Class Note for PHIL 110 at UMass(1)
Class Note for PHIL 110 at UMass(1)
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Date Created: 02/06/15
154 Hardegree Symbolic Logic 6 THE OFFICIAL INFERENCE RULES So far we have discussed only four inference rules modus ponens modus tollens and the two forms of modus tollendo ponens In the present section we add quite a few more inference rules to our list Since the new rules will be given more pictorial nonLatin names we are going to rename our original four rules in order to maintain consistency Also we are going to consolidate our original four rules into two rules In constructing the full set of inference rules we would like to pursue the following overall plan For each of the ve connectives we want two rules on the one hand we want a rule for quotintroducingquot the connective on the other hand we want a rule for quoteliminatingquot the connective An introductionrule is also called an I39mrule an eliminationrule is called an outrule Also it would be nice if the name of each rule is suggestive of what the rule does In particular the name should consist of two parts 1 reference to the spe cific connective involved and 2 indication whether the rule is an introduction in rule or an elimination out rule Thus if we were to follow the overall plan we would have a total of ten rules listed as follows AmpersandIn ampl AmpersandOut ampO Wedge ln vl WedgeOut vO DoubleArrowln lt gtl DoubleArrowOut lt gtO gtl Arrowln gtl ArrowOut gtO Tildeln NI gt TildeOut NO However for reasons of simplicity of presentation the general plan is not followed completely In particular there are three points of difference which are marked by an asterisk What we adopt instead in the derivation system SL are the following inference rules Chapter 5 Derivations in Sentential Logic 155 INFERENCE RULES INITIAL SET AmpersandIn amp1 J4 J4 B B JZI amp B B amp JZI AmpersandOut amp0 J4 amp B JZI amp B JZI B WedgeIn v1 J4 J4 14 v B B v 14 WedgeOut v0 JZI v B JZI v B NJZI NB B JZI DoubleArrowIn lt gtI JZI gt B JZI gt B B gt A B gt J4 J4 lt gt B B lt gt JZI DoubleArrowOut lt gtO JZI lt gt B JZI lt gt B JZI gt B B gt JZI ArrowOut gtO JZI gt B JZI gt B JZI NB B NJZI Double Negation DN JZI NNJZI N NJZI JZI 156 Hardegree Symbolic Logic A few notes may help clarify the above inference rules Notes 1 2 3 4 5 6 7 Arrowout gtO the rule for decomposing conditional formulas re places both modus ponens and modus tollens Wedgeout v0 the rule for decomposing disjunctions replaces both forms of modus tollendo ponens Double negation DN stands in place of both the tildein and the tilde out rule There is no arrowin rule The rule for introducing arrow is not an in ference rule but rather a showrule which is a different kind of rule to be discussed later In each of the rules J4 and B are arbitrary formulas of sentential logic Each rule is short for infinitely many substitution instances In each of the rules the order of the premises is completely irrelevant In the wedgein vl rule the formula 3 is any formula whatsoever it does not even have to be anywhere near the derivation in question There is one point that is extremely important given as follows which will be repeated as the need arises Inference rules apply to whole lines not to pieces of lines In other words what are given above are not actually the inference rules themselves but only pictures suggestive of the rules The actual rules are more properly written as follows 0 INFERENCE RULES OFFICIAL FORMULATION AmpersandIn ampl If one has available lines A and B then one is entitled to write down their conjunction in one order 1484 or the other order BampJZL AmpersandOut ampO If one has available a line of the form 1484 then one is entitled to write down either conjunct A or conjunct B Chapter 5 Derivations in Sentential Logic 157 WedgeIn vl If one has available a line A then one is entitled to write down the disjunction of A with any formula B in one order JZlvB or the other order BvJZL WedgeOut v0 If one has available a line of the form JZlvB and if one additionally has available a line which is the negation of the first disjunct NJZl then one is entitled to write down the second disjunct B Likewise if one has available a line of the form JZlvB and if one additionally has available a line which is the negation of the second disjunct NB then one is enti tled to write down the first disjunct JZL DoubleArrowln lt gtl If one has available a line that is a conditional JZl gtB and one additionally has available a line that is the converse B gtJZl