Geoinformatics

1. Geoinformatics Logics

2. First-order Logic (FOL) • Logic, as the study of entailment relations and arguments, i.e., languages, truth conditions, and rules of inference, and provides reasoning and formal semantics and language to describe facts and state of affairs about the real world • Propositional logic assumes that the world can be described by a known set of propositions that relate to specific individuals President (Obama) • Proposition: Boolean true knowledge statement or expression (specific facts) • Propositional logic uses connectives (special symbols, see below) between propositions (facts about specific objects). • However, this type of logic is not sufficient. It does not allow generalization or quantification about objects using variablese.g., it does not allow us to state: “only some sedimentary rocks have fossil”, i.e., “not all sedimentary rocks have fossils”

3. Why Predicate Logic • Assertions such as “x is less than 90” is not a proposition because the value of x is unknown, therefore we do not know if the sentence is true or false. • Statements with variables (e.g., x < 2) cannot be assessed by propositional logic because the truth value of such a statement cannot be determined (x is unknown). For example: • The predicate Lives (x,y): ”x lives in y”cannot be handled by propositional logic • Propositional logic cannot handle comparing or generalizing facts, for example for equivalency of two assertions • A predicate becomes a proposition when specific values are assigned to variables • “Obama lives in the White House” is a proposition: Lives(Obama, WhiteHouse)

4. First order (or predicate) logic (FOL) • First order logic is the principal type of logic used for complex assertions about knowledge • FOL adds predicates and quantification to propositional logic to allow writing complex statements • FOL is symbolic reasoning in which statements are broken into a subject and a predicate which defines the properties of the subject (e.g., “James is an athlete”). Note: athlete is the predicate Athelete (James) • Reasoningis the formal manipulation of symbols in a proposition to draw new facts or statements from existing ones • What is the meaning of the following? x [Mineral (x)  Crystalline (x)] Mineral (calcite) We infer that calcite is crystalline!

5. FOL … • FOL assumes that the world contains Objects (e.g., water, rock, duck), relations among objects (e.g., partOf(x,y), lessThan(x,y)), and Functions (e.g., brotherOf(x,y), minus(x,y), densityOf(x)) • We also have beliefs that these relations are true, false, or unknown, based on domain knowledge: “Water freezes below zero Celsius”, “hardness of diamond is 10”. • The predicate P(x,y): x+y=7 becomes propositionP(6,1), which is true, but P(2,3) is false; therefore, it is not a proposition • As a language that allows expressing propositions (specific facts) and formulating our knowledge, FOL has syntax rules and semantics that can be used to specify how the expressions are to be used and what they mean

6.    Logical symbols in FOL • FOL has two types of symbols: logical (punctuations, connectives, and variables) and non-logical (function and predicate) • The logicalconnectives (used to build complex formulas), include: • Unary connective: • logical negation (  or ~); reads: “not”, e.g.,  A or ~A;  Rock • Binary connectives: • logical conjunction (); reads: “and”; e.g., A  B; volcanic clastic • logical disjunction (); reads: “or”; e.g., A  B; water  Oil • logical equality (); (x=y) • logical implication or conditional ( or ; reads: “if… then…”), e.g., (A  B)  C which means if A or B is true, then C is true • logical two-way implication or biconditional (; e.g., X  Y, which reads: “X is true if and only if Y is true” or “X is equivalent to Y”)

7. Symbols … • Logical punctuations include: ‘)’, ‘(‘, and ‘.’ • These are not part of formulas, e.g., X (Y  Z) • Quantifiers: existential, ; reads: ‘there exists at least one’ or ‘there is some’ and universal, ; reads: ‘for all, if any’ xP(x) reads: for every x, P(x); for every x, P(x) is true; For all x, P(x) xP(x) there is some x, P(x); there exists an x, P(x) is true; for some x, P(x) • Functions: map individual objects to other individuals; returns a value Have arity (# of arguments). Zero-arity functions are constants. e.g., FatherOf(John) = Jack • Predicates: map individuals to truth values; • single-artity predicates represent properties, e.g., liquid (magma) • multi-arity predicates represent relations among objects, e.g.,: includes (Paleozoic, Permian) • zero-arity predicates represent proposition • Constants: individuals in the world, strings that represent objects • e.g., U.S.A., Obama, Paleozoic • Individual variables, that range over objects of any sort in reality, are denoted with x, y, z, …, are used to quantify over objects

