Piero Scaruffi

Common Sense: Engineering the Mind (These are excerpts from, or extensions to, the material published in my book "The Nature of Consciousness") The Sense of the Mind In our quest for the ultimate nature of the mind, we are confounded by the very way the mind works. The more we study it, the less it resembles a mathematical genius. On the contrary, it appears that the logic employed by the mind when it has to solve a real problem in a real situation is a very primitive logic, one that we refer to as "common sense", very different from the austere formulas of Mathematics but quite effective for the purposes of surviving in this world. If the mind was shaped by the world, then the way the mind reasons about the world is a clue to where it came from and how it works. In emergency situations, our rational thinking is often powerless. Common sense determines what we do, regardless of what we think. The puzzling aspect of common sense is that it is sometimes wrong. There are plenty of examples in the history of science of "paradoxes" about common-sense reasoning. Using common-sense reasoning, the Greek philosopher Zeno proved that Achilles could never overtake a turtle. Using common sense reasoning, one can easily prove that General Relativity is absurd (a twin that gets younger just by traveling very far is certainly a paradox for common sense). Common sense told us that the Earth is flat and at the center of the world. Physics was grounded on Mathematics and not on common sense precisely because common sense is so often wrong. There are many situations in which we teach ourselves to stay "calm", to avoid reacting impulsively, to use our brain. These are all situations in which we know our common sense would lead us to courses of actions that we would probably regret. Why don’t our brains simply use mathematical logic in all their decisions? Why does our common sense tell us things that are wrong? Why can't we often resist the power of that falsehood? Where does common sense come from, and where does its power come from? Illogical Reasoning Common sense is a key factor for acting in the real world. We rarely employ classical Logic to determine how to act in a new situation. More often, the new situation "calls" for some obvious reaction, which stems purely from common sense. If we used Logic, and only Logic, in our daily lives, we would probably be able to take only a few actions a day. Logic is too cumbersome, and allows us to reach a conclusion only when a problem is "well" formulated. In more than one way, common sense helps us deal with the complexity of the real world. Common sense provides a shortcut to making critical decisions very quickly. Common sense encompasses both reasoning methods and knowledge that are obvious to humans but that are quite distinct from the tools of classical Logic. When scientists try to formalize common sense, or when they research how to endow a machine (such as the computer) with common sense, they are faced with the limitations of classical Logic. It is extremely difficult, if not utterly impossible, to build a mathematical model for some of the simplest decisions we make. Common sense knows how to draw conclusions even in the face of incomplete or unreliable information. Common sense knows how to deal with imprecise quantities, such as "many", "red", "almost". Common sense knows how to deal with a problem that is so complex it cannot even be specified (even cooking a meal theoretically involves an infinite number of choices). Common sense knows how to revise beliefs based on facts that all of a sudden are proved false. Logic was not built for any of these scenarios. Furthermore, common sense does not have to deal with logical paradoxes. Paradoxes arising from self-referentiality (such as the liar’s paradox) have plagued Logic since the beginning. A program to ground common sense in Predicate Logic is apparently contradictory, or at least a historical paradox. Science was born out of the need to remove the erroneous beliefs of common sense: e.g., the Earth is not the center of the universe. Science checks our senses and provides us with mathematical tools to figure out the correct description of the world notwithstanding our sense’s misleading perceptions. Science was born out of the need to get rid of common sense. What was neglected is that common sense makes evolutionary sense. Its purpose is not to provide exact knowledge: its purpose is to help an individual survive. The Demise of Deduction Logic is based on deduction, a method of exact inference. Its main advantage is that its conclusions are exact. That is the reason why we use it to build bridges or airplane wings. But deduction is not the only type of inference we know. We are very familiar with "induction", which infers generalizations from a set of events, and with "abduction", which infers plausible causes of an effect. Induction has been used by any scientist who has developed a scientific theory from her experiments. Abduction is used by any doctor when she examines a patient. They are both far from being exact, so much so that many scientific theories have been proved wrong over the centuries and so much so that doctors make frequent and sometimes fatal mistakes. The power of deduction is that no mistake is possible (if you follow the rules correctly). The power of induction and abduction is that they are useful: no scientific theory can be deducted, and no disease can be deducted. If we only employed deduction, we would have no scientific disciplines and no cures. Alas, deduction works only in very favorable circumstances: when all relevant information is available, when there are no contradictions and no ambiguities. Information must be complete, precise and consistent. In practice, this is seldom the case. The information a doctor can count on, for example, is mostly