Saving Truth from Orthodoxy Joseph Norman Introduction Saving Truth from Orthodoxy Equations Better Logic Through Algebra, Probability, and Dynamical Systems Dynamical Systems Probability Conclusion References Joseph W. Norman, M.D., Ph.D. Extras University of Michigan, Ann Arbor Association for Symbolic Logic 2012 Madison, Wisconsin
� Diversity Is the Savior of Logical Truth Saving Truth from Orthodoxy Joseph Norman I. There is important mathematical diversity in what we call ‘logic’ – 4 different classes of problems: Introduction ◮ Arithmetic Equations ◮ Algebra (equations) Dynamical Systems ◮ Dynamical systems Probability ◮ Probability Conclusion II. In arithmetic every formula has an elementary value References (like 0 or 1). But the other systems return more Extras complex objects (sets, graphs, polynomials): � 1 { 0 , 1 } {} θ 3 + θ 1 θ 2 − θ 1 θ 3 0 Certain features of these objects seem paradoxical. III. But when logic problems are properly classified, and unorthodox objects like these are accepted as truth values, many paradoxes disappear. ∗ ∗ The Truth Fairy replaces paradoxes with complex truth values.
Does the Truth Fairy Exist? Saving Truth from Orthodoxy Joseph Norman Introduction Equations Dynamical Systems Probability Conclusion References Extras george boole ✶✽✶✺ – ✶✽✻✹
Lies, Damn Lies, and Interpretations Saving Truth from Orthodoxy Joseph Norman ◮ My Liar sentence: x : This sentence x is false. ◮ As a definition: x :: ¬ x , x ∈ { true , false } Introduction Equations ◮ Two interpretations: equation or recurrence relation Dynamical ◮ Many senses of = (test, assignment, constraint) Systems ◮ In C program, x==1-x and x=1-x are different. Probability ◮ Mathematica Solve[x==1-x] different again. Conclusion References ◮ As an equation: x = 1 − x , x ∈ { 0 , 1 } Extras ◮ From Boole: true � → 1; false � → 0; ¬ x � → 1 − x . ◮ As a recurrence: x t + 1 ⇐ 1 − x t , x ∈ { 0 , 1 } ◮ Assignment introduces an arrow of time. ◮ What kinds of truth values do we get from these two interpretations?
Solving Equations Gives a Set of Solutions Saving Truth from Orthodoxy Joseph Norman ◮ A logical axiom is a polynomial equation : ◮ Logical notation : A judgment ⊢ ψ that asserts Introduction Equations the truth of its content ( ψ in propositional calculus) ◮ Polynomial notation : An equation q = 0 where q Dynamical Systems is 1 − Poly ( ψ ) using Boole’s translation of logic: Probability ◮ Poly ( ¬ p ) = 1 − p ; Poly ( p → q ) = 1 − p + pq ; etc. Conclusion ◮ ψ = true � Poly ( ψ ) = 1 � 1 − Poly ( ψ ) = 0 References ◮ The solution set to polynomial equations gives the Extras possible values of an objective formula φ subject to some axioms ⊢ ψ 1 , . . . , ⊢ ψ m : S Q ( p ) { p ( x ) : x i ∈ { 0 , 1 } , q j ( x ) = 0 } ≡ with x = ( x 1 , . . . , x n ) ; p , q ∈ R [ x ] ; p = Poly ( φ ) ; each q j = 1 − Poly ( ψ j ) ; Q = { q 1 , . . . , q m } . ◮ This solution set must be a subset of the set of elementary values: for 2-valued logic S Q ( p ) ⊆ { 0 , 1 } .
Solution Sets Are Good Truth Values Saving Truth from Orthodoxy ◮ Solution set S Q ( Poly ( φ )) gives truth value of the Joseph Norman objective φ subject to the axioms ⊢ ψ j given as Q : Introduction { 1 } φ is a theorem . Equations { 0 } φ is the negation of a theorem. Dynamical Systems { 0 , 1 } φ is ambiguous Probability {} φ is unsatisfiable : the axioms ⊢ ψ j are inconsistent Conclusion ◮ Inverse sets of polynomials are useful: References ◮ S − 1 Q ( { 1 } ) : the set of all theorems entailed by axioms Q Extras ◮ S − 1 Q ( { 0 } ) : the ideal (algebraic geometry) Q generates ◮ This logic is paraconsistent and paracomplete . ◮ Some included middle: truth values come from the power set {{ 0 } , { 1 } , { 0 , 1 } , {}} of the set { 0 , 1 } . ◮ Different idea from adding new elementary objects (like 1 2 or 2), as usual in ‘multivalued’ logics. ◮ No explosion from contradiction: inconsistent axioms give every objective the empty solution set: so nothing is declared a theorem, not everything .
