SYMBOLIC LOGIC UNIT 10: SINGULAR SENTENCES
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1 SYMBOLIC LOGIC UNIT 10: SINGULAR SENTENCES
2 Singular Sentences name Paris is beautiful (monadic) predicate (monadic) predicate letter Bp individual constant
3 Singular Sentences Bp These are our new simple formulae, just like the sentence letters from Units 1 9. (Bp Fa) Gb In just the same way as sentence letters, they can be combined to form complex formulas. Note that only proper names can be symbolized as lower-case constants.
4 Relational Sentences (We ll return to this later.) (dyadic) predicate Paris is more beautiful than London names (dyadic) predicate letter Bpl constants
5 Propositional Functions individual variable x is beautiful (monadic) predicate (monadic) predicate letter Bx individual variable
6 Propositional Functions These are not sentences. They aren t true x is beautiful Bx or false. Instead, they are more like properties that can be true or false of some particular thing. Propositional functions can likewise be x is old and beautiful Ox Bx combined with sentential operators. Think of the result as complex properties. In these, the individual variables don t stand for anything in particular. They are like placeholders.
7 SYMBOLIC LOGIC UNIT 11: QUANTIFIERS
8 QUANTIFIERS Quantifiers can be put to the left of a propositional function in order to say how many things the function is true of. We ll have only two: universal (x) and existential ( x) (x)bx For every x, x is B. For all x s, x is B. Everything is a B. Everything is B. ( x)bx There is at least one x such that x is B. There exists an x such that x is B. There is at least one B. Something is B.
9 QUANTIFIER SCOPE The scope of a quantifier is the first full formula (sentence or propositional function) to its right. (Pay attention to parentheses!) ( x)bx Gx ( x)(bx Gx)
10 BOUND vs. FREE VARIABLES A variable is bound if it is within the scope of a quantifier with the same variable. This means that the variable s meaning refers back to the quantifier. A variable is free if it is not within the scope of a matching quantifier. Any formula with one or more free variables is a propositional function. ( x)bx Gx ( x)(bx Gx) Fx ( y)(bx Gy)
11 NEGATION AND QUANTIFIERS ~(x)bx It is not the case that for every x, x is B It is not the case that everything is B. (x)~bx Every x is such that x is not B. Nothing is B. Not every x is B. (At least one thing is not B.) ~( x)bx It is not the case that there is at least one x such that x is B. It is not the case that something is B. Nothing is B. ( x)~bx There is at least one x such that x is not B. At least one thing is not B. Something(s) is(/are) not B.
12 NEGATION AND QUANTIFIERS ~(x)bx ( x)~bx Not everything is B Something isn t B. ~( x)bx (x)~bx ~( x)~bx (x)bx ( x)bx ~(x)~bx Nothing is B. Everything is a non-b. Everything is B. (It isn t the case that there exists a non-b) There is at least one B. (Something is B.) It isn t the case that everything is a non-b.
13 SYMBOLIC LOGIC UNIT 12: CATEGORICAL PROPISITIONS
14 THE 4 CATEGORICAL SENTENCE FORMS A. All S are P (x)(sx Px) {x:sx} {x:px} (Every S is a P) I. Some S is P ( x)(sx Px) {x:sx} {x:px} x (At least one S is P) E. No S are P. (x)(sx ~Px) {x:sx} {x:px}= (Every S is a non-p) O. Some S are not P. (Not all S are P.) ( x)(sx ~Px) {x:sx} {x:px} x
15 QUANTIFIER+NEGATION RULES
16 SYMBOLIC LOGIC UNIT 13: COMPLEX SUBJECTS AND PREDICATES
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25 SYMBOLIC LOGIC UNIT 14: QUANTIFIER FORM AND TRUTH- FUNCTIONAL COMPOUNDS OF QUANTIFIER STATEMENTS
26 QUANTIFIER SCOPE The scope of a quantifier is the first full formula (sentence or propositional function) to its right. (Pay attention to parentheses!) ( x)bx Gx ( x)(bx Gx)
27 QUANTIFIER STATEMENTS A quantifier statement is one that (i) begins with a quantifier (ii) whose scope extends to the end of the formula. A conjunction ( x)bx Gx A negation ~( x)(bx Gx) ( x)(bx Gx) A quantifier statement
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30 SUBTLETIES These do NOT mean the same: ( x)bx ( x)gx (x)bx (x)gx ( x)(bx Gx) (x)(bx Gx) But these pairs DO mean the same: (x)bx (x)gx ( x)bx ( x)gx (x)(bx Gx) ( x)(bx Gx)
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39 SYMBOLIC LOGIC UNIT 15: PROOFS IN PREDICATE LOGIC
40 INSTANCES OF QUANTIFIED FORMULAS An instance of a quantified formula is the result of deleting the initial quantifier and replacing every "matching" variable in the propositional function uniformly with some individual constant.