incomplete and vague. The reason we can survive in a world that is mostly made of incomplete, inexact and inconsistent information is that our brain does not employ deduction in everyday life. Intuitionism The limits and inadequacies of Logic had been known for decades and numerous alternatives or improvements had been proposed. There are two main approaches: one criticizes the very concept of "truth", while the other simply extends Logic by considering more than two truth values. As an example of the first kind, "Intuitionism" (a school of thought started in 1925 by the Dutch mathematician Luitzen Brouwer) prescribes that all proofs of theorems must be constructive. Unlike classical Logic, in which the proof of a theorem is only based on rules of inference, in Intuitionistic Logic only "constructable" objects are legitimate. Classical Logic exhibits properties that are at least bizarre. For example, the logical OR operation yields "true" if at least one of the two terms is true; but this means that the proposition "my name is Piero Scaruffi or 1=2" is to be considered true, even if intuitively there is something false in it. Because of this rule, the logical implication between two terms can yield even more bizarre outcomes. A logical implication can be reduced to an OR operation between the negation of the first terms and the second term. The sentence "if x is a bird than x flies" is logically equivalent to "NOT (x is a bird) OR (x flies)". The two sentences yield the same truth values (they are both true or false at the same time). The problem is that the sentence "if the week has eight days then today is Tuesday" is to be considered true because the first term ("the week has eight days") is false, therefore its negation is true, therefore its OR with the second term is true. By the same token, the sentence "Every unicorn is an eagle" is to be considered true (because unicorns do not exist, a fact that makes that formula true). On the contrary, intuitionists accept formulas only as assertions that can be built mentally. For example, the negation of a true fact is not admissible. Since classical Logic often proves theorems by proving that the opposite of the theorem is false (an operation which is highly illegal in Intuitionistic Logic), some theorems of classical Logic are not theorems anymore. Intuitionists argue that the meaning of a statement resides not in its truth conditions but in the means of proof or verification. The "Theory of Types" introduced by the Swedish mathematician Per Martin-Lof in 1970 is an indirect consequence of this approach to demonstration. A "type" is the set of all propositions which are demonstrations of a theorem. Any element of a type can be interpreted as a computer program that can solve the problem represented (or "specified") by the type. This formalizes the obvious connection between Intuitionistic Logic and computer programs, whose task is precisely to "build" proofs. Alan Gupta's "revisionist theory of truth" also highlights how difficult it is to pin down what "true" really means. Truth is actually impossible to define: in order to determine all the sentences of a language that are true when that language includes a truth predicate (a predicate that refers to truth), one needs to determine whether that predicate is true, which in turn requires one to know what the extension of true is, while such extension is precisely the goal. The solution is to assume an initial extension of "true" and then gradually refine it. Gupta suggests that truth can only be refined step by step. An indirect, but not negligible, advantage of Gupta’s approach is that truth becomes a circular concept: therefore all paradoxes that arise from circular reasoning in classical Logic fall into normality. Frederick and Barbara Hayes-Roth’s form of opportunistic reasoning (the "blackboard model" of 1985) stems from the same principles, albeit in a computational scenario. Reasoning is viewed as a cooperative process carried out by a community of agents, each specialized in processing a type of knowledge. Each agent communicates the outcome of its inferential process to the other agents and all agents can use that information to continue their inferential process. Each agent contributes a little bit of truth, that other agents can build on. Truth is built in an incremental and opportunistic manner. Searching for truth is reduced to matching actions: the set of actions the community wants to perform (necessary actions) and the set of actions the community can perform (possible actions). An agent adds a necessary action whenever it runs out of knowledge and has to stop. An agent adds a possible action whenever new knowledge enables it. When an action is made possible that is also in the list of the necessary actions, all the agents that were waiting for it resume their processing. The search for a solution is efficient and more natural, because the only actions undertaken are those that are both possible and necessary. Furthermore, opportunistic reasoning can deal with an evolving situation, unlike classical Logic that considers the world as static. Classical Logic only admits two "values": true or false. Either a proposition or its negation are true (the "law of the excluded middle"). In 1920 the Polish mathematician Jan Łukasiewicz worked out a logic based on more than just two values. First he added “possible” to “true” and “false”. Then he extended the idea to any number of truth values. A logic with more than “true” and “false” is not as “exact” as classical Logic, but it has a higher expressive power. It can be used to better mirror the human experience. Plausible Reasoning In our daily lives, we are rarely faced with the task of finding the perfect solution to a problem. If we are running out of gasoline in the middle of the night, we are happy with finding a gas