Epistemic Logic

Epistemic Logic: A Survey of the Logic of Knowledge

Nicholas Rescher
Copyright Date: 2005
Pages: 152
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  • Book Info
    Epistemic Logic
    Book Description:

    Epistemic logic is the branch of philosophical thought that seeks to formalize the discourse about knowledge. Its object is to articulate and clarify the general principles of reasoning about claims to and attributions of knowledge. This comprehensive survey of the topic offers the first systematic account of the subject as it has developed in the journal literature over recent decades.Rescher gives an overview of the discipline by setting out the general principles for reasoning about such matters as propositional knowledge and interrogative knowledge. Aimed at graduate students and specialists,Epistemic Logic elucidates both Rescher's pragmatic view of knowledge and the field in general.

    eISBN: 978-0-8229-7092-7
    Subjects: Philosophy

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. Preface
    (pp. ix-xii)
  4. 1 Setting the Stage
    (pp. 1-7)

    Epistemic logic is that branch of philosophical logic that seeks to formalize the logic of discourse about knowledge. Its object is to articulate and clarify the general principles of reasoning about claims to and attributions of knowledge—to elucidate their inferential implications and consequences. In pursuing this goal, it deals principally with propositional knowledge (along the lines of “Smith knowsthatcoal is black”) and secondarily also with interrogative knowledge (along the lines of “Jones knowswherethe treasure is buried andwhoput it there”).¹ It is the object of this book to give an overview of the discipline...

  5. 2 Basic Principles
    (pp. 8-13)

    The distinction between four modes of propositional acceptance/assertion will be serviceable in characterizing the present system of epistemic logic (s):

    Type 1:p, that is, iff ├ (∀x)Kxp. This represents “obvious knowledge” coordinate with acceptance by our epistemic system (s) ofpas somethinguniversally recognizedamong the knowers at issue.

    Type 2: ├p,that is, iffKsp. On this basis we propose to acceptpas part of our epistemic system (s), itself now regarded as a “knower” of sorts.¹ Such acceptance represents “patent knowledge” coordinate with acceptance as certain on epistemico-logical grounds. (It, accordingly, includes the...

  6. 3 Deductivity and Knowledge Ampliation
    (pp. 14-19)

    In developing a system of epistemic logic one needs to be able to engage in deductive inference not only on one’s own account but also on account of the knowers at issue in the deliberations. To this end we shall adopt the following thesis:

    Deductivity principle:IfKxpandKx(pq), thenKxq.

    This thesis credits knowers with the known consequence of known facts. It emerges on this basis that the conception of knowledge at issue with the knowledge operatorKis what might be characterized asavailableknowledge—that is, knowledge that need not be explicitly avowed as...

  7. 4 Metaknowledge
    (pp. 20-24)

    Metaknowledge is knowledge about knowledge. Clearly, there will be very different types of metaknowledge. For one thing, such knowledge can be specific and substantive, as in “xknows thatyknows that Tokyo is the capital of Japan,” or generic and indefinite, as in “Every knower knows that something is known”:


    It is, of course, this second mode of purely formal metaknowledge that is at issue here.

    The hallmark of metaknowledge is its being a matter of knowing propositions in which oneK-operator occurs within the scope of another. Knowledge about someone’s ignorance thus counts as metaknowledge. For if...

  8. 5 For Aught That Someone Knows
    (pp. 25-29)

    The idea at issue with “For aught thatxknows (to the contrary),pis the case” is captured by the definition

    Axpiff ∼(∃q)(Kxq&Kx[q├ ∼p]).

    On this basis it becomes possible to establish that ifpobtains for aught thatxknows, thenxdoes not know not-p

    Axp├ ∼Kxp

    This can be shown as follows:

    And it is also possible to establish the converse of this theorem: ∼KxpAxpor equivalently ∼AxpKxp

    For by the definition ofAxpthis amounts to

    (∃q)(Kxq&Kx[q├ ∼p]) ├Kxp.

    And this follows at once from the...