41 INSTANCES OF QUANTIFIED FORMULAS Where φx is any propositional function, simple or complex, containing a free variable x, φa is the result of replacing every x in φx with an a.. φx Fx Gx φx (y)(fx Gy) φa Fa Ga φa (y)(fa Gy)
42 UNIVERSAL INSTANTIATION (UI) (x)φx / φa From a universal formula, we may infer any instance. *Note: this rule is unrestricted; the individual constant a can be anything you like, no matter what else has happened so far in the proof (even if a has been flagged).
43 EXISTENTIAL GENERALIZATION (EG) φa / ( x)φx Given any instance, we may infer the corresponding existential formula. *Note: this rule is unrestricted; the individual constant a can be anything you like, no matter what else has happened so far in the proof (even if a has been flagged).
44 EXISTENTIAL INSTANTIATION (EI) ( x)φx / φa (flag a) From a existential formula, we may infer an instance, provided we flag the instance letter. To flag a letter a for EI, write (flag a) after the justification.
45 UNIVERSAL GENERALIZATION (UG) flag a. φa / (x)φx If φa is the last step in an a-flagged subproof, then we may infer the universal proposition (x)φx. An a-flagged subproof begins with a line that says only flag a.
46 RESTRICTIONS ON FLAGGED LETTERS R1 A letter being flagged must be new to the proof; that is, it may not appear, either in a formula or as a letter being flagged, previous to the step in which it gets flagged. R2 A flagged letter may not appear either in the premises or in the conclusion of a proof. R3 A flagged letter may not appear outside the subproof in which it gets flagged.
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52 1. (x)(cx Dx) Pr. 2. (x)(ex ~Dx) Pr. / (x)(ex ~Cx) 3. flag a FS (UG) 4. Ca Da 1, UI 5. Ea ~Da 2, UI 6. ~Da ~Ca 4, Contra. 7. Ea ~Ca 5,6, HS 8. (x)(ex ~Cx) UG (3 7)
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54 SYMBOLIC LOGIC UNIT 17: SYMBOLIZATION IN RELATIONAL PREDICATE LOGIC
55 Singular Sentences Bp These are our new simple formulae, just like the sentence letters from Units 1 9. (Bp Fa) Gb In just the same way as sentence letters, they can be combined to form complex formulas. Note that only proper names can be symbolized as lower-case constants.
56 Relational Sentences (dyadic) predicate Paris is more beautiful than London names (dyadic) predicate letter Bpl constants
57 Multiple Quantifiers, One Formula ( x)(y)lxy (y)( x)lxy
58 Multiple Quantifiers, One Formula ( x)(y)lxy There is at least one person who loves everyone. (y)( x)lxy Everyone is loved by at least one person. *Note that we are restricting the domain to people.
59 Multiple Quantifiers, One Formula These don t have { the same meaning! ( x)(y)lxy There is at least one person who loves everyone. In general: quantifier order matters when the (y)( x)lxy Everyone is loved by at least one person. two quantifiers aren t the same. *Note that we are restricting the domain to people.
60 Scope Ambiguity I once visited an animal shelter and saw the saddest thing A dog was biting every cat.
61 Scope Ambiguity I once visited an animal shelter and saw the saddest thing A dog was biting every cat. ( x)(dx (y)(cy Bxy)) (y)(cy ( x)(dx Bxy))
62 Scope Ambiguity I once visited an animal shelter and saw the saddest thing A dog was biting every cat. ( x)(dx (y)(cy Bxy)) There is at least one dog who was biting every cat. (y)(cy ( x)(dx Bxy)) For every cat, there was at least one dog biting it. (Not necessarily the same dog.)
Formal Logic: Quantifiers, Predicates, and Validity. CS 130 Discrete Structures
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