Problem Solving and Adaptive Logics. A Logico-Philosophical Study

Transcription

1 Problem Solving and Adaptive Logics. A Logico-Philosophical Study Diderik Batens Centre for Logic and Philosophy of Science Ghent University, Belgium Diderik.Batens@UGent.be

2 CONTENTS 1 The Problem, the Claim and the Plan 2 Prospective dynamics: pushing the Heuristics into the Proofs 3 Problem-solving processes 4 Enter adaptive logics 5 Prospective dynamics for adaptive logics 6 Extensions, open problems, and the bright side of life

3 1 0 1 The Problem, the Claim and the Plan 1.1 On Solving Problems 1.2 Worries from the Philosophy of Science and from Erotetic Logic 1.3 Mastering Proof Heuristics 1.4 Unusual Logics Needed 1.5 The Traditional View On Logic 1.6 Logical Systems vs. Logical Procedures 1.7 The Plan

4 1.1 On Solving Problems problem solving is central for understanding the sciences in philosophy of science: since Kuhn,..., Laudan

5 1.1 On Solving Problems problem solving is central for understanding the sciences in philosophy of science: since Kuhn,..., Laudan from 1980s on: scientific discovery is specific kind of problem solving (cf. also scientific creativity)

6 1.1 On Solving Problems problem solving is central for understanding the sciences in philosophy of science: since Kuhn,..., Laudan from 1980s on: scientific discovery is specific kind of problem solving (cf. also scientific creativity) two kinds of contributions: (i) A.I.: set of computer programs (ii) philosophy of science: informal, often vague (Kuhn > Laudan > Nickles) Nickles: role of constraints (+ change + rational violation)

7 1.1 On Solving Problems problem solving is central for understanding the sciences in philosophy of science: since Kuhn,..., Laudan from 1980s on: scientific discovery is specific kind of problem solving (cf. also scientific creativity) two kinds of contributions: (i) A.I.: set of computer programs too specific (ii) philosophy of science: informal, often vague (Kuhn > Laudan > Nickles) Nickles: role of constraints (+ change + rational violation)

8 1.1 On Solving Problems problem solving is central for understanding the sciences in philosophy of science: since Kuhn,..., Laudan from 1980s on: scientific discovery is specific kind of problem solving (cf. also scientific creativity) two kinds of contributions: (i) A.I.: set of computer programs too specific (ii) philosophy of science: informal, often vague (Kuhn > Laudan > Nickles) Nickles: role of constraints (+ change + rational violation) nothing on the process: how proceed in order to solve

9 we need (again) a general approach here proposed: a formal approach (similar to a formal logic)

10 we need (again) a general approach here proposed: a formal approach (similar to a formal logic) Is this possible? main worries discussed in 1.2 first some more on problems

11 problem in broad sense: in principle all kinds & all domains scientific and everyday (same kind of reasoning behind them)

12 problem in broad sense: in principle all kinds & all domains scientific and everyday (same kind of reasoning behind them) problems: difficulties vs. questions

13 problem in broad sense: in principle all kinds & all domains scientific and everyday (same kind of reasoning behind them) problems: difficulties vs. questions justified questions derive from difficulties

14 problem in broad sense: in principle all kinds & all domains scientific and everyday (same kind of reasoning behind them) problems: difficulties vs. questions justified questions derive from difficulties questions answered from knowledge system / by extending it

15 problem in broad sense: in principle all kinds & all domains scientific and everyday (same kind of reasoning behind them) problems: difficulties vs. questions justified questions derive from difficulties questions answered from knowledge system / by extending it knowledge system may involve / run into difficulties

16 problem in broad sense: in principle all kinds & all domains scientific and everyday (same kind of reasoning behind them) problems: difficulties vs. questions justified questions derive from difficulties questions answered from knowledge system / by extending it knowledge system may involve / run into difficulties whether a question is difficult to answer does not depend on whether it derives from a difficulty

17 problem: will be written as a set of questions

18 problem: will be written as a set of questions consider: original problem is {?{A, A}} if B, C and D, then A

19 problem: will be written as a set of questions consider: original problem is {?{A, A}} if B, C and D, then A leads to questions?{b, B},?{C, C} and?{d, D}

20 problem: will be written as a set of questions consider: original problem is {?{A, A}} if B, C and D, then A leads to questions?{b, B},?{C, C} and?{d, D} but these are connected: if one of them receives the wrong answer, answering the others is useless with respect to the original problem so (in this context) they form a single problem: {?{B, B},?{C, C},?{D, D}} which is dropped as a whole if one of the questions has an unsuitable answer

21 problem: will be written as a set of questions consider: original problem is {?{A, A}} if B, C and D, then A leads to questions?{b, B},?{C, C} and?{d, D} but these are connected: if one of them receives the wrong answer, answering the others is useless with respect to the original problem so (in this context) they form a single problem: {?{B, B},?{C, C},?{D, D}} which is dropped as a whole if one of the questions has an unsuitable answer actually: problem = set of questions + set of pursued answers (but this will appear from the context)

22 a problem solving process (psp) has two important features: (1) it contains subsidiary and/or derived problems (derived from a previous problem derived from previous problem + premises)

23 a problem solving process (psp) has two important features: (1) it contains subsidiary and/or derived problems (derived from a previous problem derived from previous problem + premises) (2) it is goal-directed (unlike a proof on the standard definition) all steps are sensible in view of the goal (the problem solution)

24 a problem solving process (psp) has two important features: (1) it contains subsidiary and/or derived problems (derived from a previous problem derived from previous problem + premises) (2) it is goal-directed (unlike a proof on the standard definition) all steps are sensible in view of the goal (the problem solution) Note: a step may be sensible because it contributes to the solution of the problem, or because it shows that a certain road to that solution is a dead end

25 An example Galilei looking for the law of the free fall absence of adequate measuring instruments!

26 An example Galilei looking for the law of the free fall absence of adequate measuring instruments! the same force that makes the ball fall, makes it roll down the slope

27 An example Galilei looking for the law of the free fall absence of adequate measuring instruments! the same force that makes the ball fall, makes it roll down the slope

28 An example Galilei looking for the law of the free fall absence of adequate measuring instruments! the same force that makes the ball fall, makes it roll down the slope measuring the times?