8. Atomic sentences • Atomic sentence: a predicate applied to variables: P(c, d), e.g., Deforms (Laramide, Basement); Likes (Jack, car) Sued (Microsoft, Motorola) • These sentences are true or false depending on the interpretation of these arguments, which here are constants • However, if instead of constants, there were variables (x and y), it may not be possible to prove that they are true or false, e.g.,: P(x, y)

9. Syntax and Semantics in FOL • Syntax refers to the proper form and structure of the groups of symbols. The well-formed sentences that convey the propositions guarantee correct modeling of knowledge, for example: • “Normal fault is a fault” has a well-formed syntax in English and Geology, but is it well-formed in logic? No! (x) NormalFault(x)  Fault(x) this is well-formed formula • Semanticsrefers to the meaning of the well-formed formulas (wff) or expressions like the one above based on domain knowledge • While “the strike of the mineral” does not mean anything, “the strike of a fault” means something to a geologist

10. Terms • Terms and well-formed formulas (wff) are the two types of syntactic expressions in FOL • For example, x and y in the Equals(x, y) predicate are terms • A term is used to refer to something (an individual) in the domain, and can be of three types: variable, standard name, and function • One or more terms are used as arguments in predicates • If ti is a set of terms, and f is a function symbol of arity n, then f(t1, …, tn) is also a term, e.g., grade (student, course) • If arity is zero, then f is a constant, and is shown as c, without (), e.g., Atlanta

11. Well-formed sentence • A variable which is not bound (by quantifiers) in a formula is called free variable • A sentence, which is a well-formed formula (wff) with no free variable, is the most important syntactic form of FOL which takes truth value, and thus is the only one that can represent knowledge • The following well-formed sentence means that both bedding and foliation are folded by some x (e.g., folding): There exists some x where x folds bedding and foliation x folds (x, Bedding)  folds (x, Foliation) This is wff, with no free variable! • NOTE: The variable x, in this case stands for geological process entities, such as folding, i.e., the domain of values for x are entities in the structural geology domain

12. Equality; truth value • Equality of two terms is true under a given interpretation (i.e., given the universe of discourse and predicates) iff (if and only if) the two terms refer to the same object. • Universe of discourse, also called domain, is the set of values that may be assigned to the variable in a predicate e.g., Even (2,4) is true for a domain of {1,2,3,4} • The truth set of a predicate P(x) is the set of all elements t of universe (U) such that P(t) is true, for example: • Given P(x) is: “x is odd” and U={1,2,3,4,5}, then the truth set for odd numbers in this universe is: {1,3,5} Odd (1,3,5) is true in this domain or limited numbers • Given the domain of a family: the following take truth value: x,ySibling(x, y)  (x = y) wff with no free variable x,y Brother(x, y)  Sibling(x, y) wff with no free variable Note: brother implies sibling but sibling does not imply only brother ()

13. What is a predicate • Predicate is a quality (essential or accidental) that is attributed to a subject. • It describes a property of objects (individuals) or relationships among objects represented by variables, e.g.: Ice is solidsubject: ice, predicate: solid • Predicate is symbolized with a capital letter or spelled out, and is used for domain-specific properties and relations: P(x, y), ShorterThan(x, y), Analyzes(x, y) • Sometimes the letter is given a lower case subscript to represent the subject: e.g., “limestone is carbonate” is symbolized as Cl

14. Non-logical predicate symbols • Predicates can be assigned to an object (represented by variable x) as Px or P(x), e.g., “atoms (x) conduct heat (C) to other atoms (y)”x,yC(x, y) which stands for x,yConducts_heat(x, y) • Predicates have a Boolean truth value, i.e., when their variables are instantiated with the values in their domain, the predicate returns a true or false • In FOL, the domain of the variables is infinite, i.e., variables can take as many values as there are individuals!

