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Mental Metalogic: a New Paradigm in Psychology of Reasoning

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Mental Metalogic: a New Paradigm in Psychology of Reasoning Mental MetaLogic: A New Paradigm in Psychology of Reasoning Yingrui Yang & Selmer Bringsjord [email protected] [email protected] Dept. of Philosophy, Psychology &• Cognitive Science (S.B. & Y.Y.) The Minds & Machines Laboratory (S.B. & Y.Y.) Department of Computer Science (S.B.) Rensselaer Polytechnic Institute Troy NY USA Abstract mental logic theory claims that logically untrained hu- mans reason using domain-independent inference rules We introduce a new theory of reasoning, Mental Meta- or schemas (e.g., that from p q and p one can infer to q; Logic (MML), which uni®es the mental logic and men- proponents of mental logic would_ say: that something like tal models theories currently deadlocked in opposition to each other. MML, as we explain, is based on the ªpsychol- this particular rule is operative in the case of our hypo- ogizingº of all of symbolic logic Ð which includes a syn- thetical Smith). On the other hand, mental model theory tactic component (proof theory) with which mental logic holds that ordinary humans reason using not abstract syn- is associated, a semantic component (model theory and tactic rules, but semantic possibilities they in some sense its relatives) with which mental models is associated, and a ªmetaº component (metatheory or metalogic), hitherto imagine. Scientists on both sides are incontestably sem- largely ignored in the psychology of reasoning. We set inal and proli®c, and the content offered by each is pow- out the seven research questions that MML is designed to erful. But the two sides stand vehemently apart, and as a enable answers to, present the research agenda that arises result the science of reasoning is, in a real sense, funda- from the goal of answering three of these questions, and mentally paralyzed.2 MML will end this paralysis. brie¯y summarize two samples of MML-based empiri- cal/experimental work, and a sample of MML-based AI R&D. 2 The 3 Components of Symbolic Logic MML is a meta-theory because it is inspired by modern symbolic logic, which is primarily a meta-inquiry, as it is 1 Introduction devoted to the mathematical study of mathematical and Few would dispute the claim that logical reasoning is a logical systems.3 And although MML is a psychological fundamental human ability. Smith can't ®nd his car keys, theory, to present it we need a quick review of the three but knows both that he last had them minutes ago in the main components of symbolic logic. kitchen, and that he has only been in one other room In broad strokes, modern symbolic logic has three since then. After scouring every millimeter of the other main components: one is syntactic, one is semantic, and room, he reenters the kitchen and, with con®dence, pokes one is meta-theoretical. The syntactic component in- around and ®nds them. His con®dence is the result of de- cludes speci®cation of the alphabet of a given logical ductive reasoning, that much we know. But what is de- system, the grammar for building well-formed formu- ductive reasoning? In the last three decades, the psychol- las (wffs) from this alphabet, and a proof theory that ogy of reasoning has produced countless papers in psy- precisely describes how and when one formula can be chological journals (such as Psychological Review, Sci- proved from a set of formulas. The semantic compo- ence,andBehavioral & Brain Sciences) aimed at pro- nent includes a precise account of the conditions under viding an answer to this question, and aimed thereby at which a formula in a given system is true or false. The providing a scienti®c explanation of what Smith has just meta-theoretical component includes theorems, conjec- done. Alas, the answer and explanation have never ar- tures, and hypotheses concerning the syntactic compo- rived. The reason is that if one had to capture the psychol- nent, the semantic component, and connections between ogy of reasoning in a word or two, the ®rst choice would these two components. doubtless be ªThe Controversy.