29 An example Galilei looking for the law of the free fall absence of adequate measuring instruments! weigh the amount of water flowing in a vessel from the start to the point where the ball hits the wooden block compare the weights for different positions of the block (only the ratios matter)

30 1.1 1 interesting example: admittedly: no conceptual changes involved some sophistication solution is a generalization (not a singular statement) new empirical data required experiments required experiments had to be devised

31 1.2 Worries from the Philosophy of Science and from Erotetic Logic aim: devise formal procedure that explicates problem solving

32 1.2 Worries from the Philosophy of Science and from Erotetic Logic aim: devise formal procedure that explicates problem solving outdated? cf. Vienna Circle

33 1.2 Worries from the Philosophy of Science and from Erotetic Logic aim: devise formal procedure that explicates problem solving outdated? cf. Vienna Circle Nickles: no logic of discovery, only local logics of discovery

34 1.2 Worries from the Philosophy of Science and from Erotetic Logic aim: devise formal procedure that explicates problem solving outdated? cf. Vienna Circle Nickles: no logic of discovery, only local logics of discovery touchy: how do (changing) constraints surface in a formal psp? changing premises changing logics

35 Worries from the Philosophy of Science and from Erotetic Logic aim: devise formal procedure that explicates problem solving outdated? cf. Vienna Circle Nickles: no logic of discovery, only local logics of discovery touchy: how do (changing) constraints surface in a formal psp? changing premises changing logics standard erotetic logic insufficiently goal directed too restrictive (except for yes no questions)

36 1.3 Mastering Proof Heuristics logicians: good practice in solving specific type of problems: Γ A? find a proof if there is one (in most cases) see when there is no proof (in most cases)

37 1.3 Mastering Proof Heuristics logicians: good practice in solving specific type of problems: Γ A? find a proof if there is one (in most cases) see when there is no proof (in most cases) demonstrate that there is no proof if there is none (in most cases) tableau methods and other kinds of procedures (see later)

38 1.3 Mastering Proof Heuristics logicians: good practice in solving specific type of problems: Γ A? find a proof if there is one (in most cases) see when there is no proof (in most cases) demonstrate that there is no proof if there is none (in most cases) tableau methods and other kinds of procedures (see later) CL is not decidable, there only is a positive test (is partially recursive) so non-derivability cannot always be demonstrated

39 1.3 Mastering Proof Heuristics logicians: good practice in solving specific type of problems: Γ A? find a proof if there is one (in most cases) see when there is no proof (in most cases) demonstrate that there is no proof if there is none (in most cases) tableau methods and other kinds of procedures (see later) CL is not decidable, there only is a positive test (is partially recursive) so non-derivability cannot always be demonstrated usual positive tests are rather distant from proofs and so are (partial) methods for showing non-derivability

40 Ghent result: push (most of) the proof heuristics into the proof side effect of dynamic logics (prospective dynamics)

41 Ghent result: push (most of) the proof heuristics into the proof side effect of dynamic logics (prospective dynamics) simple idea: if you want to obtain A, and B A is available, look for B add to the proof: [B] A

42 Ghent result: push (most of) the proof heuristics into the proof side effect of dynamic logics (prospective dynamics) simple idea: if you want to obtain A, and B A is available, look for B add to the proof: [B] A if you want to obtain A, and A B is available, look for B add to the proof: [ B] A etc.

43 result: a procedure (see later) with the properties: (1) if Γ CL A, then the procedure leads to a proof of A from Γ (2) if the procedure leads to a proof of A from Γ, then Γ CL A (3) if the procedure stops, not providing a proof, then Γ CL A (4) for decidable fragments of CL: if Γ CL A, then the procedure stops

44 result: a procedure (see later) with the properties: (1) if Γ CL A, then the procedure leads to a proof of A from Γ (2) if the procedure leads to a proof of A from Γ, then Γ CL A (3) if the procedure stops, not providing a proof, then Γ CL A (4) for decidable fragments of CL: if Γ CL A, then the procedure stops casual comments: no way to strengthen (4)

45 result: a procedure (see later) with the properties: (1) if Γ CL A, then the procedure leads to a proof of A from Γ (2) if the procedure leads to a proof of A from Γ, then Γ CL A (3) if the procedure stops, not providing a proof, then Γ CL A (4) for decidable fragments of CL: if Γ CL A, then the procedure stops casual comments: no way to strengthen (4) algorithm for turning the prospective proof into a standard proof

46 1.3 1 result: a procedure (see later) with the properties: (1) if Γ CL A, then the procedure leads to a proof of A from Γ (2) if the procedure leads to a proof of A from Γ, then Γ CL A (3) if the procedure stops, not providing a proof, then Γ CL A (4) for decidable fragments of CL: if Γ CL A, then the procedure stops casual comments: no way to strengthen (4) algorithm for turning the prospective proof into a standard proof other (standard) logics: rather straightforward way to turn inference rules into prospective rules and to turn prospective proofs into standard proofs

47 1.4 Unusual Logics Needed problem solving requires reasoning processes for which there is no positive test (= that are not even partially recursive) inductive generalization, abduction to the best explanation, etc. traditionally seen as beyond the scope of logic

48 1.4 Unusual Logics Needed problem solving requires reasoning processes for which there is no positive test (= that are not even partially recursive) inductive generalization, abduction to the best explanation, etc. traditionally seen as beyond the scope of logic adaptive logics are capable of explicating such reasoning processes

49 Unusual Logics Needed problem solving requires reasoning processes for which there is no positive test (= that are not even partially recursive) inductive generalization, abduction to the best explanation, etc. traditionally seen as beyond the scope of logic adaptive logics are capable of explicating such reasoning processes the claim: formulating prospective proofs for adaptive logics provides us with a formal approach to problem solving

50 1.5 The Traditional View On Logic main point: adaptive logics do not suit the standard view on logic

51 1.5 The Traditional View On Logic main point: adaptive logics do not suit the standard view on logic no logic (not even CL) fits the standard view on logic of 1900 because that view was provably mistaken (and was proven to be mistaken)

52 The Traditional View On Logic main point: adaptive logics do not suit the standard view on logic no logic (not even CL) fits the standard view on logic of 1900 because that view was provably mistaken (and was proven to be mistaken) I do not claim that logics that fit the present standard view are not sensible I only claim that, in departing slightly from the standard view, one is able to decently explicate forms of reasoning that (i) are extremely important in human (scientific and other) reasoning (ii) do not fit the standard view

53 1.6 Logical Systems vs. Logical Procedures standard definition of logical system: set of rules, governing proofs any extension of a proof with an application of a rule is a proof