then one is entitled to write down either the biconditional JZllt gtB or the biconditional Blt gtJZL DoubleArrowOut lt gtO If one has available a line of the form JZllt gtB then one is entitled to write down both the conditional JZl gtB and its converse B gtJZL ArrowOut gtO If one has available a line of the form JZl gtB and if one additionally has available a line which is the antecedent JZl then one is entitled to write down the consequent B Likewise if one has available a line of the form JZl gtB and if one additionally has available a line which is the negation of the consequent NB then one is entitled to write down the negation of the antecedent NJZL Double Negation DN If one has available a line JZl then one is entitled to write down the doublenegation NNJZL Similarly if one has available a line of the form NNJZl then one is entitled to write down the formula JZL The word available is used in a technical sense that will be explained in a later section 158 Hardegree Symbolic Logic To this list we will add a few further inference rules in a later section They are not crucial to the derivation system they merely make doing derivations more convenient 7 SHOWLINES AND SHOWRULES DIRECT DERIVATION Having discussed simple derivations we now begin the official presentation of the derivation system SL ln constructing system SL we lay down a set of system rules the rules of SL It39s a bit confusing we have inference rules already presented now we have system rules as well System rules are simply the official rules for constructing derivations and include among other things all the inference rules For example we have already seen two system rules in effect They are the two principles of simple derivation which are now officially formulated as system rules System Rule 1 The Premise Rule At any point in a derivation prior to the first showline any premise may be written down The annotation is Pr System Rule 2 The InferenceRule Rule At any point in a derivation a formula may be written down if it follows from previous available lines by an inference rule The annotation cites the line numbers and the inference rule in that order System Rule 2 is actually shorthand for the list of all the inference rules as formulated at the end of Section 6 The next thing we do in elaborating system SL is to enhance the notion of simple derivation to obtain the notion of a direct derivation This enhancement is quite simple it even seems redundant at the moment But as we further elaborate system SL this enhancement will become increasingly crucial Specifically we add the following additional system rule which concerns a new kind of line called a showline which may be introduced at any point in a derivation System Rule 3 The ShowLine Rule At any point in a derivation one is entitled to write down the expression SHOW JZl for any formula JZl whatsoever Chapter 5 Derivations in Sentential Logic 159 In writing down the line SHOW JZl all one is saying is I will now attempt to show the formula J21 What the rule amounts to then is that at any point one is entitled to attempt to show anything one pleases This is very much like saying that any citizen over a certain age is entitled to run for president But rights are not guarantees you can try but you may not succeed Allowing showlines changes the derivation system quite a bit at least in the long run However at the current stage of development of system SL there is generally only one reasonable kind of showline Speci cally one writes down SHOW C where C is the conclusion of the argument one is trying to prove valid Later we will see other uses of showlines All derivations start pretty much the same way one writes down all the premises as permitted by System Rule 1 then one writes down SHOW C where C is the conclusion which is permitted by System Rule 3 Consider the following example which is the beginning of a derivation Example 1 1 P v Q gt NR Pr 2 P amp T Pr 3 RV s Pr 4 U gt s Pr 5 SHOW NU 7 These five lines may be regarded as simply stating the problem we want to show one formula given four others I write 777 in the annotation column because this still needs explaining more about this later Given the problem we can construct what is very similar to a simple deriva tion as follows 1 P v Q gt NR Pr 2 P amp T Pr 3 R v NS Pr 4 U gt S Pr 5 SHOW U 6 P 2ampO 7 P v Q 6vl 8 NR 17 gtO 9 NS 38vO 10 NU 49 gtO Notice that if we deleted the showline 5 the result is a simple derivation We are allowed to try to show anything But how do we know when we have succeeded In order to