2.4 Notational Definition

Transcription

1 24 Notational Definition Notational Definition The jdgments, propositions, and inference rles we have defined so far collectively form a system of natral dedction It is a minor variant of a system introdced by Gentzen [Gen35] One of his main motivations was to devise rles that model mathematical reasoning as directly as possible, althogh clearly in mch more detail than in a typical mathematical argment We now consider how to define negation So far, the meaning of any logical connective has been defined by its introdction rles, from which we derived its elimination rles The definitions for all the connectives are orthogonal: the rles for any of the connectives do not depend on any other connectives, only on basic jdgmental concepts Hence the meaning of a compond proposition depends only on the meaning of its constitent propositions From the point of view of nderstanding logical connectives this is a critical property: to nderstand disjnction, for example, we only need to nderstand its introdction rles and not any other connectives A freqently proposed introdction rle for not A (written A) is A tre tre I? A tre In words: A is tre if the assmption that A is tre leads to a contradiction However, this is not a satisfactory introdction rle, since the premise relies the meaning of, violating orthogonality among the connectives There are several approaches to removing this dependency One is to introdce a new jdgment, A is false, and reason explicitly abot trth and falsehood Another employs schematic jdgments, which we consider when we introdce niversal and existential qantification Here we prse a third alternative: for arbitrary propositions A, we think of A as a syntactic abbreviation for A This is called a notational definition and we write A = A This notational definition is schematic in the proposition A Implicit here is the formation rle A prop F A prop We allow silent expansion of notational definitions As an example, we prove

2 16 Propositional Logic that A and A cannot be tre simltaneosly A A tre A A tre ER EL A tre A tre tre I (A A) tre We can only nderstand this derivation if we keep in mind that A stands for A,andthat (A A) stands for (A A) As a second example, we show the proof that A A is tre w A tre A tre tre I w A tre I A A tre Next we consider A A, the so-called law of exclded middle It claims that every proposition is either tre or false This, however, contradicts or definition of disjnction: we may have evidence neither for the trth of A, nor for the falsehood of A Therefore we cannot expect A A to be tre nless we have more information abot A One has to be carefl how to interpret this statement, however There are many propositions A for which it is indeed the case that we know A A For example, ( ) is clearly tre becase tre Similarly, ( ) istre becase is tre To make this flly explicit: I tre I L ( ) tre tre I tre I R ( ) tre In mathematics and compter science, many basic relations satisfy the law of exclded middle For example, we will be able to show that for any two nmbers k and n, either k < n or (k < n) However, this reqires proof, becase for more complex A propositions we may not know if A tre or A tre We will retrn to this isse later in this corse At present we do not have the tools to show formally that A A shold not be tre for arbitrary A A proof attempt with or generic proof strategy (reason from the bottom p with introdction rles and from the top down with elimination rles) fails qickly, no matter which introdction rle for disjnction

3 25 Derived Rles of Inference 17 we start with A tre I L A A tre A tre tre I A tre I R A A tre We will see that this failre is in fact sfficient evidence to know that A A is not tre for arbitrary A 25 Derived Rles of Inference One poplar device for shortening derivations is to introdce derived rles of inference For example, A B tre A C tre B C tre is a derived rle of inference Its derivation is the following: A tre B tre A B tre C tre I A C tre B C tre Note that this is simply a hypothetical derivation, sing the premises of the derived rle as assmptions In other words, a derived rle of inference is nothing bt an evident hypothetical jdgment; its jstification is a hypothetical derivation We can freely se derived rles in proofs, since any occrrence of sch a rle can be expanded by replacing it with its jstification A second example of notational definition is logical eqivalence A if and only if B (written A B) We define (A B) =(A B) (B A) That is, two propositions A and B are logically eqivalent if A implies B and B implies A Under this definition, the following become derived rles of inference (see Exercise 21) They can also be seen as introdction and elimination rles

4 18 Propositional Logic for logical eqivalence (whence their names) w A tre B tre B tre A tre I,w A B tre A B tre B tre A tre EL A B tre A tre B tre ER 26 Logical Eqivalences We now consider several classes of logical eqivalences in order to develop some intitions regarding the trth of propositions Each eqivalence has the form A B, bt we consider only the basic connectives and constants (,,,, ) ina and B Later on we consider negation as a special case We se some standard conventions that allow s to omit some parentheses while writing propositions We se the following operator precedences > > > > where,, and are right associative For example A A A stands for ( A) ((A ( ( A))) ) In ordinary mathematical sage, A B C stands for (A B) (B C); in the formal langage we do not allow iterated eqivalences withot explicit parentheses in order to avoid confsion with propositions sch as (A A) Commtativity implication is not (C1) A B B A tre (C2) A B B A tre (C3) A B is not commtative Conjnction and disjnction are clearly commtative, while Idempotence Conjnction and disjnction are idempotent, while self-implication redces to trth (I1) A A A tre (I2) A A A tre (I3) A A tre

5 27 Smmary 19 Interaction Laws These involve two interacting connectives In principle, there are left and right interaction laws, bt becase conjnction and disjnction are commtative, some coincide and are not repeated here (L1) A (B C) (A B) C tre (L2) A A tre (L3) A (B C) donotinteract (L4) A (B C) (A B) (A C) tre (L5) A tre (L6) A (B C) (A B) (A C) tre (L7) A tre (L8) A (B C) donotinteract (L9) A (B C) (A B) C tre (L10) A A tre (L11) A (B C) (A B) (A B) tre (L12) A tre (L13) A (B C) (A B) C tre (L14) A (B C) donotinteract (L15) A do not interact (L16) (A B) C A (B C) tre (L17) C C tre (L18) (A B) C do not interact (L19) (A B) C (A C) (B C) tre (L20) C tre 27 Smmary Jdgments A prop A tre A is a proposition Proposition A is tre Propositional Constants and Connectives The following table smmarizes the introdction and elimination rles for the propositional constants (, ) and connectives (,, ) We omit the straightforward formation rles

6 20 Propositional Logic Introdction Rles Elimination Rles A tre B tre I A B tre A B tre EL A tre A B tre ER B tre A tre I L A B tre I tre A tre B tre I A B tre no I rle B tre I R A B tre A B tre A B tre no E rle B tre A tre C tre C tre tre E C tre A tre w B tre C tre E,w Notational Definitions We se the following notational definitions A = A not A A B = (A B) (B A) A if and only if B 28 Exercises Exercise 21 Show the derivations for the rles I, E L and E R nder the definition of A B as (A B) (B A)

7 Bibliography [Gen35] Gerhard Gentzen Unterschngen über das logische Schließen Mathematische Zeitschrift, 39: , , 1935 English translation in M E Szabo, editor, The Collected Papers of Gerhard Gentzen, pages , North-Holland, 1969 [ML96] Per Martin-Löf On the meanings of the logical constants and the jstifications of the logical laws Nordic Jornal of Philosophical Logic, 1(1):11 60, 1996