54 1.6 Logical Systems vs. Logical Procedures standard definition of logical system: set of rules, governing proofs any extension of a proof with an application of a rule is a proof procedure: set of rules for each rule: permission/obligation depending on stage of proof

55 1.6 Logical Systems vs. Logical Procedures standard definition of logical system: set of rules, governing proofs any extension of a proof with an application of a rule is a proof procedure: set of rules for each rule: permission/obligation depending on stage of proof standard definition: rules + universal permission this is not a sensible explication of human reasoning (goal directed)

56 Logical Systems vs. Logical Procedures standard definition of logical system: set of rules, governing proofs any extension of a proof with an application of a rule is a proof procedure: set of rules for each rule: permission/obligation depending on stage of proof standard definition: rules + universal permission this is not a sensible explication of human reasoning (goal directed) example: on the prospective-dynamics procedure, a premise cannot be added to the proof unless a present target can be obtained from the premise by means of subformulas and negations of subformulas of the premise if the target is p, p q cannot be added, but q p can

57 The Plan comment on table of contents

58 2 0 2 Prospective Dynamics: Pushing the Heuristics into the Proofs 2.1 Proofs and their Explications 2.2 Instructions vs. rules 2.3 Prospective dynamics: idea and examples 2.4 Prospective dynamics: characterization 2.5 Where went Ex Falso Quodlibet? 2.6 Some properties of CL 2.7 Afterthought

59 2.1 Proofs and their Explications CL is claimed to explicate actual proofs, for example in mathematics This presupposes: (1) specific meaning of the logical symbols in those contexts

60 2.1 Proofs and their Explications CL is claimed to explicate actual proofs, for example in mathematics This presupposes: (1) specific meaning of the logical symbols in those contexts not discussed here

61 2.1 Proofs and their Explications CL is claimed to explicate actual proofs, for example in mathematics This presupposes: (1) specific meaning of the logical symbols in those contexts not discussed here (2) correct proofs classified as correct proofs classified as correct are correct

62 2.1 Proofs and their Explications CL is claimed to explicate actual proofs, for example in mathematics This presupposes: (1) specific meaning of the logical symbols in those contexts not discussed here (2) correct proofs classified as correct OK proofs classified as correct are correct

63 2.1 Proofs and their Explications CL is claimed to explicate actual proofs, for example in mathematics This presupposes: (1) specific meaning of the logical symbols in those contexts not discussed here (2) correct proofs classified as correct OK proofs classified as correct are correct yes, but...

64 Actual proofs: actually produced result of search process actually published presentation

65 Actual proofs: actually produced actually published skip dead ends skip detours skip obvious steps... result of search process presentation

66 Actual proofs: result from goal-directed process actually produced actually published skip dead ends skip detours skip obvious steps... result of search process presentation

67 Neither produced nor published proofs are explicated adequately by CL: CL is too permissive, viz. not goal directed

68 2.1 2 Neither produced nor published proofs are explicated adequately by CL: CL is too permissive, viz. not goal directed for example 1 p Prem 2 p q 1; Add 3 p r 1; Add 4 p s 1; Add......

69 2.2 Instructions vs. rules rule: preserves truth instruction: permission/obligation to apply a rule (depending on stage of the proof)

70 2.2 Instructions vs. rules rule: preserves truth instruction: permission/obligation to apply a rule (depending on stage of the proof) official proof: procedure = rules + universal permission not goal-directed does not explicate actually produced / published proofs is border case of procedure

71 2.2 Instructions vs. rules rule: preserves truth instruction: permission/obligation to apply a rule (depending on stage of the proof) official proof: procedure = rules + universal permission not goal-directed does not explicate actually produced / published proofs is border case of procedure some procedures explicate actual proofs 2.2 2

72 2.3 Prospective dynamics: idea and examples idea: if one looks for A and, e.g., B A was derived then look for B

73 2.3 Prospective dynamics: idea and examples idea: if one looks for A and, e.g., B A was derived then look for B pushing (part of) the heuristics in the proof: if one looks for A and, e.g., B A was derived then derive [B] A indicating that one should look for B (given the premises, obtaining B is sufficient to obtain A)

74 t q, p (q r), r s, s p q

75 t q, p (q r), r s, s p q 1 [q] q Goal

76 t q, p (q r), r s, s p q 1 [q] q Goal 2 t q Prem

77 t q, p (q r), r s, s p q 1 [q] q Goal 2 t q Prem 3 [ t] q 2; E

78 t q, p (q r), r s, s p q 1 [q] q Goal 2 t q Prem 3 [ t] q 2; E t

79 t q, p (q r), r s, s p q 1 [q] q Goal 2 t q Prem 3 [ t] q 2; E t 4 p (q r) Prem

80 t q, p (q r), r s, s p q 1 [q] q Goal 2 t q Prem 3 [ t] q 2; E t 4 p (q r) Prem 5 [p] q r 4; E

81 t q, p (q r), r s, s p q 1 [q] q Goal 2 t q Prem 3 [ t] q 2; E t 4 p (q r) Prem 5 [p] q r 4; E 6 s p Prem

82 t q, p (q r), r s, s p q 1 [q] q Goal 2 t q Prem 3 [ t] q 2; E t 4 p (q r) Prem 5 [p] q r 4; E 6 s p Prem 7 [s] p 6; E

83 t q, p (q r), r s, s p q 1 [q] q Goal 2 t q Prem 3 [ t] q 2; E t 4 p (q r) Prem 5 [p] q r 4; E 6 s p Prem 7 [s] p 6; E 8 r s Prem

84 t q, p (q r), r s, s p q 1 [q] q Goal 2 t q Prem 3 [ t] q 2; E t 4 p (q r) Prem 5 [p] q r 4; E 6 s p Prem 7 [s] p 6; E 8 r s Prem 9 s 8; E

85 t q, p (q r), r s, s p q 1 [q] q Goal 2 t q Prem 3 [ t] q 2; E t 4 p (q r) Prem 5 [p] q r 4; E 6 s p Prem 7 [s] p 6; E 8 r s Prem 9 s 8; E 10 p 7, 9; Trans

86 t q, p (q r), r s, s p q 1 [q] q Goal 2 t q Prem 3 [ t] q 2; E t 4 p (q r) Prem 5 [p] q r 4; E 6 s p Prem 7 [s] p 6; E R 10 8 r s Prem 9 s 8; E 10 p 7, 9; Trans