station along our route, even if its gasoline may not be the best or the cheapest. We almost never pause to figure out the best option among the ones that are available. We pick one that leads to a desired outcome. What our mind is looking for all the time is "plausible" solutions to problems, as opposed to "exact" ones. Mathematics demands exact solutions, but in our daily lives we content ourselves with plausible ones. The reason is that sometimes a plausible solution enables us to survive, whereas looking for an exact one would jeopardize our lives. A gazelle who paused to work out the best escape route while a lion is closing in on her would stand no chances. Often, finding the perfect solution is simply pointless, because by the time we would find it the problem would have escalated, i.e. we would be dead. Classical Logic is very powerful, but lacks this basic attribute: quick, efficient response to problems when an exact solution is not necessary (and sometimes counterproductive). Several techniques have been proposed for augmenting Logic with "plausible" reasoning: degrees of belief, default rules, inference in the face of absence of information, inference about vague quantities, analogical reasoning, etc. The Impossibility of Reasoning A very powerful argument in favor of common sense is that logical reasoning alone would be utterly impossible. Classical Logic deduces all that is possible from all that is available, but in the real world the amount of information that is available is infinite: the domain must be somehow artificially "closed" to be able to do any reasoning at all. And this can be achieved in a number of ways: the "closed-world assumption" (all relations relevant to the problem are mentioned in the problem statement), "circumscription" (which extends the closed-world assumption to "non-ground" formulas as well, i.e. assumes that as few objects as possible have a given property), "default" theory (all members of a class have all the properties characteristic of the class if it is not otherwise specified). For example, a form of default theory allows us to make use of notions such as "birds fly" in our daily lives. It is obviously not true that all birds fly (think of penguins), but that statement is still very useful for practical purposes. And, in a sense, it is true, even if, in an absolute sense, it is not true. It is "plausible" to claim that birds fly (unless they are penguins). At the same time, common sense reasoning introduces new problems in the realm of Logic. For example, John McCarthy's "frame problem": it is not possible to represent what does "not" change in the universe as a result of an action, because there is always an infinite set of things that do not change. What is really important to know about the new state of the universe, after an action has been performed? Most likely, the position of the stars has not changed, my name has not changed, the color of my socks has not changed, Italy’s borders have not changed, etc. Nevertheless, any reasoning system, including our mind, must know what has changed before it can calculate the next move. A reasoning system must continuously update its model of the world, but McCarthy suggests that this is an impossible task: how does our mind manage? Complementary paradoxes are the "ramification problem" (infinite things change, because one can go into greater and greater detail of description) and the "qualification problem" (the number of preconditions to the execution of any action is also infinite, as the number of things that can go wrong is infinite). Somehow we are only interested in things that change and that can affect future actions (not just all things that change) and in things that are likely to go wrong (not just all things that can go wrong). "Circumscription" (McCarthy’s solution to the frame problem) deals with default inference by minimizing abnormality: an axiom that states what is abnormal is added to the theory of what is. This reads as: the objects that can be shown to have a certain property, from what is known of the world, are all the objects that satisfy that property. Or: the only individuals for which that property holds are those individuals for which it must hold. (This definition involves a second-order quantifier. Technically, this is analogous to Frege's method of forming the second-order definition of a set of axioms: such a definition allows both the derivation of the original recursive axioms and an induction scheme stating that nothing else satisfies those axioms). For similar reasons Raymond Reiter introduced the "closed-world axiom" (what is not true is false), or "negation as failure to derive": if a formula cannot be proven using the premises, then assume the formula's negation. In other words, everything that cannot be proven to be true must be assumed to be false. His "Default Logic" employs the following inference rule: "if A is true and it is consistent that B is true, then assume that B is also true" (or "if a premise is true, then the consequence is also true unless a condition contradicts what is known"). Ultimately, these are all tricks to account for how the mind can do any reasoning at all in the face of the gigantic complexity that surrounds it. Second Thoughts There is at least one more requirement for "plausible" reasoning. Classical logic is "monotonic": assertions cannot be retracted without compromising the entire system of beliefs. Once something has been proven to be true, it will be forever. Classical Logic was not designed to deal with "news". But our daily lives are full of events that force us to reexamine our beliefs all the time: our daily system of logic is "non-monotonic". Therefore, a crucial tool for plausible reasoning is non-monotonic logic, which allows inferences to be made provisionally and, if necessary, withdrawn at any time. A