DERIVATIONS IN SYMBOLIC LOGIC I

DERIVATIONS IN SYMBOLIC LOGIC I Tomoya Sato Department of Philosophy University of California, San Diego Phil120: Symbolic Logic Summer 2014 TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 1 / 40

WHAT IS LOGIC? LOGIC Logic is the study of formal validity. TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 2 / 40

CHAPTER 3, SECTION 6 The semantic method The proof-theoretic method Symbolization Formal Validity TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 3 / 40

SCOPE OF QUANTIFIER TWO TYPES OF OCCURRENCES OF VARIABLES Bound Variables. Free Variables. x(fx y(gy Hz)) ( Hy (Fx x( Fx Gx))) SCORE OF QUANTIFIER x ( ) y ( ) DEFINITION: SCORE OF QUANTIFIER The scope of an occurrence of a quantifier includes itself and its variable along with the formula to which it was prefixed when constructing the whole formula. TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 4 / 40

SCOPE OF QUANTIFIER EXAMPLE xfx x(fx Gx) xfx y(gy Hy) x(fx ygy) x(fx y( zgz Hy)) TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 5 / 40

BOUND VARIABLES AND FREE VARIABLES DEFINITION: BOUND VARIABLES A variable α in a formula ϕ is bound if 1 the variable is within the scope of a quantifier; 2 the variable is the same as the one that accompanies the quantifier; 3 the variable is not already bound by another quantifier occurrence within the scope of the first quantifier. x(fx Gx) xfx y(gy Hy) TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 6 / 40

BOUND VARIABLES AND FREE VARIABLES x(fx ygy) x[(fx y( zgz Hy)) Iw] x(fx y(gy Hz)) ( Hy (Fx x( Fx Gx))) DEFINITION: FREE VARIABLES A variable α in a formula ϕ is free quantifier. def it is not bound by any DEFINITION: SENTENCES A formula is a sentence def it contains only bound variables. TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 7 / 40

EXERCISES EXERCISES 1 x(fx Gx) 2 x(fx Gy) 3 x y(fx Fy) 4 x y[ z(fz Gy) (Gx ((Fx whw) Hx))] TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 8 / 40

INFERENCE RULES Universal Instantiation (ui) xϕ(x) ϕ(a) IDEA Everything is interesting. = Philosophy is interesting. = Logic is interesting. TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 9 / 40

EXAMPLE 1. Fa 2. x(fx Gx) Ga 1. Show Ga 2. Fa Ga pr2 ui 3. Ga 2 pr1 mp dd TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 10 / 40

PROBLEM 1. Socrates is a human being. 2. All human beings are mortal. Socrates is mortal. 1. P 2. Q R Invalid!! TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 11 / 40

EXAMPLE 1. Socrates is a human being. 2. All human beings are mortal. Socrates is mortal. SCHEME OF ABBREVIATION a : Socrates. Fx : x is a human being. Gx : x is mortal. 1. Fa 2. x(fx Gx) Ga TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 12 / 40

INFERENCE RULES UNIVERSAL INSTANTIATION (UI) 1 Start with a universally quantified formula; 2 Remove the quantifier phrase; 3 Replace all and only the variable occurrences bound to the quantifier phrase with occurrences of the same letter or variable. 4 Restriction: The occurrence of the variable of instantiation must be free in the symbolic formula generated by UI. TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 13 / 40

INFERENCE RULES GOOD UI x(fx Gx) Fa Ga GOOD UI x(fx Gx) Fy Gy GOOD UI x(fx Gx) Fx Gx TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 14 / 40

INFERENCE RULES BAD UI x(fx Gx) Fa Gb BAD UI x(fx Gx) Fx Gy BAD UI xfx Ga Fb Ga TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 15 / 40

INFERENCE RULES Restriction: The occurrence of the variable of instantiation must be free in the symbolic formula generated by UI. x y(fx Gy) y(fy Gy) You cannot do this! TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 16 / 40

EXERCISES 1. x(fx Gx) 2. y(hy Gy) Fa Ha 1. x Fx 2. y( Fy Gy) Fa Ga TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 17 / 40

EXERCISES 1. x y(fx Gy) 2. y(gy Hy) Fa Ha 1. x(fx Gx) Ga Fa TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 18 / 40

INFERENCE RULES Existential Generalization (eg) ϕ(a) ϕ(x) IDEA Hanzo Hattori is a Ninja. There is a Ninja. For some x, x is a Ninja. TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 19 / 40