87 t q, p (q r), r s, s p q 1 [q] q Goal 2 t q Prem 3 [ t] q 2; E t 4 p (q r) Prem 5 [p] q r 4; E 6 s p Prem 7 [s] p 6; E R 10 8 r s Prem 9 s 8; E 10 p 7, 9; Trans 11 q r 5, 10; Trans

88 t q, p (q r), r s, s p q 1 [q] q Goal 2 t q Prem 3 [ t] q 2; E t 4 p (q r) Prem 5 [p] q r 4; E R 11 6 s p Prem 7 [s] p 6; E R 10 8 r s Prem 9 s 8; E 10 p 7, 9; Trans 11 q r 5, 10; Trans

89 t q, p (q r), r s, s p q 1 [q] q Goal 2 t q Prem 3 [ t] q 2; E t 4 p (q r) Prem 5 [p] q r 4; E R 11 6 s p Prem 7 [s] p 6; E R 10 8 r s Prem 9 s 8; E 10 p 7, 9; Trans 11 q r 5, 10; Trans 12 [r] q 11; E

90 t q, p (q r), r s, s p q 1 [q] q Goal 2 t q Prem 3 [ t] q 2; E t 4 p (q r) Prem 5 [p] q r 4; E R 11 6 s p Prem 7 [s] p 6; E R 10 8 r s Prem 9 s 8; E 10 p 7, 9; Trans 11 q r 5, 10; Trans 12 [r] q 11; E 13 r 8; E

91 t q, p (q r), r s, s p q 1 [q] q Goal 2 t q Prem 3 [ t] q 2; E t 4 p (q r) Prem 5 [p] q r 4; E R 11 6 s p Prem 7 [s] p 6; E R 10 8 r s Prem 9 s 8; E 10 p 7, 9; Trans 11 q r 5, 10; Trans 12 [r] q 11; E 13 r 8; E 14 q 12, 13; Trans

92 t q, p (q r), r s, s p q 1 [q] q Goal R 14 2 t q Prem 3 [ t] q 2; E t 4 p (q r) Prem 5 [p] q r 4; E R 11 6 s p Prem 7 [s] p 6; E R 10 8 r s Prem 9 s 8; E 10 p 7, 9; Trans 11 q r 5, 10; Trans 12 [r] q 11; E R r 8; E 14 q 12, 13; Trans

93 Incidentally: algorithm: prospective proofs Fitch-style proofs 1 p (q r) Prem 2 s p Prem 3 r s Prem 4 s 3; Sim 5 p 2, 4; MP 6 q r 1, 5; MP 7 r 3; Sim 8 q 6, 7; DS

94 p q p q

95 p q p q 1 [p q] p q Goal

96 p q p q 1 [p q] p q Goal 2 [q] p q 1; C E

97 p q p q 1 [p q] p q Goal 2 [q] p q 1; C E 3 p q Prem

98 p q p q 1 [p q] p q Goal 2 [q] p q 1; C E 3 p q Prem 4 [p] q 3; E

99 p q p q 1 [p q] p q Goal 2 [q] p q 1; C E q 3 p q Prem 4 [p] q 3; E p

100 p q p q 1 [p q] p q Goal 2 [q] p q 1; C E q 3 p q Prem 4 [p] q 3; E p 5 [ p] p q 1; C E

101 p q p q 1 [p q] p q Goal 2 [q] p q 1; C E q 3 p q Prem 4 [p] q 3; E p 5 [ p] p q 1; C E 6 [ q] p 3; E

102 p q p q 1 [p q] p q Goal 2 [q] p q 1; C E q 3 p q Prem 4 [p] q 3; E p 5 [ p] p q 1; C E p 6 [ q] p 3; E q

103 p q p q 1 [p q] p q Goal 2 [q] p q 1; C E q 3 p q Prem 4 [p] q 3; E p 5 [ p] p q 1; C E p 6 [ q] p 3; E q 7 [p] p q 2, 4; Trans obtain the Goal on all non-redundant conditions

104 p q p q 1 [p q] p q Goal 2 [q] p q 1; C E q 3 p q Prem 4 [p] q 3; E p 5 [ p] p q 1; C E p 6 [ q] p 3; E q 7 [p] p q 2, 4; Trans p obtain the Goal on all non-redundant conditions

105 p q p q 1 [p q] p q Goal 2 [q] p q 1; C E q 3 p q Prem 4 [p] q 3; E p 5 [ p] p q 1; C E p 6 [ q] p 3; E q 7 [p] p q 2, 4; Trans p 8 p q 5, 7; EM obtain the Goal on all non-redundant conditions

106 2.3 2 p q p q 1 [p q] p q Goal R 8 2 [q] p q 1; C E q R 8 3 p q Prem 4 [p] q 3; E p 5 [ p] p q 1; C E p R 8 6 [ q] p 3; E q 7 [p] p q 2, 4; Trans p R 8 8 p q 5, 7; EM obtain the Goal on all non-redundant conditions

107 2.4 Prospective dynamics: characterization Rules (prospective proof for Γ G) Goal To introduce [G] G. Prem To introduce A for an A Γ. Trans EM [ {B}] A [ ] B [ ] A [ {B}] A [ { B}] A [ ] A

108 Note: the complement of a formula: if A has the form B, then A = B otherwise A = A

109 Note: the complement of a formula: if A has the form B, then A = B otherwise A = A p = p p = p p = p p = p p = p p = p

110 α α 1 α 2 β β 1 β 2 A B A B (A B) A B A B A B B A (A B) (A B) (B A) (A B) A B A B A B (A B) A B A B A B A A A Formula analysing rules: [ ] α [ ] α 1 [ ] α 2 [ ] β [ { β 2 }] β 1 [ { β 1 }] β 2

111 α α 1 α 2 β β 1 β 2 A B A B (A B) A B A B A B B A (A B) (A B) (B A) (A B) A B A B A B (A B) A B A B A B A A A Formula analysing rules: Example: [ ] α [ ] β [ ] α 1 [ ] α 2 [ { β 2 }] β 1 [ { β 1 }] β 2 [ ] p q [ ] p [ ] q [ ] p q [ { q}] p [ { p}] q

112 α α 1 α 2 β β 1 β 2 A B A B (A B) A B A B A B B A (A B) (A B) (B A) (A B) A B A B A B (A B) A B A B A B A A A Condition analysing rules: [ {α}] A [ {α 1, α 2 }] A [ {β}] A [ {β 1 }] A [ {β 2 }] A

113 α α 1 α 2 β β 1 β 2 A B A B (A B) A B A B A B B A (A B) (A B) (B A) (A B) A B A B A B (A B) A B A B A B A A A Condition analysing rules: [ {α}] A [ {α 1, α 2 }] A [ {β}] A [ {β 1 }] A [ {β 2 }] A Example: [ {q r}] p [ {q, r}] p [ {q r}] p [ {q}] p [ {r}] p

114 The permissions and obligations positive part: 1. pp(a, A). 2. pp(a, α) if pp(a, α 1 ) or pp(a, α 2 ). 3. pp(a, β) if pp(a, β 1 ) or pp(a, β 2 ).

