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  <title>NARS Tutorial 5</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-5 to NAL-6: Higher-order Inference</h1>
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<p></p>
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<h2>1. Higher-order statement</h2>
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When a statement is taken as a term, it can be used to form "higher-order" statements, i.e., statements about statements. On the other hand, a term can name a statement. In this way, the actual difference between
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statements and other terms is semantic: a statement has a truth-value. 
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<p>There are relations that take statements as components, such as "know" and
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"believe", as well as attributes of statements, such as "necessary" and
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"possible". In NAL, most of these relations are not built in (as in <a
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href="http://plato.stanford.edu/entries/logic-epistemic/">epistemic logic</a>
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or <a href="http://en.wikipedia.org/wiki/Modal_logic">modal logic</a>), but
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acquired. </p>
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<p>There is no need to further separate 2nd-order, 3rd-order, etc., nor to
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limit the maximal order allowed in structure. </p>
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Two higher-order copulas, <i>implication</i> and <i>equivalence</i>, are
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defined between statements, indicating the relation between their meanings and
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truth-values. 
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<p>There is a partial isomorphism between first-order NAL and higher-order NAL. See Table 9.1 of the textbook.
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</p>
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<p>Overall, NAL has four basic copulas that are directly recognized by the
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inference rules. They all represent a certain substitutability ("can be used
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as") relation between terms, and the syllogistic rules correspond to the
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transitivity of the copulas involved. </p>
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<h2>2. Derivation as implication</h2>
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In NAL, implication is defined by derivation. This agrees with the <a
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href="http://en.wikipedia.org/wiki/Deduction_theorem">Deduction Theorem</a> in
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classical logic. 
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<p>The higher-order copulas in IL/NAL are not defined purely by truth functions, as in
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propositional calculus. Here the two statements involved not only need to have
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truth-value relation, but also semantic relation in their contents, which is
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provided by the syllogistic nature of term logic. </p>
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<p>Semantic relation among the components is also required in
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<i>conjunction</i> and <i>disjunction</i>. </p>
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<p>Consequently, NAL is like a <a
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href="http://plato.stanford.edu/entries/logic-relevance/">relevance logic</a>,
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though it provides relevance using the intrinsic nature of term logic, rather
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than using multiple-world semantics. </p>
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<p>Using Deduction Theorem, the truth-value of a statement can be taken as the
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truth-value of a corresponding implication statement, conditioned on the
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available evidence. Using this meta-level equivalence, some new inference rules
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can be introduced into NAL, as variants of the existing rules. </p>
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<h2>3. Negative statement</h2>
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IL-5 still makes <a
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href="http://en.wikipedia.org/wiki/Closed_World_Assumption">CWA</a>, though it
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explicitly expresses negative statements, especially as substatements of
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compound statements. 
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<p>In IL-5, a consistency requirement is added on the experience of NARS. </p>
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<p>Positive and negative statements are not symmetric in NARS, either in the
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logic part or the control part. Negative observation comes from failed
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expectation. In NARS, negation is introduced when a statement has more negative than positive evidence. For a syllogistic rule, two negative premises cannot derive a
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conclusion. </p>
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<p>Equivalent statements with negations in IL may have different truth-values
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in NAL, due to the different amounts of evidence. </p>
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<h2>4. PL, IL and NAL</h2>
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IL uses connectors similar to those in <a href="http://en.wikipedia.org/wiki/Propositional_calculus">Propositional Logic</a> (PL) to build compound
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statements, and the connectors satisfy similar truth-conditions. 
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<p>PL treats inference as purely truth-functional, while IL treats inference as substitutions. Truth-conditions in PL are definitional and primary, but derived and secondary in IL. In IL, statements
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with no semantic relations won't be used as components in a compound, nor as premises in an inference step. Also, in IL (and NAL) statements with the same truth-value are not necessarily equivalent. </p>
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<p>IL is a limit case of NAL, when AIKR can be omitted. The unconditional
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truths (theorems) in IL are embedded in the (structural) inference rules of
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NAL, though not in the beliefs of the system.  There is no axiom (nor theorem) in NAL. The analytic truths are only acknowledged and accepted at the meta-level. </p>
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<p>A PL theorem becomes an IL theorem after the connective to copula
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replacement, if in the former there is semantic relations among the premises
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and conclusions. How to turn an IL theorem or rule into a NAL inference rule needs to be
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analyzed one by one, since different truth-value functions may be needed. By
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default, a true statement in IL is treated as a judgment with full positive
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evidence in NAL. </p>
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<h2>5. Reasoning with variable term</h2>
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<p>A variable term in NARS does not name a concept, and its name is local to a
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statement. Variable terms can be used to separate extensional and intensional evidence,
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as well as to indicate the existence of positive evidence. </p>
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<p>[There are 8 basic forms of variable-involved statement, corresponding to the distinctions
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extensional-intensional, positive-negative, and having-lacking. The extensional
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cases are similar to the 4 figures in <a href="http://en.wikipedia.org/wiki/Syllogism">Aristotle's Syllogistic</a> (A-E-I-O). The <a href="http://en.wikipedia.org/wiki/Square_of_opposition">square of
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opposition</a> represents immediate inference rules in IL. Also, there are
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rules based on the symmetry of I and E. It is possible to make IL partially isomorphic to Aristotle's Syllogistic, though this approach is not taken for the sake of simplicity.]</p>
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<p>Variable related inference include </p>
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<ul>
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  <li>Unification: use a variable to replace another variable; </li>
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  <li>Introduction: use a variable to replace a constant; </li>
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  <li>Elimination: use a constant to replace a variable;. </li>
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</ul>
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<h2>6. Symbolic reasoning</h2>
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<p>A variable term is a symbol that represents a constant term. It can be
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grounded on the latter via variable elimination. Similarly, abstract notions can be introduced or created, without grounding into empirical experience. Instead, <a
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href="http://en.wikipedia.org/wiki/Formal_model">formal models</a> or <a
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href="http://en.wikipedia.org/wiki/Axiomatic_system">axiomatic theories</a> are
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built around these notions, with binary deductive rules. Inference within the
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theory is theorem proving. </p>
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<p>When such a formal model is applied to a practical problem, model-theoretic
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semantics is applied to provide an interpretation to map the abstract notions
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into concrete concepts, so as to get derived conclusions efficiently. 
