[...]... etc.) i* ii* iii* iv* v* vi * vii* viii* ix * X* xi* xii* xiii* xiv * xv* A (A (A) (A + B) (A + ( (A + (B + C)) ((P & Q) + R) ((A & B) v (C + (D -( A + B) - GI)) -( P + Q) v -( Q & R) -( A) (-A) + B (-( P & P) & (P (Q v -Q))) (-( (B v P) & C) e ((D v -G) (-( Q v -( B)) v (E + H)) (D v X))) Chapter I parenthesisdropping conventions 9 Comment For ease of reading, it is often convenient to drop parentheses from... premises Chapter I 19 primitive rules Definition The ten PRIMITIVE RULES OF PROOF are the rules assumption, ampersand-introduction, ampersand-elimination, wedge-introduction, wedge-elimination, arrow-introduction, arrow-elimination, reductio ad absurdum, double-arrow-introduction, and double-arrow-elimination, as described below assumption Assume any sentence Annotation: Assumption set: Comment: Example (1)... sentential is logic, but it may be used to represent an expression of sentential logic 6 Chapter 1 well-formed formula Definition A WELL- FORMED FORMULA (WFF) of sentential logic is any expression that accords with the following seven rules: (1) A sentence letter standing alone is a wff [Definition T h e s e n t e n c e letters are the ATOMIC SENTENCES of the language of sentential logic. ] atomic... Sentential Logic formal language Comment To represent similarities a m o n g arguments of a natural language, logicians introduce formal languages The first formal language we will introduce is the language of sentential logic (also known as propositional logic) In chapter 3 we introduce a more sophisticated language: that of predicate logic vocabulary Definition The VOCABULARY OF SENTENTIAL LOGIC consists... sentential logic is any sequence of sentence letters, sentential connectives, or left and right parentheses Examples (P + Q) is an expression of sentential logic )PQ +- is also an expression of sentential logic (3 + 4) is not an expression of sentential logic I metavariable Definition Greek letters such as ( and y~ are used as METAVARIABLES They are not themselves parts of the language of sentential logic, ... together - is a UNARY CONNECTIVE, since it attaches to a single wff sentence Definition A SENTENCE of the formal language is a wff that is not part of a larger wff denial Definition The DENIAL of a wff @ that is not a negation is -@ A negation, -@ , has two DENIALS: @ and @ Example -( P + Q) has one negation: (P + Q) It has two denials: (P + Q) and (P + Q) (P + Q) has just one denial: its negation, -( P... proof using 3-instantiation assumption 1 3-instantiation Line 2 does not follow from line 1 This difference between 3elimination and 3-instantiation can be put as follows: in an 3elimination proof, you can stop at any time and still have a correct proof of some argument or other, but in an 3-instantiation proof, you cannot stop whenever you like It seems to us that these implications of 3-instantiation's... show that this is equivalent to (@ w1 v biconditionals Variants having the logical form (@w y ~ include the ) following: @ if and only if v @ is equivalent to v @ is necessary and sufficient for W @justin case v neither nor English sentences of the form 'Neither @ nor vCr' have the logical form -( @v v), or, equivalently, (-( I & -w) 14 tenses Comment In English, the sentences 'Mary is dancing' and 'Mary... of 3-Instantiation, this rule frees the student from having to remember to instantiate existential quantifications before instantiating universal quantifications It also frees the student from having to examine the not-yet-reached conclusion of the argument, to determine which instantial names are unavailable for a given application of 3-Instantiation Furthermore, at any point in a proof using 3-elimination,... Definition.A TRANSLATION SCHEME for the language of sentential logic is a pairing of sentence letters with sentences of a natural language The sentences in a translation scheme should be logically simple That is, they should not contain any of the words corresponding to the sentential connectives Chapter I logical form 11 Definition The LOGICAL FORM of a sentence of a natural language relative to a . use of the website at http://mitpress .mit. edu/LogicPrimer/). We have deliberately reduced to a minimum the amount of explanatory material, relying upon your instructor to expand on the ideas writing from the publisher. Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Allen, Colin. Logic primertcolin Allen and Michael Hand 2nd ed at the back of the book.) The comments are of two sorts. Those set in full-size type contain material we deem essential to the text. Those set in smaller type are relatively incidental-the