EXAMPLE 1. x(fx Gx) 2. Fa ygy 1. Show ygy 2. Fa Ga pr1 ui 3. Ga 2 pr2 mp 4. ygy 3 eg dd TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 20 / 40

INFERENCE RULES EXISTENTIAL GENERALIZATION (EG) 1 Start with any symbolic formula; 2 (Choice) Pick free occurrences of a term of instantiation; 3 (Choice) Pick the variable of generalization α (any variable is fine); 4 Add α to the left of the symbolic formula, and replace all picked occurrences of the term of instantiation with occurrences of α; 5 Restriction: Don t bind a free occurrence of a term other than the term of instantiation. TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 21 / 40

INFERENCE RULES GOOD EG Fa Ga x(fx Gx) GOOD EG Fx Gx y(fy Gy) GOOD EG Fa Ga y(fy Ga) TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 22 / 40

INFERENCE RULES GOOD EG Fx Ga y(fy Ga) GOOD EG Fx Ga y(fx Gy) GOOD EG x(fx Ga) z x(fx Gz) TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 23 / 40

INFERENCE RULES BAD EG Fa Gb x(fx Gx) BAD EG Fa Gy x(fx Gx) BAD EG x(fx Ga) x z(fx Gz) TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 24 / 40

INFERENCE RULES Restriction: Don t bind a free occurrence of a term other than the term of instantiation. Fx Ga x(fx Gx) You cannot do this! TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 25 / 40

EXERCISES 1. xfx xfx 1. x[( y Fy) Gx] 2. Fa Gb TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 26 / 40

INFERENCE RULES Existential Instantiation (ei) ϕ(x) ϕ(y) IDEA There is a Ninja. Nick is a Ninja. ( Nick" is a name that is temporarily assigned to that Ninja) TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 27 / 40

EXAMPLE 1. x(fx Gx) 2. yfy zgz 1. Show zgz 2. Fw pr2 ei 3. Fw Gw pr1 ui 4. Gw 2 3 mp 5. zgz 4 eg dd TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 28 / 40

INFERENCE RULES EXISTENTIAL INSTANTIATION (EI) 1 Start with an existentially quantified formula; 2 Remove the quantifier phrase; 3 Replace all and only the variable occurrences bound to the quantifier phrase with occurrences of a new variable. TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 29 / 40

INFERENCE RULES GOOD EI xfx Fw GOOD EI x(fx Ga) Fz Ga GOOD EI x y(fx Gy) x(fz Gy) TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 30 / 40

INFERENCE RULES BAD EI x(fx Qb) Fa Qb BAD EI x(fx Qy) Fy Qy TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 31 / 40

INFERENCE RULES BAD EI x y(fx Gy) x(fx Gw) BAD EI x(fx Qx) Fy Qz TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 32 / 40

EXERCISES 1. x Fx yfy 1. x Fx yfy TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 33 / 40

EXERCISES 1. x y(fx Gy) z w(fw Gz) 1. x[(bx Dx) Ex] 2. x(dx Fx) 3. x(fx Bx) y(dy Ey) TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 34 / 40

CHAPTER 3, SECTION 7 NOTE Universal derivation (ud) is not allowed in this course. TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 35 / 40

CHAPTER 3, SECTION 9 Quantifier Negation (qn) IDEA It is not case that, for all x, x is not meaningful. There exists x such that x is meaningful. x Mx. xmx. IDEA It is not case that there exists x such that x is not meaningful. For all x, x is meanigful. x Mx. xmx. TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 36 / 40

INFERENCE RULES IDEA x x. x x. αϕ α ϕ αϕ α ϕ α ϕ αϕ α ϕ αϕ α ϕ αϕ α ϕ αϕ αϕ α ϕ αϕ α ϕ TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 37 / 40

EXERCISES 1. x(fx Gx) 2. y( Fy Gy) zgz 1. x( Fx Ga) xfx Ga TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 38 / 40

ALPHABETIC VARIANCE (AV) NOTE "Alphabetic variance" is not allowed in this course. TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 39 / 40

NOTE NOTE In this course, it is not allowed to apply inference rules to a part of quantified formulas. BAD DM x(fx Qx) x (Fx Gx) x( Fx Gx) BAD CDJ x(fx Qx) x (Fx Gx) x ( Fx Gx) TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 40 / 40