DOCSLIB.ORG

Expressive Completeness

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Expressive Completeness Truth-Functional Completeness 1. A set of truth-functional operators is said to be truth-functionally complete (or expressively adequate) just in case one can take any truth-function whatsoever, and construct a formula using only operators from that set, which represents that truth-function. In what follows, we will discuss how to establish the truth-functional completeness of various sets of truth-functional operators. 2. Let us suppose that we have an arbitrary n-place truth-function. Its truth table representation will have 2n rows, some true and some false. Here, for example, is the truth table representation of some 3- place truth function, which we shall call $: Φ ψ χ $ T T T T T T F T T F T F T F F T F T T F F T F T F F T F F F F T This truth-function $ is true on 5 rows (the first, second, fourth, sixth, and eighth), and false on the remaining 3 (the third, fifth, and seventh). 3. Now consider the following procedure: For every row on which this function is true, construct the conjunctive representation of that row – the extended conjunction consisting of all the atomic sentences that are true on that row and the negations of all the atomic sentences that are false on that row. In the example above, the conjunctive representations of the true rows are as follows (ignoring some extraneous parentheses): Row 1: (P&Q&R) Row 2: (P&Q&~R) Row 4: (P&~Q&~R) Row 6: (~P&Q&~R) Row 8: (~P&~Q&~R) And now think about the formula that is disjunction of all these extended conjunctions, a formula that basically is a disjunction of all the rows that are true, which in this case would be [(Row 1) v (Row 2) v (Row 4) v (Row6) v (Row 8)] Or, [(P&Q&R) v (P&Q&~R) v (P&~Q&~R) v (P&~Q&~R) v (~P&Q&~R) v (~P&~Q&~R)] 4. It’s pretty easy to verify that this formula is bound to be true just in case the original truth function $ is true. [If the function under consideration is everywhere false, simply consider the formula (φ&~φ).] In other words, this recipe shows how one can take any truth function whatsoever, and from it construct a formula in SL that will represent that very truth function. SL is thereby shown to be “truth- functionally” or expressively complete. 5. More than that, the formulas produced by this recipe are all in what is called “Disjunctive Normal Form.” A formula in DNF is one which contains no operators other ~,&, and v, and in which no negations operate over anything other than atomic formulas, and no disjunction falls within the scope of any conjunction. Formulas in DNF are basically disjunctions of conjunctions of atomic formulas and their negations. What this shows is that a language employing just (~,&,v) is truth-functionally complete. Moreover, since any formula in SL represents some truth function or other, what this also shows is that every formula in SL has some equivalent in DNF. 6. Now we can do even better than that: Consider an approach that goes “the other way around.” Instead of focusing on those rows of the truth table representation of a function that are true, focus instead upon those rows in which the function is false. In the example above, these are rows 3, 5, and 7. One way of representing the truth function in question is simply to apply the procedure outlined above and then negate the result, saying in effect that the truth-function we want is NOT ANY of the false rows: ~[(Row 3) v (Row 5) v (Row 7)]. But those of you who are familiar with the DeMorgan’s equivalencies1 will see that this formula is equivalent to the following: [~(Row 3) & ~(Row 5) & ~(Row 7)] or more specifically, [~(P&~Q&R) & ~(~P&Q&R) & ~(~P&~Q&R)] In other words, we take the negation of the conjunctive representation of every row on which some target truth function is false, and then conjoin them all together. [If the function is everywhere true, just write down ~(P&~P).] What you get is a formula that looks something like this: ~(row i) & ~(row j) & ~(row k)….for every row where our target truth function is false.. Once again, this is a formula in SL which will be true under just those circumstances in which the original truth function is true. But notice that this formula will only contain the negation and conjunction operators! In other words, (~,&) is expressively complete all by itself. 1 The DeMorgan equivalencies basically state that “Neither A nor B” is equivalent to “Not A and Not B.” and that “Not both A and B” is equivalent to “Either Not A or Not B.” Augustus DeMorgan was a pioneering 19th century mathematical logician who attempted to show how rules of reasoning could be modeled algebraically. In short, DeMorgan’s “rules” direct us how to distribute or to extract negations to and from conjunctions and disjunctions, much as one might distribute or extract factors into or from sums. 7. By the way, we can construct yet another formula equivalent to this one by using the DeMorgan’s rule once again to “drive” the various negations into the conjunctive representation of each row. When we do (and we also use the double negation (or DN) equivalencies to rid ourselves of pesky double negations), we’ll arrive at a formula containing just (~,&,v) as functors, where negations operate only on atomic formulas and no disjunction has any conjunction within its scope. Guess what such a “conjunction of disjunctions of atomic formulas and their negations” are called? 