115 The permissions and obligations positive part: 1. pp(a, A). 2. pp(a, α) if pp(a, α 1 ) or pp(a, α 2 ). 3. pp(a, β) if pp(a, β 1 ) or pp(a, β 2 ). A line with second element [ ] A is marked as a dead end iff an element of is not a pp of any premise.

116 The permissions and obligations positive part: 1. pp(a, A). 2. pp(a, α) if pp(a, α 1 ) or pp(a, α 2 ). 3. pp(a, β) if pp(a, β 1 ) or pp(a, β 2 ). A line with second element [ ] A is marked as a dead end iff an element of is not a pp of any premise. A line with second element [ ] A is marked as a redundant iff (i) A (not the Goal line) or (ii) a line with second element [ ] A occurs and.

117 The permissions and obligations positive part: 1. pp(a, A). 2. pp(a, α) if pp(a, α 1 ) or pp(a, α 2 ). 3. pp(a, β) if pp(a, β 1 ) or pp(a, β 2 ). A line with second element [ ] A is marked as a dead end iff an element of is not a pp of any premise. A line with second element [ ] A is marked as a redundant iff (i) A (not the Goal line) or (ii) a line with second element [ ] A occurs and. more marks possible (e.g., inconsistent paths) The target is the first formula in the condition of the last unmarked line. (alternatives possible)

118 2.4 2 Phase 1: start with Goal rule apply FAR only to formula of line that has Prem-line in its path derive [B 1,..., B n ] A by FAR only if target is pp of A next, introduce a new premise A iff target is pp of A apply CAR only to target A after Prem and FAR are exhausted apply Trans only if is empty Phase 2: only: new [ ] G by EM, Trans or CAR from R-unmarked lines next return to phase 1

119 2.5 Where went Ex Falso Quodlibet? Let the logic defined by the procedure be pcl p, p pcl q

120 2.5 Where went Ex Falso Quodlibet? Let the logic defined by the procedure be pcl p, p pcl q EFQ requires, for example: EFQ To introduce [ A] G for an A Γ. This rule may be applied to every A Γ.

121 2.5 Where went Ex Falso Quodlibet? Let the logic defined by the procedure be pcl p, p pcl q EFQ requires, for example: EFQ To introduce [ A] G for an A Γ. This rule may be applied to every A Γ. EFQ is an isolated, unnatural and ad hoc rule.

122 Where pcl is (propositional) pcl + EFQ: Γ pcl A iff Γ CL A

123 2.5 2 Where pcl is (propositional) pcl + EFQ: Γ pcl A iff Γ CL A That is: If Γ CL A, the procedure will lead to a proof of A from Γ. If Γ CL A, the procedure will stop without A being derived.

124 2.6 Some properties of CL natural explication of all sensible classical proofs

125 2.6 Some properties of CL natural explication of all sensible classical proofs EFQ is absent, whence isolated, and unnatural

126 2.6 Some properties of CL natural explication of all sensible classical proofs EFQ is absent, whence isolated, and unnatural assigns same consequences as CL to consistent Γ

127 2.6 Some properties of CL natural explication of all sensible classical proofs EFQ is absent, whence isolated, and unnatural assigns same consequences as CL to consistent Γ (the intended domain of application of CL)

128 2.6 Some properties of CL natural explication of all sensible classical proofs EFQ is absent, whence isolated, and unnatural assigns same consequences as CL to consistent Γ (the intended domain of application of CL) derives a contradiction from all inconsistent Γ, but not triviality (except in border cases)

129 2.6 Some properties of CL natural explication of all sensible classical proofs EFQ is absent, whence isolated, and unnatural assigns same consequences as CL to consistent Γ (the intended domain of application of CL) derives a contradiction from all inconsistent Γ, but not triviality (except in border cases) assigns sensible consequence set to inconsistent Γ

130 2.6 Some properties of CL natural explication of all sensible classical proofs EFQ is absent, whence isolated, and unnatural assigns same consequences as CL to consistent Γ (the intended domain of application of CL) derives a contradiction from all inconsistent Γ, but not triviality (except in border cases) assigns sensible consequence set to inconsistent Γ resulting consequence relation: characterized by a semantics (and tableau method)

131 2.6 Some properties of CL natural explication of all sensible classical proofs EFQ is absent, whence isolated, and unnatural assigns same consequences as CL to consistent Γ (the intended domain of application of CL) derives a contradiction from all inconsistent Γ, but not triviality (except in border cases) assigns sensible consequence set to inconsistent Γ resulting consequence relation: characterized by a semantics (and tableau method) reflexive and monotonic

132 2.6 Some properties of CL natural explication of all sensible classical proofs EFQ is absent, whence isolated, and unnatural assigns same consequences as CL to consistent Γ (the intended domain of application of CL) derives a contradiction from all inconsistent Γ, but not triviality (except in border cases) assigns sensible consequence set to inconsistent Γ resulting consequence relation: characterized by a semantics (and tableau method) reflexive and monotonic not transitive (even weak cut does not hold)

133 2.6 Some properties of CL natural explication of all sensible classical proofs EFQ is absent, whence isolated, and unnatural assigns same consequences as CL to consistent Γ (the intended domain of application of CL) derives a contradiction from all inconsistent Γ, but not triviality (except in border cases) assigns sensible consequence set to inconsistent Γ resulting consequence relation: characterized by a semantics (and tableau method) reflexive and monotonic not transitive (even weak cut does not hold) but transitive if restricted to consistent premise sets

134 2.6 Some properties of CL natural explication of all sensible classical proofs EFQ is absent, whence isolated, and unnatural assigns same consequences as CL to consistent Γ (the intended domain of application of CL) derives a contradiction from all inconsistent Γ, but not triviality (except in border cases) assigns sensible consequence set to inconsistent Γ resulting consequence relation: characterized by a semantics (and tableau method) reflexive and monotonic not transitive (even weak cut does not hold) but transitive if restricted to consistent premise sets in an interesting (specific) sense relevant