handful of such logics became popular during the 1980s. Drew McDermott's formulation of Modal Logic is based on a coherence operator: "P is coherent with what is known" if P cannot be proven false by what is known. Robert Moore's "Autoepistemic Logic" is based on the notion of belief (related to McDermott's coherence) and models the beliefs of an agent reflecting upon his own beliefs. And so forth. Matthew Ginsberg classified formal approaches to nonmonotonic inference into: proof-based approaches (Reiter's logic), modal approaches (McDermott's logic, Moore's logic) and minimization approaches (circumscription). Ginsberg argued that a variety of approaches to nonmonotonic reasoning can be unified by resorting to multi-valued logics (logics that deal with more than just true and false statements). Uncertainty Another aspect of common sense reasoning that cannot be removed from our behavior without endangering our species is the ability (and even preference) for dealing with uncertainties. Pick any sentence that you utter at work, with friends or at home and it is likely that you will find some kind of "uncertainty" in the quantities you were dealing with. Sometimes uncertainty is explicit, as in "maybe I will go shopping" or "I almost won the game" or "I think that Italy will win the next World Cup". Sometimes it is hidden in the nature of things, as in "it is raining" (can a light shower be considered as "rain"?), or as in "this cherry is red" (how "red"?), or as in "I am a tall person" (how tall is a "tall" person?). The classic tool for representing uncertainties is Probability Theory, as formulated by Thomas Bayes in the late 18th century. Probabilities translate uncertainty into the lingo of Statistics. One can translate "I think that Italy will win the next World Cup" into a probability by examining how often Italy wins the World Cup, or how many competitions its teams have won over the last four years. One can then express a personal feeling in probabilities, as all bookmakers do. Bayes’ theorem and other formulas allow one to draw conclusions from a number of probable events. Technically, a probability simply measures "how often" an event occurs. The probability of getting tails is 50% because if you toss a coin you will get tails half of the times. But that is not the way we normally use probabilities: we use them to express a belief. A proponent of probabilities as a measure of somebody's preferences was the American mathematician Leonard Savage who in the 1950s thought of the probability of an event as not merely the frequency with which that event occurs, but also as a measure of the degree to which someone believes it "will" happen. The problems with probabilities are computational. Bayes' theorem, the main tool to propagate probabilities from one event to a related event, does not yield intuitive conclusions. For example, the accumulation of evidence tends to lower the probability, not to increase it. Also, the sum of the probabilities of all possible events must be one, and that is also not very intuitive. Our beliefs are not consistent: try assigning probabilities to a complete set of beliefs (e.g., probabilities of winning the world cup for each of the countries of the world) and see if they add up to 100%. In order to satisfy the postulates of probability theory, one has to change her belief and make them consistent, i.e. tweak them so that the sum is 100%. Bayes rule ("the probability of a hypothesis being true is proportional to the initial belief in it, multiplied by the conditional probability of an observational data, given that prior probability") would be very useful to build generalizations (or induction), but, unfortunately, it requires one to know the initial belief, or the "prior" probability, which, in the case of induction, is precisely what we are trying to assess. In 1968 mathematicians Glenn Shafer and Stuart Dempster devised a "Theory of Evidence" aimed at making Probability Theory more plausible. They introduced a "belief function" which operates on all subsets of events (not just the single events). In the throwing of a die, the possible events are only six, but the number of all subsets is 64 (all the combination of two sides, three sides, four sides and five sides). In their theory the sum of the probabilities of all subsets is one, the sum of the probabilities of all the single events is less than one. Dempster-Shafer's theory allows one to assign a probability to a group of events, even if the probability of each single event is not known. Indirectly, Dempster-Shafer's theory also allows one to represent "ignorance", as the state in which the belief of an event is not known (while the belief of a set it belongs to is known). In other words, Dempster-Shafer's theory does not require a complete probabilistic model of the domain. An advantage (and a more plausible behavior) of evidence over probabilities is its ability to narrow the hypothesis set with the accumulation of evidence. Fuzzy Logic One of the major breakthroughs in inexact reasoning came in 1965 when the Azerbaijani mathematician Lotfi Zadeh invented "Fuzzy Logic". Zadeh applied Lukasiewicz's multi-valued logic to sets. In a multi-valued logic, propositions are not only true or false but can also be partly true and partly false. A set is made of elements. Elements can belong to more than one set (e.g., I belong both to the set of authors and to the set of Italians) but each element either belongs or does not belong to a given set (I am either Italian or not). Zadeh's sets are "fuzzy" because they violate this rule. An element can belong to a fuzzy set "to some degree", just like Lukasiewicz's propositions can be true to some degree (and not necessarily completely true). The main idea behind Fuzzy Logic is that things can belong to more than one category, and they can even belong