8. Now that we have established the truth-functional completeness of certain sets of truth operators, we are in a position to show that other sets are truth-functionally complete as well. This we can do by showing that a certain set of operators has all the expressive resources of a truth functionally complete one, by “defining” all the operators of the complete set in terms of the other set. First off, it’s pretty easy to see how to represent or to “define” v in terms of (~, →). We do this by showing that the truth table representation of (φ v ψ) can be replicated by some formula that includes only negations and conditionals. As verified by the following joint truth table, the formula (~φ→ψ) fits the bill: φ ψ (φ v ψ) (~φ→ψ) T T T T T F T T F T T T F F F F Similarly, we can straightforwardly define & in terms of (~,v) by using the following Demorgan’s equivalency: (φ & ψ) : : ~(~φ v ~ψ) Once again, we can verify this equivalency by means of a truth table. What this means is that we could express the truth function represented by any conjunction by using a combination of negations and disjunction instead. In principle we could “trade in” the conjunction operator without any loss of expressive power, as long we still had the negation and disjunction operations available to us. 9. We were able to conclude in section 6. above that the set (~,&) is truth-functionally complete. Now we know that & can be defined in terms of (~ and v); we also know that ~ can be defined by those same operators because trivially, ~ can define itself. Since the set (~,v) can define all the members of a truth- functionally complete set, we may conclude that the set (~,v) is also expressively truth-functionally. And since v can be defined in terms of negation and the conditional, we also happen to be in position to say the same of (~,→). 10. Perhaps a little more intriguing is the fact that ~ can be defined in terms of (→,⊥). How? [Think of ⊥ as the falsum, a “zero-place” operator that always comes out false.] So given what we know about the set (~,→), what may we conclude from this result? [As an aside, this is the basic set of truth operators Garson (2006) deploys in his otherwise very fine text on Modal Logic.] 11. So what about sets containing just a single truth-functional operator? Can & or v be truth- functionally complete all by themselves (or even in tandem)? In order for that to be the case, one must be able to express the negation of a formula as a function of just conjunction or disjunction. Think about why you might not be able to do that. What does that suggest about the set of operators containing both (&,v). The thing that blocks conjunction and disjunction from being expressively complete all by themselves is that a truth conjoined with a truth, and a truth disjoined with a truth, will always come up true. Thus one can never construct a falsehood from just true components. Similarly, one can never construct a truth from just false components. The conditional shares the first problem though not the second. [How can one construct a true conditional out of solely false components?] 12. One might think at this point that to be truth-functionally complete, a set must contain at least two truth functors. But that would be a mistake. Think of the n/and and n/or functions, symbolized by the up arrow and down arrow respectively (↑,↓), aka the Scheffer and Peirce strokes. Notice how, unlike the conjunction and disjunction, a formula that “strokes itself” (my apologies for that!) will switch truth value. Show how each of these individually can be shown to be truth functionally complete by showing that they can individually define negation and either conjunction or disjunction.
Load more

Recommended publications
  • Physical and Mathematical Sciences 2014, № 1, P. 3–6 Mathematics ON
    PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Mathematical Sciences 2014, № 1, p. 3–6 Mathematics ON ZIGZAG DE MORGAN FUNCTIONS V. A. ASLANYAN ∗ Oxford University UK, Chair of Algebra and Geometry YSU, Armenia There are five precomplete classes of De Morgan functions, four of them are defined as sets of functions preserving some finitary relations. However, the fifth class – the class of zigzag De Morgan functions, is not defined by relations. In this paper we announce the following result: zigzag De Morgan functions can be defined as functions preserving some finitary relation. MSC2010: 06D30; 06E30. Keywords: disjunctive (conjunctive) normal form of De Morgan function; closed and complete classes; quasimonotone and zigzag De Morgan functions. 1. Introduction. It is well known that the free Boolean algebra on n free generators is isomorphic to the Boolean algebra of Boolean functions of n variables. The free bounded distributive lattice on n free generators is isomorphic to the lattice of monotone Boolean functions of n variables. Analogous to these facts we have introduced the concept of De Morgan functions and proved that the free De Morgan algebra on n free generators is isomorphic to the De Morgan algebra of De Morgan functions of n variables [1]. The Post’s functional completeness theorem for Boolean functions plays an important role in discrete mathematics [2]. In the paper [3] we have established a functional completeness criterion for De Morgan functions. In this paper we show that zigzag De Morgan functions, which are used in the formulation of the functional completeness theorem, can be defined by a finitary relation.