135 2.6 Some properties of CL natural explication of all sensible classical proofs EFQ is absent, whence isolated, and unnatural assigns same consequences as CL to consistent Γ (the intended domain of application of CL) derives a contradiction from all inconsistent Γ, but not triviality (except in border cases) assigns sensible consequence set to inconsistent Γ resulting consequence relation: characterized by a semantics (and tableau method) reflexive and monotonic not transitive (even weak cut does not hold) but transitive if restricted to consistent premise sets in an interesting (specific) sense relevant exactly the same theorems as CL

136 2.6 Some properties of CL natural explication of all sensible classical proofs EFQ is absent, whence isolated, and unnatural assigns same consequences as CL to consistent Γ (the intended domain of application of CL) derives a contradiction from all inconsistent Γ, but not triviality (except in border cases) assigns sensible consequence set to inconsistent Γ resulting consequence relation: characterized by a semantics (and tableau method) reflexive and monotonic not transitive (even weak cut does not hold) but transitive if restricted to consistent premise sets in an interesting (specific) sense relevant exactly the same theorems as CL adequate w.r.t. CL-semantics if restricted to consistent Γ

137 Sensible p q, p, q p q, p, q p p q q To derive Russell s paradox from Frege s set theory.

138 Sensible p q, p, q p q, p, q p p q q To derive Russell s paradox from Frege s set theory. not sensible p q, p, q r r To derive from Frege s set theory that the moon is and is not a blue cheese (or that ( ) = ( ) ).

139 Sensible p q, p, q p q, p, q p p q q To derive Russell s paradox from Frege s set theory. not sensible p q, p, q r r To derive from Frege s set theory that the moon is and is not a blue cheese (or that ( ) = ( ) ). In problem-solving processes, CL need to be applied.

140 A semantics (Suszko: every logic has a 2-valued semantics) v : W {0, 1} is a partial function 1. if v(a) {0, 1} and sub(b, A), then v(b), v( B) {0, 1} 2. if v(a B) = 1 then v(a) = 1 and v(b) = if v(a B) = 0 then v(a) = 0 or v(b) = if v(a B) = 1 then v(a B) = 1 and v(b A) = if v(a B) = 0 then v(a B) = 0 or v(b A) = if v( (A B)) = 1 then v( A) = 1 and v( B) = if v( (A B)) = 0 then v( A) = 0 or v( B) = if v( (A B)) = 1 then v(a) = 1 and v( B) = if v( (A B)) = 0 then v(a) = 0 or v( B) = if v( A) = 1 then v(a) = if v( A) = 0 then v(a) = if v(a B) = 1 then v( A) = 0 or v(b) = if v(a B) = 1 then v(a) = 1 or v( B) = if v(a B) = 0 then v(a) = 0 and v(b) = if v(a B) = 1 then v(a) = 0 or v(b) = if v(a B) = 1 then v( A) = 1 or v( B) = if v(a B) = 0 then v( A) = 0 and v(b) = if v( (A B)) = 1 then v(a) = 0 or v( B) = if v( (A B)) = 1 then v( A) = 1 or v(b) = if v( (A B)) = 0 then v( A) = 0 and v( B) = if v( (A B)) = 1 then v((a B)) = 0 or v( (B A)) = if v( (A B)) = 1 then v( (A B)) = 1 or v((b A)) = if v( (A B)) = 0 then v( (A B)) = v( (B A)) = if v(a) = 0 then v( A) = 1.

141 Definition A 1,..., A n B (B is a semantic consequence of A 1,..., A n ) iff all valuations that verify A 1,..., A n and for which B is determined, verify B.

142 Definition A 1,..., A n B (B is a semantic consequence of A 1,..., A n ) iff all valuations that verify A 1,..., A n and for which B is determined, verify B. That is: Definition A 1,..., A n B (B is a semantic consequence of A 1,..., A n ) iff no valuation that verifies A 1,..., A n falsifies B.

143 Definition A 1,..., A n B (B is a semantic consequence of A 1,..., A n ) iff all valuations that verify A 1,..., A n and for which B is determined, verify B. That is: Definition A 1,..., A n B (B is a semantic consequence of A 1,..., A n ) iff no valuation that verifies A 1,..., A n falsifies B. PM: three valued truth-functional semantics

144 Theorem If [A 1,..., A n ]B is derived in a pcl -proof for Γ G, then v(b) = 1 whenever v(a 1 ) =... = v(a n ) = 1 and v(b) {0, 1}. Corollary If G is derived in a pcl -proof for Γ G, then Γ CL G. (Soundness) Theorem If a prospective proof for Γ G halts without G begin derived, then Γ CL G. (Completeness)

145 Note: Tableau method T A B T A T B T A B F A T A F B T B etc. (read off from semantic clauses)

146 Non-logicians sometimes apply CL to inconsistent premises. They consider EFQ (explosion) as a logicians trick.

147 Non-logicians sometimes apply CL to inconsistent premises. They consider EFQ (explosion) as a logicians trick. Logicians know: EFQ cannot be isolated in CL

148 Non-logicians sometimes apply CL to inconsistent premises. They consider EFQ (explosion) as a logicians trick. Logicians know: EFQ cannot be isolated in CL avoiding EFQ requires avoiding: Addition or Disjunctive Syllogism A / B A or A (B B) / A A / B A or A (B B) / A or A / A etc.

149 Non-logicians sometimes apply CL to inconsistent premises. They consider EFQ (explosion) as a logicians trick. Logicians know: EFQ cannot be isolated in CL avoiding EFQ requires avoiding: Addition or Disjunctive Syllogism A / B A or A (B B) / A A / B A or A (B B) / A or A / A etc. However: That EFQ cannot be isolated in CL depends on our view on logic (mere rules vs. procedures).