to opposite categories, and that they can belong to a category only partially. For example, I belong both to the category of good writers and to the category of bad writers: I am a good writer to some extent and a bad writer to some other extent. In more precise words, I belong to the category of good writers with a certain degree of membership and to the category of bad writers with another degree of membership. I am not fully into one or the other. I am both, to some extent. Fuzzy Logic goes beyond Lukasiewicz's multi-valued logic because it allows for an infinite number of truth values: the degree of "membership" can assume any value between zero and one. Zadeh's theory of fuzzy quantities implicitly assumes that things are not necessarily true or false, but things have degrees of truth. The degree of truth is, indirectly, a measure of the coherence between a proposition about the world and the state of the world. A proposition can be true, false, or… vague with a degree of vagueness. Fuzzy Logic can explain paradoxes such as the one about removing a grain of sand from a pile of sand (when does the pile of sand stop being a pile of sand?). In Fuzzy Logic each application of the inference rule erodes the truth of the resulting proposition. Fuzzy Logic is also consistent with the principle of incompatibility stated at the beginning of the 20th century by the French physicist Pierre Duhem: the certainty that a proposition is true decreases with any increase of its precision. The power of a vague assertion rests in its being vague: the moment we try to make it more precise, it loses some of its power. A very precise assertion is almost never certain. For example, "today is a hot day" is certainly true, but its truth rests on the fact that I used the very vague word "hot". If now I restate it as "today the temperature is 36 degrees", the assertion is not certain anymore. Duhem’s principle is the analogous of Heisenberg’s principle of uncertainty: precision and uncertainty are inversely proportional. Fuzzy Logic models vagueness and reflects this principle. While mostly equivalent to Probability Theory (as proven by the American mathematician Bart Kosko), Fuzzy Logic yields different interpretations. Probability measures the likelihood of something happening (e.g., whether it is going to rain tomorrow). Fuzziness measures the degree to which it is happening (e.g., how heavily it is raining today). And, unlike probabilities, Fuzzy Logic deals with single individuals, not populations. Probability theory tells you what are the chances of finding a tall person in a crowd, whereas fuzzy logic tells you to what degree that person is tall. Technically, a fuzzy set is a set of elements that belong to a set only to some extent. Each element is characterized by a degree of membership. An object can belong (partially) to more than one set, even if they are mutually exclusive, in direct contrast with one of the pillars of classical Logic: the "law of the excluded middle". Each set can be subset of another set with a degree of membership. A set can even belong (partially) to one of its parts. Degrees of membership also imply that Fuzzy Logic admits a continuum of truth values from zero to one, unlike classical Logic that admits only true or false (one or zero). In Kosko's formulation, a fuzzy set is a point in a unitary hypercube (a multi-dimensional cube whose faces are all one). A non-fuzzy set (a traditional set) is one of the vertexes of such a cube. The paradoxes of classical Logic occur in the middle points of the hypercube. In other words, paradoxes such as the liar's or Russell's can be interpreted as "half truths" in the context of Fuzzy Logic. A fuzzy set's entropy (which could be thought of as its "ambiguity") is defined by the number of violations of the law of non-contradiction compared with the number of violations of the excluded middle. Entropy is zero when both laws hold, is maximum in the center of the hypercube. Alternatively, a fuzzy set's entropy can be defined as a measure of how a set is a subset of itself. Possibility Theory Possibility theory (formulated by Zadeh in 1977, and later expanded by French mathematicians Didier Dubois and Henri Prade) developed as a branch of the theory of fuzzy sets in order to deal with the lexical elasticity of ordinary language (i.e., the fuzziness of words such as "small" and "many"), and other forms of uncertainty which are not probabilistic in nature. The subject of possibility theory is the possible (not probable) values of a variable. Possibility theory is both a theory of imprecision (represented by fuzzy sets) and a theory of uncertainty. The uncertainty of an event is described by a pair of degrees: the degree of possibility of the event and the degree of possibility of the contrary event. The definition can be dually stated in terms of necessity, necessity being the complement to one of possibility. Its basic axioms are that: 1. the degree of possibility is one for a proposition that is true in any interpretation and is zero for a proposition that is false in any interpretation; 2. the degree of possibility of a disjunction of propositions is the maximum degree of the two. When the degree of necessity of a proposition is one, the proposition is true. When the degree of possibility of a proposition is zero, the proposition is false. When the degree of necessity is zero, or the degree of possibility is one, nothing is known about the truth of the proposition. Possibility Logic has a graded notion of possibility and necessity. A Fuzzy Physics? Unlike Probability theory, Fuzzy Logic represents the real world without any need to assume the existence of randomness. For example, relative frequency is a measure of how a set is a subset of another set. Many of Physics' laws are not reversible because otherwise causality would be violated (after a transition of state probability turns into certainty and cannot be rebuilt working backwards). If they were expressed as "ambiguity", rather than probability, they would be reversible, as the ambiguity of an event remains the same before and after the event occurred. Fuzziness is pervasive in nature (everything is a matter of degree), even if science does not admit fuzziness. Even Probability Theory still assumes that properties are crisp, while in nature they rarely are. Furthermore, Heisenberg's "uncertainty principle" (the more a quantity is accurately determined, the less accurately a conjugate quantity can be determined) can be reduced to the" Cauchy-Schwarz inequality", which is related to "Pythagoras’ theorem", which is in turn related to the "subsethood theorem", i.e. to Fuzzy Logic. One is tempted to rewrite Quantum Mechanics using Fuzzy Theory instead of Probability Theory. After all, Quantum Mechanics, upon which our description of matter is built, uses probabilities mainly for historical reasons: Probability Theory was the only theory of uncertainty available at the time. The result is that Physics has a standard interpretation of the world that is based on population thinking: we cannot talk about a single particle, but only about sets of particles. We cannot know whether a particle will end up here or there, but only how many particles will end up here or there. The interpretation of quantum phenomena would be slightly different if Quantum Mechanics were based on Fuzzy Logic: probabilities deal with populations, whereas Fuzzy Logic deals with individuals; probabilities entail uncertainty, whereas Fuzzy Logic entails ambiguity. In a fuzzy universe a particle's position would be known at all times, except that such a position would be ambiguous (a particle would be simultaneously "here" to some degree and "there" to some other degree). This might be viewed as more plausible, or at least more in line with our daily experience that in nature things are less clearly defined than they appear in a mathematical representation of them. The World of Objects Another aspect of common sense is that it deals with quantities and objects which are a tiny subset of what science deals with (or is capable of dealing with). The laws of the physical world are relatively simple and few. The daily world of humans is made of a finite set of solid objects that move in space and do not overlap. Each object has a shape, a volume, a mass distribution. For an adequate representation of the physical needs we can get by with Euclides' geometry, an ontology of space-temporal properties and a set of axioms about the way the world works. We need none of the complication of Quantum Mechanics and Relativity Theory. We need no knowledge whatsoever of elementary particles, nuclear and subnuclear forces, and so forth. Life is a lot easier for our senses than it is for laboratory physicists. What we need to know in order to survive is actually a lot less than what we need to know in order to satisfy our intellectual curiosity. We never really needed to know the gravitational laws in order to survive, as long as we were aware that objects tend to fall to the ground unless we put them on a table or in a pocket or hang them to a wall. We never really needed to be informed of the second law of Thermodynamics, as long as we realized that they can break, but they do not fix themselves. The American computer scientist Ernest Davis compiled a list of common sense domains. First, we have physical quantities, such as weight or temperature. They have values. And their values satisfy a number of properties: they can be ordered, they can be subdivided in partially ordered intervals, they can be assigned signs based on their derivatives, their relations can be expressed in the form of transition networks, their behavior can be expressed in the form of qualitative differential equations. Then we have time and space. Time operators usually operate in a world of discrete, self-contained situations and events. Space entails concepts of distance, containment, overlapping, boundaries. Physics, in the view of common sense, is a domain defined by "qualitative" rather than quantitative laws, which express the behavior of physical quantities in the context of those temporal and spatial concepts. To this scenario one must add propositional attitudes (specifically the relationship between belief and knowledge), actions (the ability to plan) and socializing (speech acts). Equipped with this basic idea of the world, an agent should be able to go about its environment and perform intelligent actions. The Measure Space In 1978 the British computer scientist Pat Hayes introduced the "measure space" for a physical quantity (length, weight, date, temperature, etc.). A measure space is simply a space in which an ordering relationship holds. Measurement spaces are usually conceived as discrete spaces, even if the quantities they measure are in theory continuous. In common use, things like birth dates, temperatures, distances, heights and weights are always rounded. For example, the height of a person is usually measured in whole centimeters (or inches), and omitting the millimeters, and it can be safely assumed that only heights over one meter and less than two meters are possible. This means that the measure space for people’s height is the set of natural numbers from 100 (centimeters) to 200 (centimeters). The measure space for driving speed can reasonably be assumed to be the set of numbers from 0 to 200 (kilometers per hour). The measure space for a shirt’s size is sometimes limited to four values: small, medium, large, very large. The measure space for jeans’ size is a (very limited) set of pairs of natural numbers. The measure space for the age of a person is the set of natural numbers from 1 to 130. The measure space for