15. Boolean Valuations of formulas • For every X and Y: • The formula X (or X), e.g.,Carbon, receives the Boolean valuation of: • true if X receives the false value; i.e., X is true if X is false (like -(-)) • false if X receives the true value, i.e., X is false if X is true (like -(+)) • The formula X  Y, (Solid  Crystalline) receives the Boolean valuation of: • true if X and Y both receive the true value; i.e., if both are true • false otherwise (i.e., either X is false or Y is false) • The formula X  Y (Black  White) receives the Boolean valuation of: • true if either X or Y or both receive the true value • false otherwise (i.e., both X and Y are false) • The formula X  Y receives the Boolean valuation of: note: X  Y =  X  Y • false if X is true and Y is false • true otherwise See table in next slide!

16. Truth Valuations1 = true 0 = false NOTE: X  Y =  X  Y

17. Non-logical function symbols • The non-logical function symbols (a, b, c, …) map from one individual to another, with or without argument, and are written in mixed case, for example, formula (a, b), which returns ‘a’ as the formula for ‘b’. • Functions return a non-Boolean (non true/false) value such as a numerical, for example, density(x) is a function that returns the value for the density of x age(x) returns age • To express that the density of x is greater than that of y, we use the GreaterThan (density(x), density(y)) predicate (that returns a Boolean value) whose arguments are the two density (x) and density (y) functions that return the values for the densities of x and y

18. Arity • Functions are often used as arguments for predicates, e.g., GreaterThan (hardness (x), hardness(y)) • Both functions and predicates have arity, which indicates the number of arguments they take (in their parentheses) • Constants (denoted by a, b, and c) have no arity, e.g., SanAndreasFault, Georgia, 4, T • Mineral (P), e.g., Mineral (calcite), is a predicate symbol of arity 1, and olderThan (Q, R) is a predicate symbol of arity 2 • OlderThan (Xenolith, IgneousBody), which means xenolith is older than the igneous body that contains it (a Boolean statement). • This is a predicate (not a function) because it returnsa Boolean rather than a non-Boolean which functions return

19. Building sentences • Sentences are built from terms and atoms: • Term: can be a constant individual, variable , or function • Atom: can be a predicate with value true or false e.g., P ∧ Q or P • Sentence: combination of atoms and variables and quantifiers, e.g., x [(Water (x) Liquid (x)] • A well-formed formula (wff) has no free variable, i.e., all variables are bound by universal or existential quantifiers or by value assignment, e.g., x is bound but y is free in the following formula (Note: it is not wff since y is not defined!): x P(x, y) • But in the following, x and y are bound while z is free in: xy (P(x, y)  Q(x, f(x), z)) • Below, the first occurrence of x is free in P(x) while the second x is bound: P(x)  x Q(x)

20. Complex statements • To simplify, we commonly assign symbols to the subject (lower case letter, subscript) and predicate (upper case letter), e.g.: • “Babaie (b) ‘breaks his teeth’ (B) if and only if he ‘eats rock’ (E), and there is meteorite (m) ‘shower at the same time in Wyoming’(S), and a cat (c) ‘sings like Andrea Bocelli‘(S)”: Bb [(Er Sm) Sc] • ”If Obama (o) is not a swimmer (S) or wrestler (W), so is Putin (p)”: (SoWo) (SpWp) • ”If Atlanta (a) ‘is hot’ (H), then ice cream (I) ‘runs out of your nose’ (R) or a ‘bird messes up’ (M) your shirt (s), and blankets (b) ‘don’t sell’ (S), or birds (b) ‘catch flies on your head’ (C)”Ha {[(Ri Ms)  Sb] Cb}