  9. 6 Group Knowledge
    (pp. 30-34)

    A great deal of knowledge is not individual and personal but collective and social, the “knower” at issue being a group rather than an individual. And it is by no means easy to describe how group knowledge is related to individual knowledge. From the epistemic standpoint a team of individuals, say {x, y}, consisting ofxandy, has to be treated as a new, different entity in its own right. Such a collectivity will know things that no individual does—and this not only in the sense of performatory “know-how” but in factual matters as well. With large physics...

  10. 7 Propositional versus Interrogative Knowledge
    (pp. 35-41)

    From a logical standpoint there is a significant difference between knowingthatand knowingwhat,betweenpropositionaland non-propositionallyinterrogativeknowledge. For one can represent “xknows thatp”—say, that the cat is sitting on the mat—by an expression of the formatKxp,wherepis a complete proposition. But more complex machinery is required for saying that “xknowswhatthe cat is doing”—and, similarly, with other interrogative pronouns such aswho, where, when, why,andhow. For, here, one will have to deal not simply withpropositionsbut withpropositional functions,and quantifying over them...

  11. 8 Collective versus Distributive Knowledge and Knower Limitedness
    (pp. 42-48)

    LetSbe a set of objects or items of some sort. Then an individualxhascollective(or composite,in sensu composito) knowledge thatS-members have a certain featureFwhen


    Here,xknows that whatever belongs toSwill have the featureFand thereby knows a certain general fact regardingS-membership at large, namely, that it involves onlyF-possessors.

    By contrast,xknowsdistributively(orin sensu diviso) of each and every object that if it is anS-member then it hasF:


    Note that in both...

  12. 9 Modality
    (pp. 49-54)

    Let us now consider quantified modal logic (QML) in an epistemic perspective. The logical modalities of possibility (◊) and necessity (□) that function in this domain are subject to the following principles:

    If □p,thenp.

    Ifp,then ◊p.

    Moreover, they are also subject to theprinciples of modal duality:

    piff ∼□∼p.

    piff ∼◊∼p.

    The validation of necessity claims inheres in exploring logico-conceptual relationships. In particular, whenpis demonstrable, then it is thereby demonstrably necessary:

    If ├p, then □p, and indeed ├ □p.

    This, however, is less a fact about necessity as such than an...

  13. 10 Problems of Epistemic Democracy
    (pp. 55-57)

    A knower’s secret is a truth known to this knower alone. Since (as maintained in chapter 4) theconjunctionof everything that is known to a given knower qualifies plausibly in this regard, we should credit every knower with a secret.

    And so since (∀x)(∃t)(xalone knowst) is a thesis of our system, we have it that

    ├ □(∀x)(∃t) (xalone knowst).

    And thus in view of the basic principles of QML we also have

    ├ (∀x)□(∃t) (xalone knowst).

    For every knower there necessarily is a truth known only to this individual.

    Moreover, we also have...

  14. 11 Possibility and Conceivability
    (pp. 58-61)

    While we learn about reality through experience, imagination is our only pathway to universal possibilities. It is thus tempting to join those theorists who hold that the merely possible stands coordinate with whatever a mind can manage to think when functioning properly—that is, to conceive coherently. They thus sought to operate with the following correlation:

    Possible ≈ thinkable/conceivable

    The philosopher Herbert Spencer (1820–1903), for one, effectively equated possibility with conceivability. However, other theorists, such as Edmund Husserl and Gottlob Frege, opposed such a coordination of possibility with conceivability. Conceivability, they insisted, is a psychological matter, whereas abstract possibility...

  15. 12 Unknowability
    (pp. 62-65)

    Clearly, we cannot, on grounds of self-inconsistency, have it that someoneknowssome truth to represent an unknown fact:

    (∃x)(∃p)Kx(p& ∼(∃y)Kyp) or equivalently (∃x)(∃p)(Kxp&Kx∼(∃y)Kyp)

    This thesis must certainly be rejected. But its weaker cousin, to the effect that there indeed is such a thing as an unknown fact—that is,

    (∃x)(∃p)(p&Kx∼(∃y)Kyp) or equivalently (∃x)(∃t)Kx∼(∃y)Kyt

    is perfectly tenable. We have to come to terms with the existence of unknowns.

