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  <title>NARS Tutorial 2</title>
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<h3 align="center"><a href="http://www.mindmakers.org/documents/13">NARS Tutorial</a></h3>
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<h1 align="center">NAL-1: The Core Logic</h1>
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<h2>1. An inheritance logic</h2>
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<p>A <i>term</i> is an internal identifier in NARS, and its simplest form is a
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word in a given alphabet. </p>
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<p>In an inheritance statement, a subject term and a predicate term are linked
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together by a <a
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href="http://en.wikipedia.org/wiki/Copula_%28linguistics%29">copula</a> called
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"inheritance". Inheritance is defined by being reflexive and transitive, and
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interpreted as the generalization-specialization relation. </p>
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<p>The language of IL-1 contains inheritance statements as sentences. </p>
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<p>A non-empty and finite set of statements in IL-1, <i>K</i>, can be treated
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as the <i>experience</i> for a system using the logic. </p>
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<p>The single inference rule of IL-1 corresponds to the transitivity of
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inheritance. The transitive closure of <i>K</i> is the system's
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<i>knowledge</i>, <i>K*</i>. </p>
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<p>[The logics in the IL-NAL family all belong to the "<a
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href="http://en.wikipedia.org/wiki/Term_logic">term logic</a>" tradition, which
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uses subject-predicate sentences and syllogistic rules, while the predicate
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logic tradition uses predicate-argument sentences and truth-functional rules.]
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</p>
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<p>Given <i>K</i>, a statement is true if it is either in <i>K*</i> or is a
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tautology, otherwise it is false. [IL-1 accepts <a
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href="http://en.wikipedia.org/wiki/Closed_world_assumption">Closed-World
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Assumption</a>.]</p>
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<p>Given <i>K</i>, the <i>extension</i> of a term includes its known
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specializations; its <i>intension</i> includes its known generalizations. The
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two together form the <i>meaning</i> of the term. </p>
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<p>The above definitions of truth-value and meaning form an
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<i>Experience-Grounded Semantics</i> (EGS). [EGS is very different from <a
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href="http://plato.stanford.edu/entries/model-theory/">Model-theoretic
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semantics</a>, while similar to <a
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href="http://en.wikipedia.org/wiki/Proof-theoretic_semantics">Proof-theoretic
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semantics</a>.]</p>
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<p>The system implementing IL-1 can answer simple questions by searching its
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knowledge. </p>
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<h2>2. Evidence in NAL-1</h2>
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NAL-1 is obtained by extending IL-1 according to AIKR. The key point is to define <i>evidence</i> for an inheritance statement. </p>
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<p>In an axiomatic system, the truth-value of a statement is determined by its
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relation with the axioms, and indicates its relation with state of affairs in a
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model; in an non-axiomatic system, the truth-value of a statement is determined
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by its relation with evidence, and indicates its relation with the experience
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of the system. </p>
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<p>Evidence is (input or derived) information that has impact on the
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truth-value of a statement in an <i>inconclusive</i> manner, while proof decides the
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truth-value a statement in a <i>conclusive</i> manner. Evidence can be positive or negative, or a mixture of them. Evidence comes to the system one piece at a time.</p>
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<p>A quantitative representation is necessary for an adaptive system, since the
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amount of evidence matters when a selection is made among competitive
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conclusions. The advantage of a numerical measurement of evidence is its generality, not
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its accuracy. Furthermore, an interpretation of the measurement is required,
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which usually defines the measurement in an idealized situation. </p>
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<p>A IL-1 theorem: an inheritance statement is equivalent to the inclusion of the
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extension of the subject in the extension of the predicate, as well as to the
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inclusion of the intension of the predicate in the intension of the subject. Therefore, an inheritance statement summarizes many pairs of inheritance
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statements, each of which provides a piece of evidence. </p>
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<p>In ideal situations (ignoring fuzziness, inaccuracy, etc.), amount of
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evidence can be represented by a pair of non-negative integers. It can be
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generalized into a pair of non-negative real numbers <i>w</i><sup>+</sup> and
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<i>w</i><sup>-</sup>. We use <i>w</i> for "all available evidence", which is
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the sum of <i>w</i><sup>+</sup> and <i>w</i><sup>-</sup>. </p>
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<h2>3. The semantics of NAL-1</h2>
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<p>When truth is evaluated according to experience, "truth-value" measures