    [Show full text]
  • Completeness
    Completeness The strange case of Dr. Skolem and Mr. G¨odel∗ Gabriele Lolli The completeness theorem has a history; such is the destiny of the impor- tant theorems, those for which for a long time one does not know (whether there is anything to prove and) what to prove. In its history, one can di- stinguish at least two main paths; the first one covers the slow and difficult comprehension of the problem in (what historians consider) the traditional development of mathematical logic canon, up to G¨odel'sproof in 1930; the second path follows the L¨owenheim-Skolem theorem. Although at certain points the two paths crossed each other, they started and continued with their own aims and problems. A classical topos of the history of mathema- tical logic concerns the how and the why L¨owenheim, Skolem and Herbrand did not discover the completeness theorem, though they proved it, or whe- ther they really proved, or perhaps they actually discovered, completeness. In following these two paths, we will not always respect strict chronology, keeping the two stories quite separate, until the crossing becomes decisive. In modern pre-mathematical logic, the notion of completeness does not appear. There are some interesting speculations in Kant which, by some stretching, could be realized as bearing some relation with the problem; Kant's remarks, however, are probably more related with incompleteness, in connection with his thoughts on the derivability of transcendental ideas (or concepts of reason) from categories (the intellect's concepts) through a pas- sage to the limit; thus, for instance, the causa prima, or the idea of causality, is the limit of implication, or God is the limit of disjunction, viz., the catego- ry of \comunance".
    [Show full text]
  • Some Analogues of the Sheffer Stroke Function in N-Valued Logic
    MA THEMA TICS SOME ANALOGUES OF THE SHEFFER STROKE FUNCTION IN n-VALUED LOGIe BY NORMAN M. MARTIN (Communicated by Prof. A. HEYTING at the meeting of June 24, 1950) The interpretation of the ordinary (two-valued) propositional logic in terms of a truth-table system with values "true" and "false" or, more abstractly, the numbers "I" and "2" has become customary. This has been useful in giving an algorithm for the concept "analytic" , in giving an adequacy criterion for the definability of one function by another and it has made possible the proof of adequacy of a list of primitive terms for the definition of all truth-functions (functional completeness). If we generalize the concept of truth-function so as to allow for systems of functions of 3, 4, etc. values (preserving the "extensionality" requirement on functions examined) we obtain systems of functions of more than 2 values analogous to the truth-table interpretation of the usual pro­ positional calculus. The problem of functional completeness (the term is due to TURQUETTE) arises in each ofthe resulting systems. Strictly speaking, this problem is notclosely connected with problems of deducibility but is rather a combinatorial question. It will be the purpose of this paper to examine the problem of functional completeness of functions in n-valued logic where by n-valued logic we mean that system of functions such that each function of the system determines, by substitution of an arbitrary numeral a for the symbol n in the definition of the n-valued function, a function in the truth table system of a values.
    [Show full text]
  • Boolean Logic
    Boolean logic Lecture 12 Contents . Propositions . Logical connectives and truth tables . Compound propositions . Disjunctive normal form (DNF) . Logical equivalence . Laws of logic . Predicate logic . Post's Functional Completeness Theorem Propositions . A proposition is a statement that is either true or false. Whichever of these (true or false) is the case is called the truth value of the proposition. ‘Canberra is the capital of Australia’ ‘There are 8 day in a week.’ . The first and third of these propositions are true, and the second and fourth are false. The following sentences are not propositions: ‘Where are you going?’ ‘Come here.’ ‘This sentence is false.’ Propositions . Propositions are conventionally symbolized using the letters Any of these may be used to symbolize specific propositions, e.g. :, Manchester, , … . is in Scotland, : Mammoths are extinct. The previous propositions are simple propositions since they make only a single statement. Logical connectives and truth tables . Simple propositions can be combined to form more complicated propositions called compound propositions. .The devices which are used to link pairs of propositions are called logical connectives and the truth value of any compound proposition is completely determined by the truth values of its component simple propositions, and the particular connective, or connectives, used to link them. ‘If Brian and Angela are not both happy, then either Brian is not happy or Angela is not happy.’ .The sentence about Brian and Angela is an example of a compound proposition. It is built up from the atomic propositions ‘Brian is happy’ and ‘Angela is happy’ using the words and, or, not and if-then.