    With finite knowers we can never have a thesis of the format (∀p)KxF(p) or (∀t)KxF(t), owing to the infinitude of that quantificational range. Accordingly, we have it that


    so that in...

  16. 13 Fitch’s Theorem and Its Consequences
    (pp. 66-70)

    The findings on unknowability discussed in the preceding chapter have interesting ramifications. Thus, consider yet another application of the Modal Collapse theorem, namely, that arising with the following specification ofF:


    The theorem’s grounding principles (of conjunction and veracity) now both obtain once again, so that

    If (∀t)◊Kxt,then (∀t)Kxt.

    Now one certainly does not have it in general that

    If (∀u)◊Fu,then (∀u)Fu.

    In this light the aforementioned implication thesis is extraordinary and may seem surprising.

    The implication thesis at issue—initially established by Fitch in 1963—might be called Fitch’s theorem. For all intents and...

  17. 14 Finite and Infinite Knowers
    (pp. 71-75)

    The difference between a finite and an infinite knower is of farreaching importance and requires careful elucidation. For an “infinite knower” should not be construed as anomniscientknower—one from whom nothing knowable is concealed (and so who knows, for example, who will be elected U.S. president in the year 2200). Rather, what is now at issue is a knower who can manage to know in individualized detail an infinite number of independent facts. Such a knower might, for example, be able to answer such a question as “Will the decimal expansion of π always continue to agree for...

  18. 15 Vagrant Predicates and Noninstantiability
    (pp. 76-84)

    One can refer to an item in two distinctly different ways: either specifically and individually by means of naming or identifying characterizations (“George Washington, the father of our country”), or obliquely and sortally as an item of a certain type or kind (“an American male born in the eighteenth century”). Now, a peculiar and interesting mode of reference occurs when an item is referred to obliquely in such a way that its specific identification is flat-out precluded as a matter of principle. This phenomenon is illustrated by claims to the existence of

    a thing whose identity will never be known...

  19. 16 Unanswerable Questions and Insolubilia
    (pp. 85-93)

    It is instructive to adopt an erotetic—that is, question-oriented—view of knowledge and ignorance. It can be supposed, without loss of generality, that the answers to questions are always complete propositions. Often, to be sure, it appears on the surface that a specific item is merely at issue with a question, as in the following example:

    Q: “Who is that man?”

    A: “Tom Jones.”

    Q: “When will he come?”

    A: “At two o’clock.”

    Q: “What prime numbers lie between two and eight?”

    A: “Three, five, and seven.”

    But, throughout, the answers can be recast as completed propositions, respectively: “That...

  20. 17 Unknowable Truth
    (pp. 94-99)

    As already noted, we can move from the by-xunanswerability of “What is an example of truth that you,x, do not know to be so?” to the universally insoluble “What is an example of a truth that no one whatsoever knows to be so?” For in a finite society of imperfect knowers, the existence of such a truth is guaranteed by the conjunctivity principle mooted earlier, that ift1is a truth thatx1does not know it to be so andt2is a truth thatx2does not know to be so, thent1&t2is a truth...

  21. 18 Implications of Cognitive Limitation
    (pp. 100-104)

    While there are, indeed, cognitive insolubilia—and we can plausibly identify some of them—the fact remains that detailed knowledge about theextentof our ignorance is unavailable to us. For what is at stake with this issue of extent is the size-ratio of the manifold of what one does know to the manifold of that what one does not. And getting a clear fix on the latter is not possible. For the actual situation is not a crossword puzzle or a geographic exploration where we can somehow measure the size of the terra incognita in advance. We can form...

  22. Appendix 1: A Survey of Thesis Acceptability
    (pp. 105-108)
  23. Appendix 2: On Quantifying Knowledge (and the Gulf Between Linguistic Truth and Objective Fact)
    (pp. 109-124)
  24. Notes
    (pp. 125-134)
  25. Bibliography
    (pp. 135-138)
  26. Index
    (pp. 139-140)