1、ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersOnline Course: Foundations of LogicDag Westerst ahlTsinghua LogicND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersThis is Chapter
2、 3.12This week we will be occupied with the syntactic aspects oflogical consequence.We will say that a conclusion follows from premises in = f 1;:; ng if there is a derivation of from 1;:; n, in some inference system S.In symbols: 1;:; n S What a derivation looks like depends on the system S.We will
3、 take a quick look at two such systems: NaturalDeduction (ND) and (next time) a Hilbert type system (H).1 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersNatural Deduction: the ideaA derivation in ND starts from assumptions.Some assu
4、mptions are temporary, and get discharged duringthe derivation; others (the premises) remain when thederivation is nished.The inference rules come in pairs: an introduction rule and anelimination rule for each logical constant.The derivation has a tree form, with the assumptions asleaves and the con
5、clusion as the root.2 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersRules for We write the rules for coinjunction rules like this:(I) (E) The correctness of these rules is obvious.3 of 33ND: the idea ND: rules for connectives ND: n
6、egation ND: rules for quanti ers ND: rules for identity ParametersHow to read the rulesConsider again the rule I.(I). Suppose we have a derivation ending with , and a derivationending with .The rules says we can join these two derivations, adding as a new root, to form a new derivation.The new deriv
7、ation depends on the assumptions that the twogiven derivations depend on.4 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersAn assumption ruleInstead of derivation, we often say proof tree.All derivations in ND start from assumptions.
8、So we have an Assumption Rule:(A) Any sentence may be introduced as an assumption. That is,the tree consisting of a single sentence occurrence is a prooftree (with as both assumption/premise and conclusion).The sentences in branches with an assumption at the topdepend on | unless has been discharged
9、.5 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersRules for !: (!I)! ! (!E) !I says: assume and derive (possibly with otherassumptions as well). Then we can write ! on the nextline, at the same time discharging the assumption .We ma
10、rk discharged assumptions with . Sentences below nolonger depend on them.!E is the familar rule Modus Ponens.6 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersInference dynamics7 of 33ND: the idea ND: rules for connectives ND: negati
11、on ND: rules for quanti ers ND: rules for identity ParametersDerivations in NDDe nition 1;:; n ND if there is a proof tree (a derivation) built with the ND rules,ending with , and whose undischarged assumptions are among 1;:; n.8 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for
12、 quanti ers ND: rules for identity ParametersDerivations in ND: rst examplesWe show thatND ! ( ! ( )that is, that ! ( ! ( ) is derivable in ND withoutassumptions:2 1 I !I1 ! ( )!I2! ( ! ( )Note how the numbering of rules and discharged assumptionshelps us keep track!9 of 33ND: the idea ND: rules for
13、 connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersDerivations in ND: rst examplesNext, show thatND ! ( !)The rst application of !I looks odd: was not needed toderive , so there is nothing to discharge, but we can stillconclude !.But this is in accordance with the ru
14、le !I, and it is correct: if is true, then ! must be true.10 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersRules for _The introduction rules are obvious, but how to eliminate _?We use reasoning by cases: if derives , and also deriv
15、es , then _ derives .(_I)_ _ _ : : (_E) 11 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersThe rules in actionShow; _ ND ( ) _ ( )12 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for iden
16、tity ParametersRules for negationIn ND is it practical to assume the PL language has a falsumsymbol ? , standing for a contradiction.Then we can de ne : in terms of ?: =def !?Now the following rules become special cases of !I and !E:?(:I) : :(:E)?These rules are very useful, but not su cient for neg
17、ation.13 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersReductio Ad AbsurdumWhat more do we need?One possibility is to add a reductio rule: if you can derive acontradiction from :, then conclude :?(RAA) Note the di erence from :I.14
18、 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersDNE implies RAAIf we had a rule of Double Negation Elimination, then RAAfollows:(DNE)15 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for