Sets of Solution Sets Give Modal Logic Saving Truth from Orthodoxy s ≡ S Q ( Poly ( φ )) Description of φ given axioms ⊢ ψ j in Q Joseph Norman { 0 } { 1 } { 0 , 1 } {} Set Modal Natural language Introduction • s = { 1 } � ( φ ) necessarily true Equations • • • s � = { 1 } ¬ � ( φ ) not necessarily true Dynamical • s = { 0 } � ( ¬ φ ) necessarily false Systems • • • s � = { 0 } ¬ � ( ¬ φ ) not necessarily false Probability • • 1 ∈ s ♦ ( φ ) possibly true Conclusion 1 / not possibly true • • ∈ s ¬ ♦ ( φ ) References 0 ∈ s ♦ ( ¬ φ ) possibly false • • Extras • • 0 / ∈ s ¬ ♦ ( ¬ φ ) not possibly false • s = {} ⊘ ( φ ) necessarily unsatisfiable • | s | > 1 ⊳ ( φ ) necessarily ambiguous ⊲ • • | s | = 1 � ( φ ) determinate • • • • s ⊆ { 0 , 1 } ♥ ( φ ) { 0 , 1 } -compatible ◮ We reject ♦ ( φ ) ≡ ¬ � ( ¬ φ ) : ‘not necessarily false’ allows inconsistent axioms, ‘possibly true’ does not. ◮ We reject � ( φ ) ≡ φ : � ( φ ) depends on axioms in Q but φ itself does not. Better: � ( φ | Q ) , ♦ ( φ | Q ) , etc.
� � Curry’s Dynamical System Saving Truth from Orthodoxy Joseph Norman x : If this sentence x is true then y is true. Introduction Equations 0. Definition: x :: x → y with x , y ∈ { true , false } Dynamical ◮ Boolean Poly ( x → y ) = 1 − x + xy with x , y ∈ { 0 , 1 } Systems 1. As recurrence: x t + 1 ⇐ 1 − x t + x t y t with x , y ∈ { 0 , 1 } Probability ◮ Dynamical system with state ( x , y ) , transition graph: Conclusion References � 1 , 1 � 0 , 1 Extras 0 , 0 1 , 0 ◮ There is one fixed point ( x , y ) = ( 1 , 1 ) . ◮ Because of the periodic cycle it seems paradoxical when y = 0 (i.e. the consequent in x → y is false). 2. As equation: x = 1 − x + xy with x , y ∈ { 0 , 1 } ◮ Solution sets S Q ( x ) = { 1 } and S Q ( y ) = { 1 } ◮ Interpretation #1 adds value: shows oscillating cycle
� � � Kripke’s Watergate Dynamical System Saving Truth from Orthodoxy Joseph Norman x : That sentence y is false (J: “Most Nixon assertions false”) y : That sentence x is true (N: “Everything Jones says true”) Introduction Equations Dynamical 0. Definition: x :: ¬ y , y :: x with x , y ∈ { true , false } Systems ◮ Boolean Poly ( ¬ y ) = 1 − y Probability Conclusion 1. As recurrences: x t + 1 ⇐ 1 − y t , y t + 1 ⇐ x t ; x , y ∈ { 0 , 1 } References ◮ Dynamical system with state ( x , y ) , transition graph: Extras 0 , 1 1 , 1 � 1 , 0 0 , 0 ◮ There are no fixed points. ◮ Periodic cycle: every state seems paradoxical . 2. As equations: x = 1 − y and y = x with x , y ∈ { 0 , 1 } ◮ Solution sets S Q ( x ) = {} and S Q ( y ) = {} ◮ Interpretation #1 adds value: pattern of infeasibility
� Gödel’s Dynamical System Saving Truth from Orthodoxy Joseph Norman x : This formula x is true if and only if it is not provable. Introduction Equations 0. Definition: x :: ¬ Provable ( x ) ; x ∈ { true , false } Dynamical ◮ Solution set { 1 } means ‘provable’; here no axioms. Systems ◮ Revised definition x :: � � S {} ( x ) � = { 1 } or x :: ¬ � ( x ) . Probability ◮ But S {} ( x ) is just { x } . Then ( { x } � = { 1 } ) is true when Conclusion x = 0 and false when x = 1: its value is 1 − x . References ◮ Re-revised definition x :: 1 − x (as the Liar x :: ¬ x ). Extras 1. As recurrence: x t + 1 ⇐ 1 − x t with x ∈ { 0 , 1 } ◮ Dynamical system with state x and transition graph: � 1 0 ◮ There are no fixed points. ◮ Periodic cycle: Gödel called paradox undecidable . 2. As equation: x = 1 − x with x ∈ { 0 , 1 } ◮ Solution set S Q ( x ) = {}
The Logic of Parametric Probability Saving Truth from Orthodoxy ◮ My parametric probability analysis method Joseph Norman solves many problems in logic, including: Introduction ◮ Counterfactual conditionals Equations ◮ Probabilities of formulas in the propositional calculus Dynamical Systems ◮ Aristotle’s syllogisms Probability ◮ Smullyan’s puzzles with liars and truth-tellers Conclusion ◮ Solutions to probability queries are polynomials in References the parameters θ 1 , θ 2 , . . . used to specify probabilities Extras (distinct from the primary variables x 1 , . . . , x n whose probabilities are specified). ◮ These θ -polynomials can be used for secondary analysis such as linear and nonlinear optimization, search, and general algebra. ◮ Details are on arXiv.org. With some probability 0 � θ � 1, I will present at LC2012 in Manchester.
Recommend
More recommend
