DOCSLIB.ORG

Logical Equivalence, Logical Truths, and Contradictions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Logical Equivalence, Logical Truths, and Contradictions 3-1. LOGICAL EQUIVALENCE I introduced logic as the science of arguments. But before turning to ar- guments, we need to extend and practice our understanding of logic's. basic tools as I introduced them in chapter 1. For starters, let's look at the truth table for 'A', '-A', and the negation of the negation of 'A', namely, '--A' This uuth table exhibits the special situation which I mentioned at the end of the last chapter: The auth value of '--A' is always the same as that of 'A'. Logicians say that 'A' and '--A' are Logicdl' Equrualtnt. As we will see in a moment, much more complicated sentences can be logically equivalent to each other. To get dear on what this means, let us review some of the things that truth tables do for us. Suppose we are looking at a compound sentence, perhaps a very complicated one which uses many sentence letters. When we write out the truth table for such a 30 Logical Equivdmcc, Logical Truths, and Contradictions sentence, we write out all the possible cases, that is, all the possible assign- ments of truth values to sentence letters in all possible combinations. In 1 each one of these possible cases our original sentence has the truth value i t or the truth value f. Now suppose that we look at a second sentence which uses the same sentence letters, or perhaps only some of the sentence letters that the first sentence uses, and no new sentence letters. We say that the two sentences are logically equivalent if in each possible case, that is, for each line of the these eventualities, we must draw the circles representing X and Y as truth table, they have the same truth value. overlapping, as in Figure 3-2 and 3-3. Two sentences of sentence logic are Logically Equivalent if and only if in each possible case (for each assignment of truth values to sentence letters) the two sentences have the same truth value. What we said about the double negation of 'A' naturally holds quite generally: I X&Y XvY The Law of Double Negation (DN): For any sentence X, X and --X are logically equivalent. Figure 3-2 Figure 3-3 Here are two more laws of logical equivalence: The conjunction X&Y is true in just those cases represented by points that lie inside both the X and Y circles, that is, the shaded area in Figure De Morgan's Laws (DM): For any sentences X and Y, -(X&Y) is logically 3-2. The disjunction XVY is true in just those cases represented by points equivalent to -XV-Y. And -(XvY) is logically equivalent to -X&-Y. that lie inside either the X or the Y circle (or both), that is, the shaded area in Figure 3-3. Now we can use Venn diagrams to check De Morgan's laws. Consider Thus 'Adam is not both ugly and dumb.' is logically equivalent to 'Either first a negated conjunction. Look for the area, shown in Figure 3-4, Adam is not ugly or Adam is not dumb.' And 'Adam is not either ugly or which represents -(X&Y). This is just the area outside the shaded lens in dumb.' is logically equivalent to 'Adam is not ugly and Adam is not Figure 3-2. dumb.' You should check these laws with truth tables. But I also want to show you a second, informal way of checking them which allows you to "see" the laws. This method uses something called Van Diagram. A Venn diagram begins with a box. You are to think of each point inside the box as a possible case in which a sentence might be true or FIgule 3-4 false. That is, think of each point as an assignment of truth values to sentence letters, or as a line of a truth table. Next we draw a circle in the box and label it with the letter 'X', which is supposed to stand for some arbitrary sentence, atomic or compound. The idea is that each point in- side the circle represents a possible case in which X is true, and each point Let us compare this with the area which represents -XV-Y. We draw outside the circle represents a possible case in which X is false. overlapping X and Y circles. Then we take the area outside the first circle Look at Figure 3-1. What area represents the sentence -X? The area (which represents -X; see Figure 3-5), and we take the area outside the outside the circle, because these are the possible cases in which X is false. second (which represents -Y; see Figure 3-6). - Now let's consider how Venn diagrams work for compound sentences Finally, we put these two areas together to get the area