alpha} {contains}{beta}) [or alternatively of {similar to}M({alpha}{dot} {similar to}{beta})]

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 {contains}p

2. L(p {contains}q) {contains}(Lp {contains}Lq)

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 {alpha}is a theorem so is L{alpha} (the rule of necessitation). The intuitive rationale of this rule is that, in a sound axiomatic system, it is expected that every instance of a theorem {alpha}will be not merely true but necessarily true--and in that case every instance of L{alpha} will be true

Among the simpler theorems of T are

p {contains}Mp,

L(p{dot}q) {congruence}(Lp{dot} Lq),

M(p {logical or}q) {congruence}(Mp {logical or}Mq),

(Lp {logical or}Lq) {contains}L(p {logical or}q) (but not its converse),

M(p{dot}q) {contains}(Mp{dot} Mq) (but not its converse),


LMp {congruence}{similar to}ML{similar to} p,

(p {dyadic operator}q) {contains}(Mp {contains}Mq),

({similar to}p {dyadic operator}p) {congruence}Lp,

L(p {logical or}q) {contains}(Lp {logical or}Mq)

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 {contains}LLp, Mp {contains}LMp, and p {contains}LMp

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 {contains}LLp to T is known as S4; that obtained by adding Mp {contains}LMp to T is known as S5; and the addition of p {contains}LMp to T gives the Brouwerian system, here called B for short

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 {contains}LMp is added to T then Lp {contains}LLp can be derived as a theorem but that if one merely adds the latter to T the former cannot then be derived

Examples of theorems of S4 that are not theorems of T are Mp {congruence}MMp, MLMp {contains}Mp, and (p {dyadic operator}q) {contains}(Lp {dyadic operator}Lq). Examples of theorems of S5 that are not theorems of S4 are Lp {congruence}MLp, L(p {logical or}Mq) {congruence}(Lp {logical or}Mq), M(p{dot}Lq) {congruence}(Mp{dot} Lq), and (Lp {dyadic operator}Lq) {logical or}(Lq {dyadic operator}Lp). One important feature of S5 but not of the other systems mentioned is that any wff that contains an unbroken sequence of monadic modal operators (Ls or Ms or both) is provably equivalent to the same wff with all these operators deleted except the last

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

Other Deductive Systems

Fuzzy logics

Multi-valued logics

Many logics, though useful, reduce to standard two valued logics


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

The equation is characteristic of mathematics [Wittgenstein]

Mathematics uses logic in deduction

Inductive Logics

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


[1] Going back to John Grote, William James and Bertrand Russell

[2] Is a mood knowledge? It would seem that a mood is a subjective state rather than being about anything. That seems to imply that a mood is not knowledge. However the measure of whether a mental state is about something in the world may be analyzed, in this case, as direct, explicit, verbal description. A mood relates the individual to the world in a certain way. Some moods involve belonging, others involve alienation… Further, moods may be in response to events and states in the world and may condition behavior to be appropriate to those events and states. In this way moods can be about the world.

[3] To use Kant’s term

[4] For further information on tracing back –details, remoteness– see Evolution and Design, 1987 and the literature on evolutionary epistemology.

[5] This system is due to John Searle

[6] There is no suggestion that we are or are not, in any sense, the end result of evolution. There is no suggestion that language or the linguistic ability begins with humans; or that the language function in humans is the universal language function. In the beginning there is one line of evolution that branches. It is not suggested that evolution is a one dimensional progression. Appropriate metaphors may be an envelope, a tree, or a wave. Any metaphor will have limitations. The primary interest in the study of human nature is self interest not superiority. Self interest has a number of expressions. I am an individual; and I am a human being; and a mammal, a vertebrate, an animal…

[7] H. Maturana and F. Varela, The Tree of Knowledge: The Biological Roots of Understanding, 1987.

[8] This is not a claim that invertebrates do not have experience. I see no reason and find no motive to make that claim.

[9] It is common to use “declaration” to mean the same as “assertion”. This is a departure from the earlier identification of different kinds of speech act.

[10] A number of detailed descriptions of logic systems and related items are from Encyclopedia Britannica.

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