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evidential support, and shows degree of belief of the system. Though in principle, the information is already carried by the amount of
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evidence, very often a relative and bounded measurement is preferred. </p>
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<p>A natural indicator of truth is the frequency (proportion) of positive
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evidence in all evidence, that is, <i>f = w<sup>+</sup> / w</i>. The limit <i>f</i>, if exists, is the <a
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href="http://en.wikipedia.org/wiki/Probability_theory">probability</a> for the
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statement. However, from the value of <i>f</i> alone, whether its limit exists
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cannot be determined, not to mention where it is. </p>
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<p>In an open system, all frequency values may be changed by new evidence, and
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this is a major type of uncertainty &mdash; <i>ignorance</i> about the future
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frequency value. [Related approaches include <a
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href="http://plato.stanford.edu/entries/reasoning-defeasible/suppl5.html">higher-order
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probability</a>, <a href="http://www.statisticat.com/credible.html">probability
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interval</a>, <a
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href="http://en.wikipedia.org/wiki/Imprecise_probability">imprecise
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probability</a>, <a
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href="http://en.wikipedia.org/wiki/Dempster%E2%80%93Shafer_theory">Dempster-Shafer
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theory</a>, etc, though none of them fully satisfies the needs of a
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non-axiomatic system.]</p>
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<p>While frequency compares <i>positive</i> and <i>negative</i> evidence, a
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second measurement, <i>confidence</i>, can compare <i>past</i> and
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<i>future</i> evidence, in the same manner. Here the key idea is to only
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consider to a constant horizon in the future, that is, <i>c = w / (w + k)</i>. A high confidence value means the statement is supported by more evidence,
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so less sensitive to new evidence. It doesn't mean that the statement is
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"closer to the reality", or the frequency is "closer to the true probability".
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</p>
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<p>The &lang;frequency, confidence&rang; pair can be used as the truth-value of
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a statement in a non-axiomatic system. It is fully defined on available
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evidence, without any assumption about future evidence. Also, it captures the
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uncertainty caused by negative evidence and future evidence. </p>
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<p>The frequency <i>f</i> value will be in the interval [<i>lower, upper</i>]
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in the near future specified by the horizon <i>k</i>. </p>
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<p>The three representations: {<i>w<sup>+</sup>, w</i>}, &lang;<i>f,
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c</i>&rang;, and [<i>l, u</i>] can be transformed into each other. They all
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represent the system's degree of belief on the statement, or its evidential
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support. </p>
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<p>For the normal statements in an non-axiomatic system, the amount of
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supporting evidence is finite. There are two limit cases that are discussed in
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meta-language only: null (zero) evidence and full (infinite, or no future)
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evidence. </p>
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<p>NAL uses an <i>idealized</i> experience in IL to define its semantic
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notions, while the <i>actual</i> experience of the system contains Narsese
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sentences. At the input/output interface, the truth-value of a statement can also be
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represented imprecisely. If there are N verbal labels, the [0, 1] interval can
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be divided into N equal-width subintervals. Or, default values can be used, so
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that the users can omit the numbers. </p>
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<p>There are reasons to use high-accuracy representations inside the system,
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while allow low-accuracy representations outside the system.</p>
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[Many traditional problems are solved by this treatment of evidence and truth:
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<ul>
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<li>The <a href="http://en.wikipedia.org/wiki/Raven_paradox">Confirmation
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Paradox</a> does not appear here, because the Equivalence Condition is no
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longer held in NAL-1.</li>
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<li><a href="http://en.wikipedia.org/wiki/Wason_selection_task">Wason's
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Selection Task</a> can be re-analyzed similarly. The common response is not a
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fallacy if truth-value depends on both positive and negative evidence.</li>
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<li>Popper's proposed <a
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href="http://en.wikipedia.org/wiki/Critical_rationalism">asymmetry between
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falsification and verification</a> can be criticized in the same way: on its
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assumption that a "theory" is logically a universally quantified proposition.]</li>
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</ul>
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[NAL-1 also achieves a unification in the representation of uncertainty:
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randomness comes from extension; fuzziness comes from intension; ignorance
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comes from the future.]