21. Universal statements • We symbolize universals with the use of individual variables and the universal quantifier • For example, all waters are clean • We write: for all (any, every) x, if x is water, then x is clean, i.e., we use x as an individual variable over the whole range of individuals in the universe: x [(Water (x)  Clean (x)] Alternatively: x [(Water (x) Clean (x)] • Note that x in clean (x) is a free variable • But x in Water (x) is a bound variable (preceded by a quantifier)

22. Universal quantifier • x (P(x)) reads: “for all x, P(x) holds”, i.e. the predicate P(x) is true for all objects in the universe of discourse • For example, if W stands for the following predicate: “all water molecules have oxygen”, then: x (W(x)) holds • If the formula has more than one variable, then quantification is done from the inside: x y (P(x, y)) is read as x [y (P(x,y)] and reads as: “There is some (an) x such that for all y, P(x, y) holds” x y z (P(x, y, z))  y x z (P(x, y, z)) x y z (P(x, y, z))  y x z (P(x, y, z)), but: x y z (P(x, y, z))  y x z (P(x, y, z)) • Typically, the implication  (also shown with the  symbol) is the main connective with the universal quantifier , not the  symbol x Ocean(x)  Large(x), but not:x Ocean(x)  Large(x)

23. Existential quantifier • There exists an x, where x is something and Px holds. x(Px), e.g., x(Lx): “there is an x where x is a lake” • Note: always make sure to translate the meaning of the sentence, not the grammar. x (Sx Px), i.e., there is some x, where x is an S and x is a P, where S and P stand for predicates (e.g., being spider and poisonous) • We use conjunction with the existential quantifier (i.e., particulars), but use implication for universal quantifier, e.g.: x (Lx  Px)some lakes (L) are polluted (P): i.e., there is some x and it is polluted x (SxPx) all spiders (S) are poisonous (P) • Not all lakes are polluted: x (Lx  Px) • Silicates are minerals x (SxMx)

24. Universal & Existential quantifiers • The statement x<2 can be quantified in predicate logic by stating: “for all x, x<2” or:x (x<2), which can be true or false in the domain of discourse if it is defined • For example, if universe (U) = {0, 1} then x (x<2) it is true • The same statement can be quantified with the existential quantifier by stating: “for some x, x<2” or: x (x<2), which can be true or false depending on the value of x • NOTE: If xP(x) is true for at least one element in the domain, then xP(x) is true. Otherwise it is false

25. Universe of discourse - U • Is the set of objects in a domain (e.g., Mineralogy); or the values that a variable can take • Predicate logic deals with statements about the objects in a given universe (domain, area of interest) such as Mineralogy, e.g.: • if predicate D stands for “Diamond is harder than Gypsum”. Then x (Dx) this is true in the Mineralogy domain • Hot magma (x) is ‘less viscous’ (L) than cold magma (y) x L(x, y) • Let the universe U be the sub-set of mafic silicates:U={olivine, pyroxene, amphibole}, thenif M(x) means x ‘is mafic’ x M(x) M(olivine) M(pyroxene)  M(amphibole) x M(x) M(olivine) M(pyroxene)  M(amphibole)

26. Rules of inference in FOL • Universal elimination (Universal Instantiation): • If x P(x) is true, then x P(c) is true (where c is a constant) • From x Likes (John, x) infer: x Likes (John, frog) • Universal Generalization: • If there is an individual P(c) in a universe U, thenx P(x) is true • Universal Modus Ponens - i.e., if P then QP implies Q; • P  Q. P is asserted to be true, therefore Q must be true x P(x)  Q(x), e.g., “all politicians lie” x Politician (x)  Lies (x) Politician (Obama)  Lies (Obama) x Calcite (x)  Fizzes_with_HCl (x) x Have_password(x)  Login (x) x Snake (x)  Bites (x)