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The notions in an formal model are "symbols" whose meaning depends on the
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interpretation. This is not the case for the ordinary terms in the system. </p>
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<p>NAL can use acquired relations like <i>define</i> and <i>represent</i> to
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learn (or create) a language, and to relate it to its empirical concepts,
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respectively. NARS can emulate an arbitrary logic, by representing its truth-values and
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propositions as terms, and its inference rules as implication statements.</p>
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<p>[Production systems like <a
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href="http://sitemaker.umich.edu/soar/home">Soar</a> can also implement
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arbitrary inference rules, though NARS is different from these systems, given
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its other inference rules, memory structure, and control mechanism.]</p>
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NARS uses a similar idea to handle natural language. Some preliminary <a href="http://code.google.com/p/open-nars/source/browse/trunk/nars-dist/Examples/Example-NLP-edited.txt">results</a> are discussed in this <a href="http://www.cis.temple.edu/~pwang/Publication/NLP.pdf">paper</a>.
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<p>[The ability of formal inference is related to <a
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href="http://en.wikipedia.org/wiki/Piaget%27s_theory_of_cognitive_development#Formal_operational_stage">Piaget's notion of "formal operation"</a>.]</p>
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<p>[The analytic vs. empirical distinction can be re-established in this way. It
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is related to <a href="http://en.wikipedia.org/wiki/Modal_logic">modal
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logic</a> and <a href="http://en.wikipedia.org/wiki/Possible_world">possible-world theory</a>.
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In NARS, "analytic" is with respect to a set of assumptions made in an abstract
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theory, while "empirical" is with respect to the system's experience.]</p>
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<h2>[7. Traditional and classical logics re-evaluated]</h2>
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<p>IL-6 is comparable with Syllogistic and FOL. </p>
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<p>Key differences between IL-6 and classical <a
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href="http://en.wikipedia.org/wiki/Term_logic">term logic</a>: </p>
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  <li>derivative copulas, </li>
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  <li>compound terms, </li>
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  <li>statements as terms. </li>
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</ul>
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<p>Key differences between IL-6 and classical predicate logic: </p>
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<ul>
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  <li>categorical sentences (relative subject-predicate distinction), </li>
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  <li>meaning as relation (extensional and intensional), </li>
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  <li>syllogistic rules (relevant premise-conclusion). </li>
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</ul>
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<p>IL gets ideas from set theory, propositional logic, and predicate logic. </p>
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<p><a href="http://en.wikipedia.org/wiki/Description_logic">Description
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logic</a> can be seen as a combination of set theory and predicate logic,
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though not very deep. </p>
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<p>NAL is built by fitting IL onto AIKR, with multi-strategy inference, both
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strong (deductive) and weak (non-deductive). </p>
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<p>Many previous discussions on rationality take classical logics and
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probability theory as normative models with universal authority. The same
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happens to the psychological study of human reasoning. </p>
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<p>Peirce initially introduced the <a
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href="http://plato.stanford.edu/entries/peirce/#dia">deduction-induction-abduction
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trio</a> in term logic at the inference rule level, though later use the terms
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informally at the inference process level, aimed at the cognitive functions.
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Now they are usually formalized in predicate logic, so the rules become
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under-specified. </p>
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<p>After all, the differences from NAL and other logics come from AIKR. </p>
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<p></p>
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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. 9-10</li>
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  <li><i><a href="http://code.google.com/p/open-nars/source/browse/trunk/nars-dist/Examples/Example-NAL5-edited.txt">Examples of NAL-5</a>, <a href="http://code.google.com/p/open-nars/source/browse/trunk/nars-dist/Examples/Example-NAL6-edited.txt">Examples of NAL-6</a></i>
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  </li>
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</ul>
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