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The Logic of Quantifiers

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The Logic of Quantifiers Introduction FO Validity FO Consequence Introduction FO Validity FO Consequence Announcements 03.31 The Logic of Quantifiers Logical Truth & Consequence in Full Fol 1 The Midterm has been returned If you haven't gotten yours back, see me after class William Starr 03.31.09 William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 1/34 William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 2/34 Introduction FO Validity FO Consequence Introduction FO Validity FO Consequence Outline Overview The Big Picture Now that we've added 8 and 9, we have introduced every connective of fol: 1 Introduction 8 9 $ ! _ ^ : = For six of these symbols we've studied: 2 FO Validity 1 It's semantics: truth-tables, satisfaction, game rules 2 How to translate English sentences using it 3 FO Consequence 3 It's role in logic: which sentences containing it are logical truths and which arguments containing it are valid 4 It's role in proofs: which inference steps and methods of proof it supports and how these can be formalized For 8 and 9, we've only done the first two Today, we'll get started on the third! William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 3/34 William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 5/34 Introduction FO Validity FO Consequence Introduction FO Validity FO Consequence Overview The Logical Concepts Today Logical Truth & Logical Consequence Logical Truth A is a logical truth iff it is impossible for A to be false given the meaning of the logical vocabulary it contains So today we'll be interested in two questions: Which sentences containing quantifiers are logical Logical Consequence truths? C is a logical consequence of P1;:::; Pn iff it is impossible Which arguments containing quantifiers are valid? for P1;:::; Pn to be true while C is false We'll start by reviewing our past discussion of logical truths and logical consequence Both of these concepts are at the very heart of logic But, they are annoyingly vague and imprecise What exactly is meant by impossible? In the first half of the class we explored one method for making logical possibility precise: truth tables William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 6/34 William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 8/34 Introduction FO Validity FO Consequence Introduction FO Validity FO Consequence Truth Tables Truth Tables Their Spoils Their Drawbacks Truth tables allowed us to define the following These definitions are a step towards better understanding logical truth and consequence: concepts: Every tautology is an (intuitive) logical truth Tautology Every tautological consequence is an (intuitive) A is a tautology iff every row the truth logical consequence table assigns t to A But the step is not complete: Tautological Consequence Some logical truths are not tautologies C is a tautological consequence of Some logical consequences are not tautological P1;:::; Pn iff every row of their joint truth consequences table which assigns t to P1;:::; Pn also The difficulty was that the notion of logical possibility assigns t to C used in truth tables was not discerning enough William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 9/34 William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 10/34 Introduction FO Validity FO Consequence Introduction FO Validity FO Consequence Truth Tables Truth Tables Not Discerning Enough Not Discerning Enough Recall the procedure for building a truth-table: 1 Build ref. col's Truth Table 2 Fill ref. col's 1 Build ref. col's Truth Table a = a b = b a = a ^ b = b a = a b = b a = a ^ b = b 3 Fill col's under t t t 2 Fill ref. col's t t t connectives t f f 3 Fill col's under t f f f t f connectives f t f f f f f f f The problem is caused by the fact that in building This table shows that a = a ^ b = b is not a tautology: truth tables, possibilities are included which are not there are some f's in the main column genuine logical possibilities But it is, intuitively, a logical truth It is not logically possible for a = a or b = b to be f! William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 11/34 William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 12/34 Introduction FO Validity FO Consequence Introduction FO Validity FO Consequence Discussion Tying In Quantification Truth Tables & Logical Possibility We Need That Better Analysis Even More The same deficiency causes there to be logical consequences which are not tautological consequences Example: a = c is a logical but not a tautological In case you weren't already convinced that truth tables consequence of a = b ^ b = c left something to be desired, think about how few of the quantificational logical truths are tautologies Why not just leave rows out if they aren't genuine logical possibilities? 8x (Cube(x) ! Cube(x)) (Not a Tautology) 8x (Cube(x) _:Cube(x)) (Not a Tautology) This robs truth tables of their purpose: 9x (x = x) (Not a Tautology) They were supposed to be a precise way of analyzing Although some logical truths with quantifiers are logical possibility tautologies: If we can just appeal to our intuitions about logical possibility in building the columns, our analysis gets 8x Cube(x) _ :8x Cube(x) (Tautology) us nowhere :(9x Cube(x) ^ :9x Cube(x)) (Tautology) So, we want to develop a better analysis of logical possibility William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 13/34 William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 14/34 Introduction FO Validity FO Consequence Introduction FO Validity FO Consequence FO Validity FO Validity A Small Step An Idea Logical Truth We need to be more clear about the notion of logical A is a logical truth iff it is impossible for A to be false possibility used to define FO validity given the meaning of the logical vocabulary it contains Here's the insight we'll build on We are only interested in 8; 9; $; !; _; ^; : and =, so The FO validities are sentences which are true purely we are interested in a more limited concept in virtue of the meaning of 8; 9; $; !; _; ^; : and = If their truth derives solely from the logical symbols, First-Order Validity (FO Validity) then you should be able vary the meaning of any of its A sentence A is a first-order validity just in case it is predicates (other than =) and names and still get a impossible for A to be false, given the meanings of true sentence 8; 9; $; !; _; ^; : and = Any variation of the meaning of the non-logical symbols is a logical possibility Better named First-Order Logical Truth William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 16/34 William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 17/34 Introduction FO Validity FO Consequence Introduction FO Validity FO Consequence An Example Another Example Use a Non-Sense Predicate Use a Non-Sense Predicate (1) 8x (Cube(x) ! Cube(x)) It sounds true even with a non-sense predicate: (4) 8x Rich(x) ! Rich(mc:hammer) (2) 8x (Blornk(x) ! Blornk(x)) This sounds true even with non-sense predicates and (3) All blornks are blornks names: There's no interpretation of `Blornk' according to which (5) 8x Rorg(x) ! Rorg(dude) (2) isn't true (6) If everything is a rorg, then dude is a rorg So (1) remains true no matter how we interpret its So (4) must be a FO validity non-logical symbols So (1) must be a FO validity William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 18/34 William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 19/34 Introduction FO Validity FO Consequence Introduction FO Validity FO Consequence Yet Another Example Counterexamples Use a Non-Sense Predicate What They Are (7) :9x LeftOf(x; x) (7) :9x LeftOf(x; x) We saw that, intuitively, (7) is not a logical truth Replace meaningful predicate with meaningless one: But we want to have a more precise way of showing this (8) :9x Glirs(x; x) Here's our new method: Is this obviously true? 1 Replace predicates and names with non-sense names No, it would depend on whether or not something can when checking for FO validity glir itself 2 Then consider whether or not there is any This is a not a fact about the meaning of logical reinterpretations of the formula that falsify it 3 If there are, specify such an interpretation symbols, so this is not a FO validity This specification is called a counterexample 4 If there is no such specification, then the formula is a logical truth William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 20/34 William Starr | The Logic of Quantifiers (Phil 201.02) | Rutgers University 21/34 Introduction FO Validity FO Consequence Introduction FO Validity FO Consequence Counterexamples FO Validity How to Formulate Them The Replacement Method for FO Validities (7) :9x LeftOf(x; x) The Replacement Method (FO Validities) The following method can be used to check whether or not Let's provide a counterexample to this S is a FO Validity 1 Systematically replace all of the predicates, other than 1 Replace predicates & names w/non-sense ones: =, and names with new, meaningless predicates and (8) :9x Glirs(x; x) names 2 Try to reinterpret the non-sense and find a 2 Try to describe a circumstance, along with circumstance under which the reinterpreted formula is interpretations for the names and predicates, in which false: S is false.
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