Introduction to Deductive Logic Summer 2004 87 Introduction Rule for ' ' (p. 157/173) The ' ' Introduction rule allows us to write down a sentence of the form P Q whenever we have two previous and accessible subderivations, one starting with the assumption of P and ending with the sentence Q and the other starting with the assumption of Q and ending with the sentence P. It looks like this: i Q Assumption j P n Q Assumption m P k P Q i–j, n–m I The subderivations i–j and n–m can occur in either order. Introduction to Deductive Logic Summer 2004 88 Introduction and Elimination Rules for '~' (p. 152/168) The Introduction and Elimination rules for '~' are the most difficult to know how to apply. In all of the rules we have looked at so far, every SL-sentence that appeared in the rule-schema was a sentence that was either part of the input or the output of the inference rule. (Of course, many other sentences may be involved between the SL- sentences that explicitly appear in the schema.) This makes the rules relatively easy to use—your goal is in effect determined by the rule. The negation rules are not like that. The rules for adding or removing the '~' from some sentence P or ~P. involve the use of some sentence R that need bear no relation at all to P. This makes it harder to see what to aim for when you are using the negation rules. Introduction to Deductive Logic Summer 2004 89 Introduction Rule for '~' The '~' Introduction rule allows us to write down a sentence of the form ~P immediately after a subderivation which starts with an assumption of P and includes, under the scope of P, some sentence R and its negation ~R. It looks like this: i P Assumption j R h ~R k ~P i–h ~ I The lines j and h can occur in either order. But one of the two must be immediately before line k and thus the last line of the subderivation. Introduction to Deductive Logic Summer 2004 90 Elimination Rule for '~' The '~' Elimination rule allows us to write down a sentence of the form P immediately after a subderivation which starts with an assumption of ~P and includes, under the scope of ~P, some sentence R and its negation ~R. It looks like this: i ~P Assumption j R h ~R k P i–h ~ E The lines j and h can occur in either order. But one of the two must be immediately before line k and thus the last line of the subderivation. Introduction to Deductive Logic Summer 2004 91 Elimination Rule for '' The '' Elimination rule is our most complicated rule. It allows us to write down a sentence of the form R if we have: 1) a disjunction P Q, 2) a subderivation which starts with an assumption of P and concludes with the sentence R, and 3) a subderivation which starts with an assumption of Q and concludes with the sentence R. It looks like this: i P Q j P Assumption h R m Q Assumption n R k R i, j–h, m–n I The subderivations j–h and m–n can occur in either order. Introduction to Deductive Logic Summer 2004 92 Derivability in SD (p. 170/185) A sentence P of SL is derivable in SD from a set of sentences of SL if and only if there is a derivation in SD in which all of the primary assumptions are members of and P occurs in the scope of only those assumptions. ├SD P =df P is derivable from in SD. ├SD P =df P is not derivable from in SD. ├SD P =df ├SD P. Introduction to Deductive Logic Summer 2004 93 Validity in SD (p. 171/186) An argument of SL is valid in SD if and only if the conclusion of the argument is derivable in SD from the set consisting of the premises. An argument of SL is invalid in SD if and only if it is not valid in SD. Theorem in SD (p. 172/187) A sentence P of SL is a theorem in SD if and only if P is derivable in SD from the empty set. (If an SL-sentence P is a theorem in SD, we also call it an SD-theorem.) Thus, an SL-sentence P is an SD-theorem if and only if ├SD P. Introduction to Deductive Logic Summer 2004 94 Equivalence in SD (p. 173/187) Sentences P and Q of SL are equivalent in SD if and only if Q is derivable in SD from { P } and P is derivable in SD from { Q }. (If SL- sentences P and Q are equivalent in SD, we also call them SD- equivalent.) Thus SL-sentences P and Q are equivalent in SD if and only if {P } ├SD Q and {Q } ├SD P. Inconsistency in SD (p. 174/188) A set of sentences of SL is inconsistent in SD if and only if both a sentence P of SL and its negation ~P are derivable in SD from (i.e. iff for some SL-sentence P, both ├SD P and ├SD ~P ). A set of sentences of SL is consistent in SD if and only if it is not inconsistent in SD. We also call a set of SL-sentences that is consistent (inconsistent) in SD SD-consistent (SD-inconsistent). Introduction to Deductive Logic Summer 2004 95 Soundness of a Logical System (p. 229/248) For some logical language L and for some logical system S based upon that language to say that S is sound is to say: If some L-sentence P is derivable in S from a set of L-sentences , then entails P. Put symbolically, for any L-sentence P and any set of L- sentences : If ├S P then ╞L P. Or, more symbolically still, for any L-sentence P and any set of L-sentences : ├S P ╞L P. Metatheorem 6.2/6.3.1 Soundness of SD For any SL-sentence P and any set of SL-sentences , ├SD P ╞SL P Introduction to Deductive Logic Summer 2004 96 Strong versus Weak Soundness Some presentations of logic distinguish between strong and weak soundness. In such presentations, a systems S is said to be weakly sound iff ├S P ╞L P. Whereas a system S is said to be strongly sound iff ├S P ╞L P. We won't be drawing upon this distinction in this course. (But it is worth noting in passing that there are two different notions here.) Introduction to Deductive Logic Summer 2004 97 Completeness of a Logical System (p. 236/256) For some logical language L and for some logical system S based upon that language to say that S is complete is to say: If some L-sentence P is entailed by a set of L- sentences , then P is derivable from in S. Put symbolically, for any L-sentence P and any set of L-sentences : If ╞L P then ├S P Or, more symbolically still, for any L-sentence P and any set of L- sentences : ╞L P ├S P Metatheorem 6.3/6.4.1 Completeness of SD For any SL-sentence P and any set of SL-sentences , ╞SL P ├SD P Introduction to Deductive Logic Summer 2004 98 Completeness of a Logical System Continued: There are also notions of strong and weak completeness in many presentations of logic. (These are defined in the same way as the parallel notions of strong and weak soundness.) So, in such presentations, a systems S is said to be weakly complete iff ╞L P ├S P. Whereas a system S is said to be strongly sound iff ╞L P ├S P. We won't be drawing upon this distinction in this course, either. (But, once again, it is worth noting in passing that there are two different notions here.) Introduction to Deductive Logic Summer 2004 99 Linking Completeness and Soundness Once we have proved both the Completeness and Soundness of SD we have this result: ╞SL P ├SD P, i.e. a sentence P is truth-functionally entailed by a set iff there is an SD derivation of P from . This in turn means that our syntactic system SD “hooks up with” our semantics for SL in just the way that we would like. In less vague terms, we will have shown that the entailment and derivability notions pick out exactly the same set of SL- sentences/SL-sentence pairs. Thus, our derivability notion will coincide exactly with the entailment notion such that we can derive all and only the sentences that are entailed. Introduction to Deductive Logic Summer 2004 100 Decidability for SL (p. 247/267) We have in the truth-table method, a mechanical test for semantic properties of SL like truth-functional truth, truth-functional validity, truth-functional consistency, and truth-functional entailment. Further, since we have1 both Completeness and Soundness for SD, we can use the truth-table method to test whether the corresponding SD-properties hold of sentences of SL. (E.g. we can test whether a sentence P is a theorem of SD and whether a sentence P is derivable in SD from a set of SL-sentences .) 1 “Have both Completeness and Soundness for SD” in the sense that they are provable (we shall sketch this shortly), though we have not yet shown them to hold. Introduction to Deductive Logic Summer 2004 101 Truth-Function (p.