150 2.6 2 The upgrade to predicate logic (minus EFQ): is straightforward if the procedure stops (not for all Γ and A), with A derived: then Γ CL A with A not derived: then Γ CL A

151 2.7 Afterthought Hintikka: distinction between rules and heuristics comparison with game of chess

152 2.7 Afterthought Hintikka: distinction between rules and heuristics comparison with game of chess This is a mistake: heuristic reasoning leads to sensible proofs part of this reasoning can be pushed into the (object-language) proofs

153 Moreover: truth-in-a-model is a touchy matter: given CL-models one can distinguish valid consequences from sensible consequences

154 Moreover: truth-in-a-model is a touchy matter: given CL-models one can distinguish valid consequences from sensible consequences BUT: there is a semantics that is adequate for sensible reasoning in CL

155 Moreover: truth-in-a-model is a touchy matter: given CL-models one can distinguish valid consequences from sensible consequences BUT: there is a semantics that is adequate for sensible reasoning in CL A is a CL -consequence of Γ (no CL -model of Γ falsifies A) iff A is a sensible CL-consequence of Γ

156 2.8 2 Moreover: truth-in-a-model is a touchy matter: given CL-models one can distinguish valid consequences from sensible consequences BUT: there is a semantics that is adequate for sensible reasoning in CL A is a CL -consequence of Γ (no CL -model of Γ falsifies A) iff A is a sensible CL-consequence of Γ in other words: sensibility can be incorporated into truth

157 3 0 3 Problem-solving processes 3.1 Aim and introductory remarks 3.2 Problem-solving processes: first elements 3.3 An example 3.4 The rules and the permissions and obligations 3.5 Answerable questions 3.6 Variants, extensions and comments

158 3.1 Aim and introductory remarks backbone of formal approach to problem solving

159 3.1 Aim and introductory remarks backbone of formal approach to problem solving aim: explication of problem solving processes (psps)

160 3.1 Aim and introductory remarks backbone of formal approach to problem solving aim: explication of problem solving processes (psps) backbone: solve {?{A, A}} by deriving A or A from Γ by CL

161 3.1 Aim and introductory remarks backbone of formal approach to problem solving aim: explication of problem solving processes (psps) backbone: solve {?{A, A}} by deriving A or A from Γ by CL empirical means (observation and experiment)

162 3.1 Aim and introductory remarks backbone of formal approach to problem solving aim: explication of problem solving processes (psps) backbone: solve {?{A, A}} by deriving A or A from Γ by CL empirical means (observation and experiment) + new available information (not originally seen as relevant) (easy extension)

163 3.1 Aim and introductory remarks backbone of formal approach to problem solving aim: explication of problem solving processes (psps) backbone: solve {?{A, A}} by deriving A or A from Γ by CL empirical means (observation and experiment) + new available information (not originally seen as relevant) (easy extension) + corrective and ampliative logics, handling inconsistency,... includes forming new hypotheses adaptive logics: control by conditions and marking definition (easy extension)

164 3.1 Aim and introductory remarks backbone of formal approach to problem solving aim: explication of problem solving processes (psps) backbone: solve {?{A, A}} by deriving A or A from Γ by CL empirical means (observation and experiment) + new available information (not originally seen as relevant) (easy extension) + corrective and ampliative logics, handling inconsistency,... includes forming new hypotheses adaptive logics: control by conditions and marking definition (easy extension) + devise new empirical means: future research (seems within reach)

165 3.1 Aim and introductory remarks backbone of formal approach to problem solving aim: explication of problem solving processes (psps) backbone: solve {?{A, A}} by deriving A or A from Γ by CL empirical means (observation and experiment) + new available information (not originally seen as relevant) (easy extension) + corrective and ampliative logics, handling inconsistency,... includes forming new hypotheses adaptive logics: control by conditions and marking definition (easy extension) + devise new empirical means: future research (seems within reach) + model-based reasoning,... : future research

166 Plan given a logic (or a set of logics) L, we can handle the heuristics (see previous lecture)

167 Plan given a logic (or a set of logics) L, we can handle the heuristics (see previous lecture) viz. define the procedure for solving {?{A, A}} by deriving A or A from Γ by L

168 Plan given a logic (or a set of logics) L, we can handle the heuristics (see previous lecture) viz. define the procedure for solving {?{A, A}} by deriving A or A from Γ by L adaptive logics enable us to explicate the reasoning behind many psps (see next lecture)

169 Background philosophy of science: Nickles, Meheus, Batens erotetic logic (varying on Wiśniewski) logic adaptive logics prospective dynamics procedures

170 Background philosophy of science: Nickles, Meheus, Batens erotetic logic (varying on Wiśniewski) logic adaptive logics prospective dynamics procedures problem determined by (changing) constraints conditions on the solution methodological instructions / heuristics / examples certainties (conceptual system... )

171 Formal approach formal but not logic with the usual connotations

172 Formal approach formal but not logic with the usual connotations proofs success psp (problem solving process) success?

173 Formal approach formal but not logic with the usual connotations proofs success arbitrary sequence of applications of rules psp (problem solving process) success? goal directed

174 Formal approach formal but not logic with the usual connotations proofs success arbitrary sequence of applications of rules infinite consequence set psp (problem solving process) success? goal directed unique aim (possibly unspecified at outset)

175 Formal approach formal but not logic with the usual connotations proofs success arbitrary sequence of applications of rules infinite consequence set useless subsequences psp (problem solving process) success? goal directed unique aim (possibly unspecified at outset) unsuccessful subsequences

176 Formal approach formal but not logic with the usual connotations proofs success arbitrary sequence of applications of rules infinite consequence set useless subsequences deductive psp (problem solving process) success? goal directed unique aim (possibly unspecified at outset) unsuccessful subsequences also other forms of reasoning

177 Formal approach formal but not logic with the usual connotations proofs success arbitrary sequence of applications of rules infinite consequence set useless subsequences deductive CL psp (problem solving process) success? goal directed unique aim (possibly unspecified at outset) unsuccessful subsequences also other forms of reasoning multiplicity of logics

178 3.1 3 differences partly rely on confusion proof search is goal-directed process (and is a psp) proof search is not always successful no arbitrary sequences result of proof search proof search for one formula from given premises (but set of problems solvable by certain means) unsuccessful subsequences in proof search no useless subsequences in goal-directed proofs that all logic is deductive (or is CL) is a plain prejudice

179 3.2 Problem-solving processes: first elements terminology: psp refers to explicandum and to explicatum

180 3.2 Problem-solving processes: first elements terminology: psp refers to explicandum and to explicatum psps contain unsuccessful subsequences justified at some point in the psp not justified any more at later point and vice versa unsuccessful is a dynamic property

181 psps require prospective dynamics + derived problems

182 psps require prospective dynamics + derived problems prospective dynamics (previous lecture) now breath first (better w.r.t. problems)