º 2 The Controversy is well-known to many of our read- There are ways of avoiding the Controversy. For example, one can lump all abstract ªanalyticº reasoning (deductive rea- ers. In brief, it arises from two competing theories 1 soning of all kinds, statistical reasoning, and so on) together, for modeling and explaining ordinary human reason- and give this reasoning a name, and then give another name ing (especially deductive reasoning). On the one hand, to intuitive, concrete, non-symbolic thinking thinking. This is something Stanovich & West (2000) do. But such a move really 1We say `ordinary' here to exclude those experts who are is avoidance; it makes no progress in the face of the challenge to extensively educated in the ways of formal deductive reason- give a precise scienti®c account of all the various forms of an- ing, and those humans apparently gifted to the degree that their alytic reasoning, including that species of most concern to us: performance on deductive reasoning tasks is what would be ex- deductive reasoning. pected from such experts. For a discussion of such experts, see 3Tellingly, mathematical logic, a part of symbolic logic, is (Bringsjord, Bringsjord & Noel 1998). sometimes called `meta-mathematics.' As an example, consider the familiar logical system Mental MetaLogic known as the propositional calculus (also known as sen- Mental Mental tential logic). The alphabet in this case is simply an in- Logic-like Model-like ®nite list p1;p2;:::;pn;pn+1;:::of propositional vari- Content Content ables (according to tradition p1 is p, p2 is q,andp3 is r), and the ®ve familiar truth-functional connectives ; ; ; ; . The grammar for the propositional calculus: ! is $ ^ _ straightforward and well-known. in untrained reasoners Various proof theories for the propositional calculus "Naturally Occurring" Reasoning are possible. One such theory, a so-called natural deduc- in math/logic, & experts thereof tion theory, includes the inference rule which says that from a pair of formulas having the form φ and φ one ! can infer ; this rule is known as modus ponens.Given Proof Formal a full set of rules of inference, one can speak in general Theory Semantics about an individual formula φ being derivable from a set Metatheory Φ of formulas that serve as suppositions; this is tradition- Symbolic Logic ally written Φ φ. ` Figure 1: Overview of Mental MetaLogic Symmetry A formula derivable from the null set is said to be a the- with Symbolic Logic orem, and where φ is such a formula, we follow custom and write φ holds. When the ®rst direction holds, a logical system is ` to express such a fact. said to be sound, and this fact can be expressed in the no- The semantic side of the propositional calculus is of tation we've invoked as course based on the notion of a truth-value assignment, If Φ φ then Φ = φ. an assignment of a truth-value (T or F , or sometimes, in- ` j stead, 1 or 0) to each propositional variable p , and upon i Roughly put, a logical system is sound if it's guaranteed the truth-functions expressed by the ®ve connectives al- that true formulas can only yield (through derivations) luded to just above; usually these truth-functions are de- true formulas; one cannot pass from the true to the false. ®ned via truth tables. When the ªother directionº is true of a system that system Given a truth-value assignment v, the truth or falsity is said to be complete; in our notation this is expressed by of a given propositional formula, no matter how com- plicated, can be mechanically calculated. Armed with a If Φ = φ then Φ φ. truth-value assignment v, we can say that v ªmakes trueº j ` or ªmodelsº or ªsatis®esº a given formula φ; this is stan- As a matter of fact, the propositional calculus is both dardly written provably sound and complete. One consequence of this v = φ. j is that all theorems in the propositionalcalculus are valid, Some formulas are true on all models. For example, the and all validities are theorems.4 formula ((p q) q) pis in this category. Such formulas are_ said to^: be valid!and are sometimes referred 3 The Components & ML, MM, MML to as validities. To indicate that a formula φ is valid we Now, using the review provided in the previous section, write we characterize mental logic, mental models, and Mental = φ. MetaLogic. This characterization is summed up in Figure j Another important semantic notion is consequence.An 1. individualformulaφ is said to be a consequence of a set Φ The representational system of current mental logic of formulas provided that all the truth-value assignments theory can be viewed as (at least to some degree) a psy- on which all of Φ are true is also one on which φ is true; chological selection from the syntactic components of this is customarily written systems studied in modern symbolic logic. For exam- ple, in mental logic modus ponens is often selected as a Φ = φ. schema while modus tollens5 isn't; yet both are valid in- j ferences in most standard logical systems. Another ex- At this point it's easy enough to describe, through ample can be found right at the heart of Lance Rips' sys- the example of the propositional calculus, the meta- tem of mental logic, PSYCOP, set out in (Rips 1994), theoretical component of modern symbolic logic that is for this system includes conditional proof (p. 116), but particularly relevant to Mental MetaLogic: Meta-theory not the rule which sanctions passing from the denial of would deploy logical and mathematical techniques in or- der to answer such questions as whether or not provabil- 4A fact captured by: = φ if and only if φ.
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