the date of an historical event is the set of integer numbers from -3,000 (roughly the time when writing was invented) to the number of the year we live in. And so forth. A measure space is a discrete representation of a continuous space that takes into account only the significant values that determine boundaries of behavior. Hayes' program was more ambitious than just measurement spaces. Hayes set out to write down in the language of Predicate Logic everything that we take for granted about the world, all of our common-sense knowledge about physical objects. For example, we know that water is contained in something and that, if it overflows, it will run out, but it will not run upward. We know that wood floats in water but iron sinks. We know that a heavy object placed on top of a light object may crash it. We know that an object will not move if placed on a table, but it will fall if pushed beyond the edge. This is what Hayes called "Naïve Physics" and it is the physics that we employ in our daily lives. Histories of the World During the 1980s several techniques were proposed for re-founding Physics on a more practical basis. John McCarthy's "Situation Calculus" represents temporally limited events as "situations" (snapshots of the world at a given time), by associating a situation of the world (a set of facts that are true) to each moment in time. Actions and events are represented mathematically as mathematical functions from states to states. An interval of time is a sequence of situations, a "chronicle" of the world. The history of the world is a partially ordered sequence of states and actions. A state is expressed by means of logical expressions that relate objects in that state. An action is expressed by a function that relates each state to another state. The property of states is permanence, the property of actions is change. Each situation is expressed by a formula of first-order Predicate Logic. The advantage of this logical apparatus is that causal relations between two situations can be computed. The elementary unit of measure for common sense is not the point, but the interval. Which interval makes sense depends on the domain: history is satisfied with years (and sometimes centuries), but birth dates require the day. Points require Physics' differential equations, but intervals can be handled with a logic of time that deals with their ordering relationship. Qualitative Reasoning "Qualitative" reasoning is the discipline that aims at describing a physical system through something closer to common sense than Physics' dynamic equations. In "Qualitative" Physics, a physical system is conceived as made of parts that contribute to the overall behavior through local interactions, and its behavior is represented inside some variation of Hayes' measure space. Ultimately, qualitative reasoning is a set of methods for representing and reasoning with incomplete knowledge about physical systems. A qualitative description of a system allows for common-sense reasoning that overcomes the limitations of classical Logic. Qualitative descriptions capture the essential aspects of structure, function and behavior, at the expense of others. Since most phenomena that matter to ordinary people depend only on those essential aspects, qualitative descriptions are enough for moving about in the world. Several approaches are possible, depending on the preferred ontology: Benjamin Kuipers adopts qualitative constraints among state variables; Johan DeKleer focuses on the devices (pipes, valves, springs) connected in a network of constraints; Kenneth Forbus deals with processes by extending the notion of history. Ultimately, a system's behavior is almost always described by constraint propagation. DeKleer describes a phenomenon in a measure space through "qualitative differential equations", or "confluences". His "envisionment" is the set of all possible future behaviors. Forbus defines a "quantity space" as a partially ordered set of numbers. Common sense is interested in knowing that quantities "increase" and "decrease" rather than in formulas yielding the quantities’ values in time. In other words, the sign of the derivative is more important than the exact value of a quantity. Kuipers formalizes qualitative analysis as a sequence of formal descriptions. From the structural description the behavioral description (or "envisionment") can be derived, and from this the functional description can be derived. In his quantity space, besides the signs of the derivatives, what matters most are critical or "landmark" values, such as the temperature at which water undergoes a phase transition. Change is handled by discrete state graphs and qualitative differential equations. A qualitative differential equation is a quadruple of variables, quantity spaces (one for each variable), constraints (that apply to the variables) and transitions (rules to define the domain boundaries). Each of these three frameworks prescribes a number of constraint propagation techniques, which can be applied to a discrete model of the physical system. Physics is a science of laws of nature which are continuous and exact. Things move because they are subject to these laws. Qualitative Physics is a science of laws of common sense that are discrete and approximate. Things move because other things make them move. Qualitative Physics may not be suitable for studying galaxies and electrons, but can work wonders at analyzing a piece of equipment, a machine, and, in general, a physical system made of components. For example, it has been applied at troubleshooting machines: a model of behavior of a system makes it easier to figure out what must be wrong in order for the system to work the way it is working, i.e. which component is not doing its job properly. Heuretics The physical world is only one part of the scenario. There is