27. Universal Modus Tollens If P implies Q, then, if something is not Q, it is not P x [P(x)  Q(x)], e.g., x [Duck(x)  Quacks(x)] “all ducks quack”  Quacks (Donald) “Donald does not quack”, therefore Duck (Donald) “Donald is not a duck” x [Water (x)  Freezes_below_0o_C (x)]“all waters freeze below 0oC”  Freezes_below_0o_C (fluid), therefore  Water (fluid)

28. Universal hypothetical syllogisms If x [P(x)  Q(x)] and x [Q(x)  R(x)], then: x [P(x)  R(x)] it is transitive x [Release_into_atmosphere (CO2)  Rise (temperature)] x [Rise (temperature)  Melt (glacier)] x [Release_into_atmosphere (CO2)  Melt (glacier)] x [Precipitate_fast_in_Montana (rain)  Flood (MissouriRiver] x [Flood (MissouriRiver)  Flood (MississippiRiver)] x [Precipitate_fast_in_Montana (rain)  Flood (MississippiRiver)]

29. Rules of inference … • Existential Introduction (Existential Generalization): • Specific statement to a quantified generalized statement • If P(c) is true, then we can infer: x P(x). Likes (Peggy, IceCream)x Likes (Peggy, x)i.e., Peggy likes ice cream, there is something that Peggy likes Melt (Grinnell_glacier )  x Melt (x) • If P(x) is a predicate and c is a constant (individual) in domain U: x P(x) P(c) implies P(c)  x P(x)

30. Existential Elimination (Existential Instantiation) If x P(x) is true, then we can infer: P(c) where c is a new term Given x loves (Joe, x) we infer: Loves (Joe, eggPlant)

31. Truth value • A predicate such as P(x, y, z) has no truth value on its own, but takes a true or false value when the connectives are used and the variables are instantiated with values from the domains of x, y, and z • The xy P(x, y), for example, reads: there exists some x, for all y, that P(x, y) is true • That is, no matter what value we choose for y, P(x, y) is true for some value of x • The relative position of the connectives in the sentence makes a difference in the meaning of the sentence, for example, xy P(x, y) is not the same as xy P(x, y) ory x P(x, y); the changes modify the meaning • The special predicate equal (x, y) returns true if x=y, and false for inequality of x and y (i.e., equal(x, y))

32. Interpretation • We get an interpretation for the formula when we specify the universe (that specifies the range of quantifiers) and predicates, and assign a value for the free variable in the well formed formula (wff) • It means that each sentence is assigned an object that it represents, and each sentence is assigned a truth value • An interpretation provides semantics to the sentence! • For example, if the universe (U) is the subset of minerals in the Mohs hardness scale, i.e., U={1. …10}, if the predicate S(x, y) reads: “is softer than”, and y is diamond, then the interpretation of x S(x, y) reads: “for all x (i.e., minerals) in the set {1. …9}, x is softer than (S) diamond (y) xSofterThan (x, Diamond) • A wff is satisfiableif there is an interpretation that makes it true, i.e., makes it a proposition • A wff is valid if it is true for every interpretation, e.g., x Px x Px is valid because it covers everything (note: x Px is the negation of x Px)

33. Disjunction vs. conjunction • Conjunction“Mafic minerals are heavy and dark”; For all x, if x is a mineral (M) and it is mafic (F), then, it is heavy (H) and dark (D): x (Mx  Fx)  (Hx  Dx) • Disjunction“Some mountains (M) are high (H) and rugged (R)”, meaning: “some mountains are high, or rugged, or both” x [Mx  (Hx Rx)] • Note: the statement has ‘, i.e., and’ in it because this is an inclusive disjunction (either this or that or both)

34. Negation “Samples (S) can be analyzed (A) unless they are heavily altered (H). For all x if x is a sample, then () if it is not heavily altered, then () it can be analyzed x [Sx  (HxAx)] • Or alternatively: if there is a sample it is either analyzed or it is heavily altered: x [Sx (Hx Ax)] • When an existential statement is negated, the existential quantifier changes to the universal quantifier, and we negate the property or change the condition to the complement of the statement: • DeMorgan’s Laws for quantifiers: x P(x)  x P(x) x P(x)  x P(x)