representing the built up from two components, X and Y. Depending on what the sen- disjunction -XV-Y, as represented in Figure 3-7. tences X and Y happen to be, both of them might be true, neither might Notice that the shaded area of Figure 3-7, representing -XV-Y, is the be true, or either one but not the other might be true. Not to omit any of same as that of Figure 3-4, representing -(X&Y). The fact that the same 32 Logical EquivaJcnce, Logical Truths, and Contradietions 3-2. Subrtirurion of Logical Equiualrntr and Some Mon Lcnus 33 'Fill in the areas to represent YvZ, and then indicate the area which represents the conjunction of this with X. In a separate diagram, first fill in the areas representing X&Y and X&Z, and then find the area corre- sponding to the disjunction of these. If the areas agree, you will have demonstrated logical equivalence. Do the second of the distributive laws similarly. Also, if you feel you need more practice with truth tables, prove Figure 3-5 Figure 3-6 these laws using truth tables. EXERCISES 3-1. Prove the second of De Morgan's laws and the two distributive laws using Venn diagrams. Do this in the same way that I proved the first of De Morgan's laws in the text, by drawing a Venn diagram Figure 3-7 for each proof, labeling the circles in the diagram, and explaining in a few sentences how the alternate ways of getting the final area give shaded area represents both -XV-Y and -(X&Y) means that the two the same result. Use more than one diagram if you find that helpful sentences are true in exactly the same cases and false in the same cases. in explaining your proof. In other words, they always have the same truth value. And that is just what we mean by two sentences being logically equivalent. Now try to prove the other of De Morgan's laws for yourself using Venn diagrams. 3-2. SUBSTITUTION OF LOGICAL EQUIVALENTS Here are two more laws of logical equivalence: AND SOME MORE LAWS The Dishibutive Laws: For any three sentences, X, Y, and Z, X&(YvZ) is We can't do much with our laws of logical equivalence without using a logically equivalent to (X&Y)v(X&Z).And Xv(Y&Z)is logically equivalent to very simple fact, which our next example illustrates. Consider (XW&(XvZ). For example, 'Adam is both bold and either clever or lucky.' comes to the same thing as 'Adam is either both bold and clever or both bold and '--A' is logically equivalent to 'A'. This makes us think that (1) is logically lucky.' You should prove these two laws for yourself using Venn dia- equivalent to grams. To do so, you will need a diagram with three circles, one each representing X, Y, and Z. Again, to make sure that you omit no possible (2) AVB. combination of truth values, you must draw these so that they all overlap, as in Figure 3-8. This is right. But it is important to understand why this is right. A compound sentence is made up of component sentences, which in turn may be made up of further component sentences. How do subsentences (components, or components of components, or the like) affect the truth value of the original sentence? Only through their truth values. The only way that a subsentence has any effect on the truth values of a larger sen- tence is through the subsentence's truth value. (This, again, is just what we mean by saying that compound sentences are truth -functions.) But if only the truth values matter, then substituting another sentence which always has the same truth value as the first can't make any difference. 34 Logical Equiwlmee, Logical Truth, and ConhndictiOnr I'll say it again in different words: Suppose that X is a subsentence of (--A&--B)v(--A&-B) DM,SLE some larger sentence. Suppose that Y is logically equivalent to X, which means that Y and X always have the same truth value. X affects the truth (As before, 'DM' on the right means that we have used one of De Mor- value of the larger sentence only through its (i.e., X's) truth value. So, if gan's laws. 'SLE' means that we have also used the law of substitution of we substitute Y for X, there will be no change in the larger sentence's logical equivalents in getting the last line from the previous one.) truth value. Now we can apply the law of double negation (abbreviated 'DN') to But this last fact is just what we need to show our general point about '--A' and to '--B' and once more substitute the results into the larger logical equivalence.
Recommended publications
  • CS 512, Spring 2017, Handout 10 [1Ex] Propositional Logic: [2Ex