19、identity ParametersRAA and DNE 2Conversely, DNE follows from RAA:This is a typical use of RAA: we assume : and derive from it, andpossibly other assumptions, a contradiction. Then we may conclude and discharge the assumption :, but of course the otherassumptions remain.16 of 33ND: the idea ND: rules
20、 for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersEx Falso Quodlibet 1Everything follows from a contradiction.This has a well-known medieval name, EFQ; and is also calledthe Law of Explosion:?(EFQ) It can be derived with RAA (or DNE).In fact, it is an instance o
21、f RAA!17 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersEx Falso Quodlibet 2EFQ allows us to prove many facts about negation, such asDisjunctive Syllogism:_ ;: ND But EFQ does not su ce to show, for example,: !:ND ! which requires R
22、AA (or DNE).18 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersLaw of Excluded Middle(LEM) ND _:The proof tree for this is a little less obvious:19 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND:
23、 rules for identity ParametersLEM + EFQ implies RAAThis is not so trivial either. We are assuming:?and must derive while discharging :. Since LEM is adisjunction we must somehow get the result with _E.20 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for
24、identity Parameters21 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersClassical and intuitionistic logicIntuitionistic Propositional Logic (IPL) is the rule system for ;_;!just presented, with : de ned as !?, plus the rule EFQ.Classi
25、cal Propositional Logic (CPL) is obtained by instead addingRAA (or DNE, or LEM).We will see that CPL exactly matches j= (in Chapter 4).Exactly matching IPL requires a di erent kind of semantics, onethat does not make e.g. p _:p valid!Very roughly: IPL formalizes warranted assertibility, not truth.Th
26、ere are interesting connections between derivability in IPL and inCPL; see Exercises 3.1.47 and 3.3.10. For example:If CPL , then IPL :.!22 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersRules for 8:(c)(8I)8x(x)8x(x)(8E)(t)(x) has a
27、t most x free.t can be any closed term (no variables).But c in 8I must be new: cannot occur in (x) or in anyundischarged assumptions in the derivation of (c).The idea: if we can derive (c) for an arbitrary c, then we haveshown that (x) holds for all x.(Remember: if j= (c) where c is new, then j= 8x(
28、x).)23 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersRules for 9(t)(9I)9x(x) 9x(x)(c): (9E) 9I is unproblematic.Again, in 9E, c must be new: cannot occur in (x), or in thesentence , or in any undischarged assumptions in the derivat
29、ion of , except in the assumption (c).The idea: We want to conclude some sentence from 9x(x). Sothere is some x such that (x); call it c! If we can derive from(c), we are done.(Remember: if ;(c) j= where c is new, then ;9x(x) j= .)24 of 33ND: the idea ND: rules for connectives ND: negation ND: rules
30、 for quanti ers ND: rules for identity ParametersExamplesShowND 8x:(Px :Px)Idea: Derive :(Pc :Pc) (with no assumptions), then use 8I.25 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersMore examplesShow:9xPx ND 8x:PxIdea: Derive :Pc f
31、or a new c. This should work if we assumePc, which yields 9xPx, which contradicts our assumption.Note that RAA is not used, not even EFQ.26 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersMore examplesShow:8xPx ND 9x:PxThis is harder
32、, and requires RAA.Idea: We must derive a contradiction from :9x:Px (and then useRAA). To do this, we derive a contradiction from :Pc, which yieldsPc, which gives 8xPx, and we have our contradiction.27 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for id
33、entity ParametersRules for =(=)1 t = t s = t (s)(=)2(t)t = s (s)(t)Here (t) results from (s) by replacing some occurrences ofthe closed term s with the closed term t.For example:ND 8x8y(x = y ! y = x)(For the derivation, see textbook p. 57!)28 of 33ND: the idea ND: rules for connectives ND: negation
34、 ND: rules for quanti ers ND: rules for identity ParametersOne more (easy) exampleShowPc;:Pd ND 9x9y x 6= y29 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersIndividual constants as parametersTo apply the quanti er rules, we need a s
35、upply (in fact, anin nite supply) of individual constants.These constants dont appear in the premises or theconclusions we want to obtain.But they are used during the derivations, as parameters.So in what follows well assume that such an in nite supply ofindividual constants is always available.(An
36、alternative would be to use free variables instead, andthus allow formulas with free variables in derivations.)30 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity ParametersNext Thursday:Chapter 3.331 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity Parameters32 of 33ND: the idea ND: rules for connectives ND: negation ND: rules for quanti ers ND: rules for identity Parameters33 of 33