35. Negation … (x y (P(x,y)))  x y (P(x,y)) becomes (after negating the existential quantifier and moving the negation to the inside): (x y (P(x,y)))  x y (P(x,y))  x y (P(x,y)) Negate: “There are some minerals that are liquid” to “There is no mineral which is liquid” i.e., negate: x (Mx  Lx) to: x (Mx  Lx) Negate “Some geoscientists eat mud” to “No geoscientist eats mud”: i.e., negate x (Gx  Ex) to: x (Gx  Ex) The negation of a universal statement: For all x, if P(x) holds changes to “there exists an x such that P(x) does not hold, i.e.,: x P(x)  x P(x) DeMorgan’s Law

36. Using DeMorgan’s Law Push negation through multiple quantifiers using:x P(x)  x P(x) and x P(x)  x P(x) ∃x∀y∀z P(x, y, z) ≡ ∀x∀y∀z P(x, y, z) ≡ ∀x∃y∀zP(x, y, z) ≡ ∀x∃y∃zP(x, y, z) • Note the equivalence of the following sentences: • “not all oceans are warm” is equivalent to: • “some oceans are not warm” • “not all folds are isoclinal” is equivalent to: • “some folds are not isoclinal” • “not all countries are rich” is equivalent to: • “some countries are not rich”

37. Using DeMorgan’s Law Push negation through multiple quantifiers using:x P(x)  x P(x) and x P(x)  x P(x) ∃x∀y∀z P(x, y, z) ≡ ∀x∀y∀z P(x, y, z) ≡ ∀x∃y∀zP(x, y, z) ≡ ∀x∃y∃zP(x, y, z) • Note the equivalence of the following sentences: • “not all oceans are warm” is equivalent to: • “some oceans are not warm” • “not all folds are isoclinal” is equivalent to: • “some folds are not isoclinal” • “not all countries are rich” is equivalent to: • “some countries are not rich”

38. Order of quantifiers • Everybody has a mother (i.e., for all x, there is a y who is the mother of x): x y [M(y, x)] where M(y, x) is: “y is the mother of x” Order matters: The following are not equivalent x y [M(y, x)]  y x [M(y, x)] for example: x y [Marries(y, x)]  y x [Marries (y, x)] Any problem? Scope of a quantifier • Is part of the statement to which the quantifier applies, i.e., where variables are bound by the quantifier ∃x {Rock(x) ∧ ∀x[Igneous (x)]} states that all rocks are igneous (x in Igneous (x) is universally quantified

39. Relations worth to know! (A  B)  (B  A) commutativity of  (A  B)  (B  A) commutativity of  [(A  B)  C ] [(A  (B  C)] associativity of  [(A  B)  C ] [(A  (B  C)] associativity of  ( A)  A double negation elimination (A  B)  ( A   B) contraposition (A  B)  ( A B) implication elimination (A  B)  [(A  B)  (B  A)] biconditional elimination (A  B)  (A B) DeMorgan (A  B)  (A B) DeMorgan [A  (B  C )]  (A  B)  (A  C) distributivity of  over  [A  (B  C )]  (A  B) (A  C) distributivity of  over 

40. Logical relationships with quantifiers • ∀ distributes over : ∀x[P(x) ∧ Q(x)] ≡ ∀xP(x) ∧ ∀xQ(x), i.e., the left and right sentences have the same truth value! • ∀ does NOT distribute over ∨: ∀x[P(x) ∨ Q(x)]  ∀xP(x) ∨ ∀xQ(x) • ∃ does NOT distributes over : ∃x[P(x) ∧ Q(x)] ∃xP(x) ∧ ∃xQ(x) • ∃ distributes over ∨: ∃x[P(x) ∨ Q(x)] ≡ ∃xP(x) ∨ ∃xQ(x) • The following are true: ∃x[P(x) ∧ Q(x)] → ∃xP(x) ∧ ∃xQ(x) [∀xP(x) ∨ ∀xQ(x)] → ∀x(P(x) ∨ Q(x))