183 psps require prospective dynamics + derived problems prospective dynamics (previous lecture) now breath first (better w.r.t. problems) derived problems: {?{A, A}} (problem)... [B 1,..., B n ] A (if B 1,..., B n true, then also A)

184 psps require prospective dynamics + derived problems prospective dynamics (previous lecture) now breath first (better w.r.t. problems) derived problems: {?{A, A}} (problem)... [B 1,..., B n ] A (if B 1,..., B n true, then also A) {?{B 1, B 1 },...,?{B n, B n }} derived problem

185 Lines occurring in a psp problem lines: problem = non-empty set of questions

186 Lines occurring in a psp problem lines: problem = non-empty set of questions declarative lines conditional: [B 1,..., B n ] A unconditional: [ ] A, viz. A

187 a stage of a psp: sequence of lines a psp: chain of stages next stage obtained by adding new line marks may change (governed by marking definitions) relation between stages governed by procedure

188 rehearsal the complement of a formula a-formulas and b-formulas (a and b) formula analysing rules and condition analysing rules pp(a, B) (A is a positive part of B) the Prem rule EM, EM0 and Trans direct answer to a question / problem

189 Specific rules Where {?{M, M}} is the main (or original) problem: Main Start a psp with the line: 1 {?{M, M}} Main

190 Specific rules Where {?{M, M}} is the main (or original) problem: Main Start a psp with the line: 1 {?{M, M}} Main Target rule (to choose a target that one tries to obtain) Target If P is the problem of an unmarked problem line, and A is a direct answer of a member of P, then one may add: k [A] A Target

191 3.2 3 Specific rules Where {?{M, M}} is the main (or original) problem: Main Start a psp with the line: 1 {?{M, M}} Main Target rule (to choose a target that one tries to obtain) Target If P is the problem of an unmarked problem line, and A is a direct answer of a member of P, then one may add: k [A] A Target Derive problems: DP If A is an unmarked target from problem line i and [B 1,..., B n ] A is the formula of an unmarked line j, then one may add: k {?{B 1, B 1 },...,?{B n, B n }} i, j; DP

192 3.3 An example main problem:?{p q, (p q)} premise set: { s, u r, (r t) s, (q u) ( t q), t u}

193 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main

194 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main 2 [ (p q)] (p q) Target

195 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main 2 [ (p q)] (p q) Target D 3 3 [ p, q] (p q) 2; C E D 3

196 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main 2 [ (p q)] (p q) Target D 3 3 [ p, q] (p q) 2; C E D 3 4 [p q] p q Target

197 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main 2 [ (p q)] (p q) Target D 3 3 [ p, q] (p q) 2; C E D 3 4 [p q] p q Target 5 [p] p q 4; C E D 5

198 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main 2 [ (p q)] (p q) Target D 3 3 [ p, q] (p q) 2; C E D 3 4 [p q] p q Target 5 [p] p q 4; C E D 5 6 [q] p q 4; C E

199 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main 4 [p q] p q Target 6 [q] p q 4; C E

200 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main 4 [p q] p q Target 6 [q] p q 4; C E 7 {?{q, q}} 4, 6; DP pursued answer: q

201 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main 4 [p q] p q Target 6 [q] p q 4; C E 7 {?{q, q}} 4, 6; DP 8 [q] q Target

202 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main 4 [p q] p q Target 6 [q] p q 4; C E 7 {?{q, q}} 4, 6; DP 8 [q] q Target 9 (q u) ( t q) Prem 4 and 6 have no premise in their path

203 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main 4 [p q] p q Target 6 [q] p q 4; C E 7 {?{q, q}} 4, 6; DP 8 [q] q Target 9 (q u) ( t q) Prem 10 [q u] t q 9; E

204 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main 4 [p q] p q Target 6 [q] p q 4; C E 7 {?{q, q}} 4, 6; DP 8 [q] q Target 9 (q u) ( t q) Prem 10 [q u] t q 9; E 11 [q] t q 10; C E

205 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main 4 [p q] p q Target 6 [q] p q 4; C E 7 {?{q, q}} 4, 6; DP 8 [q] q Target 9 (q u) ( t q) Prem 10 [q u] t q 9; E 11 [q] t q 10; C E D [q, t] q 11; E I 12 12: inoperative line

206 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main 4 [p q] p q Target 6 [q] p q 4; C E 7 {?{q, q}} 4, 6; DP 8 [q] q Target 9 (q u) ( t q) Prem 10 [q u] t q 9; E 11 [q] t q 10; C E D [q, t] q 11; E I [u] t q 10; C E

207 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main 4 [p q] p q Target 6 [q] p q 4; C E 7 {?{q, q}} 4, 6; DP 8 [q] q Target 9 (q u) ( t q) Prem 10 [q u] t q 9; E 11 [q] t q 10; C E D [q, t] q 11; E I [u] t q 10; C E 14 [u, t] q 13; E

208 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main 4 [p q] p q Target 6 [q] p q 4; C E 7 {?{q, q}} 4, 6; DP 8 [q] q Target 9 (q u) ( t q) Prem 10 [q u] t q 9; E 11 [q] t q 10; C E D [q, t] q 11; E I [u] t q 10; C E 14 [u, t] q 13; E 15 {?{u, u},?{t, t}} 8, 14; DP

209 { s, u r, (r t) s, (q u) ( t q), t u} 1 {?{p q, (p q)}} Main 4 [p q] p q Target 6 [q] p q 4; C E 7 {?{q, q}} 4, 6; DP 8 [q] q Target 9 (q u) ( t q) Prem 10 [q u] t q 9; E 11 [q] t q 10; C E D [q, t] q 11; E I [u] t q 10; C E 14 [u, t] q 13; E 15 {?{u, u},?{t, t}} 8, 14; DP 16 [t] t Target cleaning up for lack of space

210 { s, u r, (r t) s, (q u) ( t q), t u} 8 [q] q Target [u, t] q 13; E 15 {?{u, u},?{t, t}} 8, 14; DP 16 [t] t Target

211 { s, u r, (r t) s, (q u) ( t q), t u} 8 [q] q Target [u, t] q 13; E 15 {?{u, u},?{t, t}} 8, 14; DP 16 [t] t Target 17 (r t) s Prem

212 { s, u r, (r t) s, (q u) ( t q), t u} 8 [q] q Target [u, t] q 13; E 15 {?{u, u},?{t, t}} 8, 14; DP 16 [t] t Target 17 (r t) s Prem 18 [ s] r t 17; E