    [Show full text]
  • The Logic of Internal Rational Agent 1 Introduction
    Australasian Journal of Logic Yaroslav Petrukhin The Logic of Internal Rational Agent Abstract: In this paper, we introduce a new four-valued logic which may be viewed as a variation on the theme of Kubyshkina and Zaitsev’s Logic of Rational Agent LRA [16]. We call our logic LIRA (Logic of Internal Rational Agency). In contrast to LRA, it has three designated values instead of one and a different interpretation of truth values, the same as in Zaitsev and Shramko’s bi-facial truth logic [42]. This logic may be useful in a situation when according to an agent’s point of view (i.e. internal point of view) her/his reasoning is rational, while from the external one it might be not the case. One may use LIRA, if one wants to reconstruct an agent’s way of thinking, compare it with respect to the real state of affairs, and understand why an agent thought in this or that way. Moreover, we discuss Kubyshkina and Zaitsev’s necessity and possibility operators for LRA definable by means of four-valued Kripke-style semantics and show that, due to two negations (as well as their combination) of LRA, two more possibility operators for LRA can be defined. Then we slightly modify all these modalities to be appropriate for LIRA. Finally, we formalize all the truth-functional n-ary extensions of the negation fragment of LIRA (including LIRA itself) as well as their basic modal extension via linear-type natural deduction systems. Keywords: Logic of rational agent, logic of internal rational agency, four-valued logic, logic of generalized truth values, modal logic, natural deduction, correspondence analysis.
    [Show full text]
  • 6C Lecture 2: April 3, 2014
    6c Lecture 2: April 3, 2014 2.1 Functional completeness, normal forms, and struc- tural induction Before we begin, lets give a formal definition of a truth table. Definition 2.1. A truth table for a set of propositional variables, is a function which assigns each valuation of these variables either the value true or false. Given a formula φ, the truth table of φ is the truth table assigning each valuation of the variables of v the corresponding truth value of φ. We are ready to begin: Definition 2.2. We say that a set S of logical connective is functionally com- plete if for every finite set of propositional variables p1; : : : ; pn and every truth table for the variables p1; : : : ; pn, there exists a propositional formula φ using the variables p1; : : : ; pn and connectives only from S so that φ has the given truth table. We will soon show that the set f:; ^; _g is functionally complete. Before this, lets do a quick example. Here is an example of a truth table for the variables p1; p2; p3: p1 p2 p3 T T T T T T F F T F T T T F F T F T T F F T F F F F T F F F F F If f:; ^; _g is functionally complete, then we must be able to find a formula using only f:; ^; _g which has this given truth table (indeed, we must be able to do this for every truth able). In this case, one such a formula is p1 ^ (:p2 _ p3).