also the "human" world, the huge mass of knowledge that humans tend to share in a natural way: rain is wet, lions are dangerous, most politicians are crooks and carpets get stained. "Heuristics" is the proper name for most of what we call common sense. Heuristics is the body of knowledge that allows us to find quick and efficient solutions to complex problems without having to resort to mathematical Logic. Heuristics is, for example, the set of "rules of thumb" that most people employ in their daily lives. The intellectual power of our brain is rarely utilized, as in most cases we can find a rule of thumb that will make it unnecessary. We truly reason only when we cannot find any rule of thumb to help us. A human being who did not know any rules of thumb, who did not have any heuristics, would treat each single daily problem as a mathematical theorem to prove and would probably starve to death before understanding where and how to buy food. We tend to employ heuristics even when we solve mathematical problems. And countless games (such as chess) are about our ability to apply heuristics, rather than mere mathematical reasoning. The American computer scientist Douglas Lenat developed a global ontology of common knowledge and a set of first principles (or reasoning methods) to work with it. Units of knowledge for common sense are units of "reality by consensus": all the things we know and we assume everybody knows; i.e., all that is implicit in our acts of communication. World regularities belong to this tacitly accepted knowledge. And "regularity" may be a key to understand how we construct and why we believe in heuristics. Lenat’s "principle of economy of communications" states the need to minimize the acts of communication and maximize the information that is transmitted. Another open issue is whether common sense is learned or innate, or: to what extent it is learned and to what extent it is innate. If it is learned, how is it learned? In the 1940s the Swiss mathematician George Polya studied how mathematicians solve mathematical problems. Far from being the mechanical procedure envisioned by the proponents of the logistic program, he realized that solving a problem required heuristics. Later, he envisioned "Heuretics", a discipline that would aim at understanding the nature, power and behavior of heuristics: where it comes from, how it becomes so convincing, how it changes over time, etc. One of the intriguing properties of heuristics is, for example, the impressive degree to which we rely on it: the moment we realize that a rule of thumb applies, we abandon our line of reasoning. What makes us so confident about the effectiveness of heuristics? Maybe the fact that heuristics is "acquired effectiveness"? The scope of Heuretics is, ultimately, the scientific study of wisdom. Stupidity Artificial Intelligence has been researching human intelligence and machine intelligence, possibly abusing the term "intelligence" from the beginning. It can be educational to focus for a few minutes on the opposed quantity, stupidity. Common sense seems to have a perfectly clear understanding of what stupidity is. Most people would agree at once that some statements are stupid. "Which is the shortest river in the world?" There is no shortest river, because one can always find a shorter stream of water, all the way down to the leak in your bath tub and to a single drop of water in the kitchen sink. While it makes sense to ask which is the longest river in the world, it makes no sense to ask which is the shortest one. As the length gets shorter, the number of rivers increases exponentially. " Is everybody here?" The question is stupid because if somebody is missing she won’t be able to answer the question. "Does everybody understand English?" exhibits the same type of stupidity. What do these questions have in common?   Further Reading Bobrow Daniel: QUALITATIVE REASONING ABOUT PHYSICAL SYSTEMS (MIT Press, 1985) Bobrow Daniel: ARTIFICIAL INTELLIGENCE IN PERSPECTIVE (MIT Press, 1994) Brachman Ronald: READINGS IN KNOWLEDGE REPRESENTATION (Morgan Kaufman, 1985) Davis Ernest: REPRESENTATION OF COMMON-SENSE KNOWLEDGE (Morgan Kaufman, 1990) Dubois Didier & Prade Henri: POSSIBILITY THEORY (Plenum Press, 1988) Dubois Didier, Prade Henri & Yager Ronald: READINGS IN FUZZY SETS (Morgan Kaufmann, 1993) Forbus Kenneth & DeKleer Johan: BUILDING PROBLEM SOLVERS (MIT Press, 1993) Gigerenzer Gerd & Todd Peter: SIMPLE HEURISTICS THAT MAKES US SMART (Oxford Univ Press, 1999) Gupta Anil & Belnap Nuel: THE REVISION THEORY OF TRUTH (MIT Press, 1993) Heyting Arend: INTUITIONISM (North Holland, 1956) Hobbs Jerry & Moore Robert: FORMAL THEORIES OF THE COMMONSENSE WORLD (Ablex Publishing, 1985) Kandell Abraham: FUZZY MATHEMATICAL TECHNIQUES (Addison Wesley, 1986) Kosko Bart: NEURAL NETWORKS AND FUZZY SYSTEMS (Prentice Hall, 1992) Kosko Bart: FUZZY THINKING (Hyperion, 1993) Kuipers Benjamin: QUALITATIVE REASONING (MIT Press, 1994) Lenat Douglas: BUILDING LARGE KNOWLEDGE-BASED SYSTEMS (Addison-Wesley, 1990) Lukaszewicz Witold: NON-MONOTONIC REASONING (Ellis Harwood, 1990) Marek Wiktor & Truszczynski Miroslav: NON-MONOTONIC LOGIC (Springer Verlag, 1991) Martin-Lof Per: INTUITIONISTIC TYPE THEORY (Bibliopolis, 1984) Polya George: MATHEMATICS AND PLAUSIBLE REASONING (Princeton Univ Press, 1954) Savage Leonard: THE FOUNDATIONS OF STATISTICS (John Wiley, 1954) Shafer Glenn: A MATHEMATICAL THEORY OF EVIDENCE (Princeton Univ Press, 1976) Sowa John: PRINCIPLES OF SEMANTIC NETWORKS (Morgan Kaufman, 1991) Turner Raymond: LOGICS FOR ARTIFICIAL INTELLIGENCE (Ellis Horwood, 1985) Tversky Amos, Kahnemann Daniel & Slovic Paul: JUDGMENT UNDER UNCERTAINTY (Cambridge University Press, 1982) Weld Daniel & DeKleer Johan: QUALITATIVE REASONING ABOUT PHYSICAL SYSTEMS (Morgan Kaufman, 1990) Zimmermann Hans: FUZZY SET THEORY (Kluwer Academics, 1985)