decide when a formula has in fact been shown we need additional system rules which we call quotshowrulesquot The first showrule is so simple it barely requires mentioning Nevertheless in order to make system SL completely clear and precise we must make this rule explicit 160 Hardegree Symbolic Logic The first showrule may be intuitively formulated as follows Direct Derivation Intuitive Formulation If one is trying to show formula J4 and one actually obtains A as a later line then one has succeeded The intuitive formulation is unfortunately not sufficiently precise for the purposes to which it will ultimately be put So we formulate the following official system rule of derivation System Rule 4 a showrule Direct Derivation DD If one has a showline SHOW 14 and one obtains A as a later available line and there are no intervening uncancelled showlines then one is entitled to box and cancel SHOW 14 The annotation is DD As it is officially written direct derivation is a very complicated rule Don39t worry about it now The subtleties of the rule don39t come into play until later For the moment however we do need to understand the idea of cancelling a showline and boxing off the associated subderivation Cancelling a showline simply amounts to striking through the word SHOW to obtain SHGW This indi cates that the formula has in fact been shown Now the formula JZl can be used The tradeoff is that one must box off the associated derivation No line inside a box can be further used One in effect trades the derivation for the formula shown More about this restriction later The intuitive content of direct derivation is pictorially presented as follows Direct Derivation DD SHGW le JZl The box is of little importance right now but later it becomes very important in helping organize very complex derivations ones that involve several showlines For the moment simply think of the box as a decoration a ourish if you like to celebrate having shown the formula Chapter 5 Derivations in Sentential Logic 161 Let us return to our original derivation problem Completing it according to the strict rules yields the following l P v Q gt NR Pr 2 P amp T Pr 3 R v NS Pr 4 U gt S Pr 5 SHGW NU DD 6 P 2ampO 7 P v Q 6vl 8 NR l7 gtO 9 NS 38VO 10 NU 49 gtO Note that SHOW has been struck through resulting in SHGW Note the annotation for line 5 DD indicates that the showline has been cancelled in accordance with the showrule Direct Derivation Finally note that every formula below the showline has been boxed off Later we will have other more complicated showrules For the moment however we just have direct derivation 8 EXAMPLES OF DIRECT DERIVATIONS In the present section we look at several examples of direct derivations Example 1 1 MP gt Q v R Pr 2 P gt Q Pr 3 Q Pr 4 SHGW R DD 5 NP 23 gtO 6 Q v R 15 gto 7 R 36vO Example 2 1 P amp Q Pr 2 SHGW P amp Q DD 3 P 1ampo 4 Q 1ampO 5 P 3DN 6 Q 4DN 7 P amp Q 56ampI 162 Hardegree Symbolic Logic Example 3 1 2 3 4 5 6 7 8 PampQ QvR gtS SHGW PampS P 2 QvR S PampS ExmnMe4 1 2 3 4 5 6 7 8 9 AampB AvE gtC D gtC SHGWD A AVE C NNC ND ExmnMeS 1 2 3 4 5 6 7 8 9 10 11 12 AampB vaam ampEHD WMAampC A NB A gtD D D gtCampE CampE C AampC Chapter 5 Derivations in Sentential Logic 163 Example 6 1 A gt B 2 A gtB gtB gtA 3 A lt gt B gt A 4 SHGW A amp B 5 B gt A 6 A lt gt B 7 A 8 B 9 A amp B Example 7 1 NA amp B 2 CvB gt ND gtA 3 ND lt gt E 4 SHGW NE 5 NA 6 B 7 C v B 8 MD gt A 9 D 10 E gt ND 11 NE Pr Pr DD 12 gtO 15lt gtI 36 gtO 17 gtO 78ampI Pr Pr Pr DD lampO lampO 6vl 27 gtO 58 gtO 3lt gtO 910 gtO NOTE From now on for the sake of typographical neatness we will draw boxes in a purely skeletal fashion In particular we will only draw the left side of each box the remaining sides of each box should be mentally lled in using skeletal boxes the last two derivations are written as follows Example 6 rewritten 1 2 3 4 5 6 7 8 9 A gtB A gtB gtB gtA Alt gtB gtA SHGWAampB B gtA Alt gtB A B AampB For example 164 Hardegree Symbolic Logic Example 7 rewritten 1 NA amp B Pr 2 C v B gt ND gt A Pr 3 ND lt gt E Pr 4 SHGW NE DD 5 NA 1ampO 6 B 1ampO 7 C v B 6vl 8 ND gt A 27 gtO 9 D 58 gtO 10 E gt ND 3lt gtO 11 NE 910 gtO NOTE In your own derivations you can draw as much or as little of a box as you like so long as you include at a minimum its left side For example you can use any of the following schemes WVWL Finally we end this section by rewriting the Direct Derivation Picture in accor dance with our minimal boxing scheme SHGW Direct Derivation DD SHGW J21 DD B O O O O O O O
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