    [Show full text]
  • Warren Goldfarb, Notes on Metamathematics
    Notes on Metamathematics Warren Goldfarb W.B. Pearson Professor of Modern Mathematics and Mathematical Logic Department of Philosophy Harvard University DRAFT: January 1, 2018 In Memory of Burton Dreben (1927{1999), whose spirited teaching on G¨odeliantopics provided the original inspiration for these Notes. Contents 1 Axiomatics 1 1.1 Formal languages . 1 1.2 Axioms and rules of inference . 5 1.3 Natural numbers: the successor function . 9 1.4 General notions . 13 1.5 Peano Arithmetic. 15 1.6 Basic laws of arithmetic . 18 2 G¨odel'sProof 23 2.1 G¨odelnumbering . 23 2.2 Primitive recursive functions and relations . 25 2.3 Arithmetization of syntax . 30 2.4 Numeralwise representability . 35 2.5 Proof of incompleteness . 37 2.6 `I am not derivable' . 40 3 Formalized Metamathematics 43 3.1 The Fixed Point Lemma . 43 3.2 G¨odel'sSecond Incompleteness Theorem . 47 3.3 The First Incompleteness Theorem Sharpened . 52 3.4 L¨ob'sTheorem . 55 4 Formalizing Primitive Recursion 59 4.1 ∆0,Σ1, and Π1 formulas . 59 4.2 Σ1-completeness and Σ1-soundness . 61 4.3 Proof of Representability . 63 3 5 Formalized Semantics 69 5.1 Tarski's Theorem . 69 5.2 Defining truth for LPA .......................... 72 5.3 Uses of the truth-definition . 74 5.4 Second-order Arithmetic . 76 5.5 Partial truth predicates . 79 5.6 Truth for other languages . 81 6 Computability 85 6.1 Computability . 85 6.2 Recursive and partial recursive functions . 87 6.3 The Normal Form Theorem and the Halting Problem . 91 6.4 Turing Machines .
    [Show full text]
  • CS40-S13: Functional Completeness 1 Introduction
    CS40-S13: Functional Completeness Victor Amelkin [email protected] April 12, 2013 In class, we have briefly discussed what functional completeness means and how to prove that a certain system (a set) of logical operators is functionally complete. In this note, I will review what functional completeness is, how to prove that a system of logical operators is complete as well as how to prove that a system is incomplete. 1 Introduction When we talk about logical operators, we mean operators like ^, _, :, !, ⊕. Logical operators work on propositions, i.e. if we write a ^ b, then both a and b are propositions. The mentioned propositions may be either simple or compound, as in a ^ :(x ! y) { here, a is simple, while b is compound. | {z } b For a logical operator it does not matter whether it operates on simple or compound propositions; the only thing that matters is that all the arguments are propositions, i.e., either True of False. Since logical operators apply to propositions and \return" either True or False, they can be seen as boolean functions. For example, conjunction ^(x1; x2) = x1 ^ x2 is a binary boolean function, i.e., it accepts two arguments (hence, \binary") that can be either True of False and returns either True of False (hence, \boolean"). As another example, negation can be seen as a unary (accepts only one argument) boolean function :(x) = :x. We, however, rarely use the functional notation ^(x1; x2) (like in f(x)); instead, we usually use the operator notation, e.g., x1 _ x2. (Similarly, when you do algebra, you probably write a + b instead of +(a; b), unless you do algebra in Lisp.) Further, instead of talking about logical operators, we will talk about boolean functions, but, having read the material above, you should understand that these two are the same.
    [Show full text]
  • Logical Connectives in Natural Languages
    Logical connectives in natural languages Leo´ Picat Introduction Natural languages all have common points. These shared features define and constrain them. A key property of all natural languages is their learnability. As long as children are exposed to it, they can learn any spoken or signed language. This process takes place in environments in which children are not exposed to the full diversity of their mother tongue: it is with only few evidences that they master the ability to express themselves through language. As children are not exposed to negative evidences, learning a language would not be possible if con- straints were not present to limit the space of possible languages. Input-based theories predict that these constraints result from general cognition mechanisms. Knowledge-based theories predict that they are language specific and are part of a Universal Grammar [1]. Which approach is the right one is still a matter of debate and the implementation of these constraints within our brain is still to be determined. Yet, one can get a taste of them as they surface in linguistic universals. Universals have been formulated for most domains of linguistics. The first person to open the way was Greenberg [2] who described 50 statements about syntax and morphology across languages. In the phono- logical domain, all languages have stops for example [3]. Though, few universals are this absolute; most of them are more statistical truth or take the form of conditional statements. Universals have also been defined for semantics with more or less success [4]. Some of them shed light on non-linguistic abilities.