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<p>[It is not easy to revise predicate logic for this purpose, while to do it in
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a term logic is easy and natural, given the subject-copula-predicate structure.]
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</p>
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<h2>4. Revision and choice</h2>
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<p>Each judgment in the system has an evidential base in the system's experience. </p>
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<p>The <i>revision</i> rule combines distinct evidential bases for the same
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statements. The amount of (positive or negative) evidence is the same of those of the premises.</p>
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<p>[Compared to <a
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href="http://en.wikipedia.org/wiki/Dempster%E2%80%93Shafer_theory#Dempster.27s_rule_of_combination">Dempster's
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rule of combination</a>: there are different interpretations of "evidence
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combination". In NARS, it is a form of average, while in D-S theory, it is a
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form of conjunction.]</p>
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<p>[NAL tolerates inconsistency in knowledge, though it is different from the
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existing <a
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href="http://plato.stanford.edu/entries/logic-paraconsistent/">paraconsistent
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logic</a> and <a href="http://en.wikipedia.org/wiki/Belief_revision">belief
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revision</a>.]</p>
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<p>The <i>choice</i> rule chooses among competing answers to a question. </p>
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<p>For an evaluative question, the statement with a higher confidence value is
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preferred; for a selective question, the statement with a higher expectation
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value is preferred. </p>
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<p>Given two competing answers, one with a confirmation record of 19 out of 20,
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and the other by all <i>n</i> cases, when we should prefer the latter when
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<i>n</i> gets larger? It depends on the value of <i>k</i>. </p>
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<h2>5. Forward inference</h2>
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<p>A syllogistic rule requires the two premises to share one term, and produces
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a conclusion between the other two terms. </p>
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<p>For the inheritance copula, the two premises have four possible
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combinations, and only one of them corresponds to a valid rule in IL-1. In
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NAL-1, all can be valid when associated with a proper truth-value function. </p>
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<p>Truth-value functions are determined according to the semantics, by treating
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the involved measurements as extended Boolean variable. </p>
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<p>A variant of syllogistic rule is a rule for <i>immediate inference</i>,
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which only takes one premise. </p>
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<p>An inference rule of NAL can be either "strong" or "weak", depending on
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whether it converges to an inference rule in IL. [This distinction is similar to
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the traditional distinction between "deductive" and "inductive" inference, or
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between "explicative" and "ampliative" inference.]</p>
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<h2>6. Backward inference</h2>
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<p>A question and a judgment can be used as premises to derive another
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question, if and only if the answer of the derived question and the judgment can be used
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as premises to derive an answer to the original question. </p>
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<p>The syllogistic rules of NAL-1 are <i>reversible</i>, in the sense that to
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exchange a conclusion and a premise leads to another rule. </p>
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<p>Because of the reversibility of the rules, the rule tables for forward and
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backward inference are the same, except truth-values. </p>
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<p>The conclusion of any rule can be used as a premise of any other rule. To
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avoid circular inference, the two premises must have distinct evidential bases.
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<hr>
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<h2>Reference</h2>
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<ul>
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  <li><i><a href="NAL-Wang.pdf">Non-Axiomatic Logic: A Model of Intelligent Reasoning</a></i>, Ch. 2-4</li>
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  <li><i><a href="http://code.google.com/p/open-nars/source/browse/trunk/nars-dist/Examples/Example-NAL1-edited.txt">Examples of NAL-1</a></i>
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  </li>
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</ul>
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