41. Universal terms • Universal terms can be connected to reality by defining the relationship between their instances (i.e., particulars) • For example, the universal PartOfand HasPart relations (note the upper case first letter for the relation among universals!) can be defined by the relation between their instances • In other words, we are able to quantify over both real-world universals and their instances

42. Universal PartOf • We assert that universal A PartOf B or B HasPart A, if every instance of A is partOfsome instance of B or every instance of B has some instance of A as part • Note the lower case first letter for relation among individuals! • Formally, ‘A PartOf B’ is defined in FOL as: • A PartOf B = def. x [inst(x, A) y (inst(y, B)  x partOf y)]

43. Meaning of the sentence x [inst(x, A) y (inst(y, B)  x partOf y)] • The expression reads: for all x, if x is an instance of A (i.e., if x instantiates A), then there exists a y, where y is an instance of B (i.e. y instantiates B), and x is part of y • For example, the following expresses the relations among the Fault and FaultSegment universals: FaultSegmentPartOf Fault Fault HasPartFaultSegment

44. Universal B HasPart A • The universal relation ‘B HasPart A’ is defined as: B HasPart Ay [inst(y, B) x (inst(x, A)  x partOf y)] • Meaning: for every individual y, if y instantiates B, there is an individual x that instantiates A, and x is part of y

45. Universal isA • The universal isA relation is similarly defined as: A isA Bx [inst(x, A)  inst (x, B)] • Meaning: A isA B if every instance of A is also an instance of B • For example, the universal NormalFaultisAFaultif every instance of NormalFault is also an instance of the Fault universal

46. Processes • We can write the same for processes (occurrents, e.g., deformation) which involve continuants such as a mineral: x [occurrent (x) y (y isA continuant z inst(z, y)  (z participates-in x)] which reads: • For all x, if x is an occurrent, then there exists a universal y, where y is a continuants, and there is at least one z where z is an instance of y, and z participates in x i.e., a rock or a mineral participates in a deformation

47. Example 1 • Every mineral is solid and inorganic • The predicates and variables are: • Solid (x), Mineral (x), and Inorganic (x). • Having these predicates, we write: x [Mineral (x)  Solid (x)  Inorganic (x)] • Translation: for all x, if x is a mineral, then x is solid and inorganic

48. Example 2 • Not all minerals are altered and broken • In this case, the predicates are: Mineral (x), Altered (x), Broken (x). The FOL expression is: [x Mineral (x)  broken (x)  altered (x)] • For all x, if x is a mineral, then it is not broken or altered • Alternatively, the expression can also be rewritten by pushing the negation through (the negation changes the  symbol into , and the conjunction symbol () into disjunction (): x Mineral (x) broken (x) altered (x)] •  There is some x, which is a mineral, and it is not broken or altered

49. Example 3 Only metamorphic rocks are exposed • Predicates: Metamorphic (x), Rock (x), and Exposed (x). The expression is: x[Rock (x) Metarmorphic (x) exposed (x) y ((x=y)  Rock (y)) Exposed (y)] • There exists some x, that if it is a rock, metamorphic, and exposed, then for all y, if y is not the same as x, and y is a rock, then y is not exposed

50. Example 4 Only older rocks are faulted and metamorphosed • Predicates: Rock (x) Older(x, y), Faulted (x), and Metamorphosed (x) x [Rock (x) Faulted (x)  Metamorphosed (x) y((x=y)  Rock(y)  Older(x, y))faulted(y) metamorphosed (y)] • There exists some x, that if x is a rock, faulted, and metamorphosed, then for all y, if y is not the same as as x, and y is a rock, and x is older than t, then y is not faulted or metamorphosed