    [Show full text]
  • Switching Circuits and Logic Design
    Contents 1 Boolean Algebra F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 28, 2020 1 / 12 Boolean Algebra Section outline Boolean expression manipulation 1 Boolean Algebra Exclusive OR SOP from sets Series-parallel switching Boolean expressions circuits Functional completeness Shannon decomposition Distinct Boolean functions F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 28, 2020 2 / 12 1^2 A \ B (A \ B \ C) [ (A \ B \ C) abc + abc = ab 1^2^3^5 A (A \ B \ C) [ (A \ B \ C) [ (A \ B \ C) [ (A \ B \ C) abc + abc + abc + abc = ab + ab = a a I have an item from A a I don’t have an item from A ab + c I have an item from A but not from B or an item from C Boolean Algebra SOP from sets Sum of products from sets Selections Regions 1 A \ B \ C B 2 A \ B \ C 6 8 3 A \ B \ C 2 4 4 A \ B \ C 1 5 3 7 5 A \ B \ C A C 6 A \ B \ C 7 A \ B \ C 8 A \ B \ C F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 28, 2020 3 / 12 (A \ B \ C) [ (A \ B \ C) abc + abc = ab 1^2^3^5 A (A \ B \ C) [ (A \ B \ C) [ (A \ B \ C) [ (A \ B \ C) abc + abc + abc + abc = ab + ab = a a I have an item from A a I don’t have an item from A ab + c I have an item from A but not from B or an item from C Boolean Algebra SOP from sets Sum of products from sets Selections Regions 1^2 A \ B 1 A \ B \ C B 2 A \ B \ C 6
    [Show full text]
  • TMM023- DISCRETE MATHEMATICS (March 2021)
    TMM023- DISCRETE MATHEMATICS (March 2021) Typical Range: 3-4 Semester Hours A course in Discrete Mathematics specializes in the application of mathematics for students interested in information technology, computer science, and related fields. This college-level mathematics course introduces students to the logic and mathematical structures required in these fields of interest. TMM023 Discrete Mathematics introduces mathematical reasoning and several topics from discrete mathematics that underlie, inform, or elucidate the development, study, and practice of related fields. Topics include logic, proof techniques, set theory, functions and relations, counting and probability, elementary number theory, graphs and tree theory, base-n arithmetic, and Boolean algebra. To qualify for TMM023 (Discrete Mathematics), a course must achieve all the following essential learning outcomes listed in this document (marked with an asterisk). In order to provide flexibility, institutions should also include the non-essential outcomes that are most appropriate for their course. It is up to individual institutions to determine if further adaptation of additional course learning outcomes beyond the ones in this document are necessary to support their students’ needs. In addition, individual institutions will determine their own level of student engagement and manner of implementation. These guidelines simply seek to foster thinking in this direction. 1. Formal Logic – Successful discrete mathematics students are able to construct and analyze arguments with logical precision. These students are able to apply rules from logical reasoning both symbolically and in the context of everyday language and are able to translate between them. The successful Discrete Mathematics student can: 1.1. Propositional Logic: 1.1a. Translate English sentences into propositional logic notation and vice-versa.
    [Show full text]
  • Form and Content: an Introduction to Formal Logic
    Connecticut College Digital Commons @ Connecticut College Open Educational Resources 2020 Form and Content: An Introduction to Formal Logic Derek D. Turner Connecticut College, [email protected] Follow this and additional works at: https://digitalcommons.conncoll.edu/oer Recommended Citation Turner, Derek D., "Form and Content: An Introduction to Formal Logic" (2020). Open Educational Resources. 1. https://digitalcommons.conncoll.edu/oer/1 This Book is brought to you for free and open access by Digital Commons @ Connecticut College. It has been accepted for inclusion in Open Educational Resources by an authorized administrator of Digital Commons @ Connecticut College. For more information, please contact [email protected]. The views expressed in this paper are solely those of the author. Form & Content Form and Content An Introduction to Formal Logic Derek Turner Connecticut College 2020 Susanne K. Langer. This bust is in Shain Library at Connecticut College, New London, CT. Photo by the author. 1 Form & Content ©2020 Derek D. Turner This work is published in 2020 under a Creative Commons Attribution- NonCommercial-NoDerivatives 4.0 International License. You may share this text in any format or medium. You may not use it for commercial purposes. If you share it, you must give appropriate credit. If you remix, transform, add to, or modify the text in any way, you may not then redistribute the modified text. 2 Form & Content A Note to Students For many years, I had relied on one of the standard popular logic textbooks for my introductory formal logic course at Connecticut College. It is a perfectly good book, used in countless logic classes across the country.
    [Show full text]