    CS 512, Spring 2017, Handout 10 [1Ex] Propositional Logic: [2Ex

    CS 512, Spring 2017, Handout 10 Propositional Logic: Conjunctive Normal Forms, Disjunctive Normal Forms, Horn Formulas, and other special forms Assaf Kfoury 5 February 2017 Assaf Kfoury, CS 512, Spring 2017, Handout 10 page 1 of 28 CNF L ::= p j :p literal D ::= L j L _ D disjuntion of literals C ::= D j D ^ C conjunction of disjunctions DNF L ::= p j :p literal C ::= L j L ^ C conjunction of literals D ::= C j C _ D disjunction of conjunctions conjunctive normal form & disjunctive normal form Assaf Kfoury, CS 512, Spring 2017, Handout 10 page 2 of 28 DNF L ::= p j :p literal C ::= L j L ^ C conjunction of literals D ::= C j C _ D disjunction of conjunctions conjunctive normal form & disjunctive normal form CNF L ::= p j :p literal D ::= L j L _ D disjuntion of literals C ::= D j D ^ C conjunction of disjunctions Assaf Kfoury, CS 512, Spring 2017, Handout 10 page 3 of 28 conjunctive normal form & disjunctive normal form CNF L ::= p j :p literal D ::= L j L _ D disjuntion of literals C ::= D j D ^ C conjunction of disjunctions DNF L ::= p j :p literal C ::= L j L ^ C conjunction of literals D ::= C j C _ D disjunction of conjunctions Assaf Kfoury, CS 512, Spring 2017, Handout 10 page 4 of 28 I A disjunction of literals L1 _···_ Lm is valid (equivalently, is a tautology) iff there are 1 6 i; j 6 m with i 6= j such that Li is :Lj.
  • 7.1 Rules of Implication I

    7.1 Rules of Implication I

    Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic. The method consists of using sets of Rules of Inference (valid argument forms) to derive either a conclusion or a series of intermediate conclusions that link the premises of an argument with the stated conclusion. The First Four Rules of Inference: ◦ Modus Ponens (MP): p q p q ◦ Modus Tollens (MT): p q ~q ~p ◦ Pure Hypothetical Syllogism (HS): p q q r p r ◦ Disjunctive Syllogism (DS): p v q ~p q Common strategies for constructing a proof involving the first four rules: ◦ Always begin by attempting to find the conclusion in the premises. If the conclusion is not present in its entirely in the premises, look at the main operator of the conclusion. This will provide a clue as to how the conclusion should be derived. ◦ If the conclusion contains a letter that appears in the consequent of a conditional statement in the premises, consider obtaining that letter via modus ponens. ◦ If the conclusion contains a negated letter and that letter appears in the antecedent of a conditional statement in the premises, consider obtaining the negated letter via modus tollens. ◦ If the conclusion is a conditional statement, consider obtaining it via pure hypothetical syllogism. ◦ If the conclusion contains a letter that appears in a disjunctive statement in the premises, consider obtaining that letter via disjunctive syllogism. Four Additional Rules of Inference: ◦ Constructive Dilemma (CD): (p q) • (r s) p v r q v s ◦ Simplification (Simp): p • q p ◦ Conjunction (Conj): p q p • q ◦ Addition (Add): p p v q Common Misapplications Common strategies involving the additional rules of inference: ◦ If the conclusion contains a letter that appears in a conjunctive statement in the premises, consider obtaining that letter via simplification.
  • Two Sources of Explosion

    Two Sources of Explosion

    Two sources of explosion Eric Kao Computer Science Department Stanford University Stanford, CA 94305 United States of America Abstract. In pursuit of enhancing the deductive power of Direct Logic while avoiding explosiveness, Hewitt has proposed including the law of excluded middle and proof by self-refutation. In this paper, I show that the inclusion of either one of these inference patterns causes paracon- sistent logics such as Hewitt's Direct Logic and Besnard and Hunter's quasi-classical logic to become explosive. 1 Introduction A central goal of a paraconsistent logic is to avoid explosiveness { the inference of any arbitrary sentence β from an inconsistent premise set fp; :pg (ex falso quodlibet). Hewitt [2] Direct Logic and Besnard and Hunter's quasi-classical logic (QC) [1, 5, 4] both seek to preserve the deductive power of classical logic \as much as pos- sible" while still avoiding explosiveness. Their work fits into the ongoing research program of identifying some \reasonable" and \maximal" subsets of classically valid rules and axioms that do not lead to explosiveness. To this end, it is natural to consider which classically sound deductive rules and axioms one can introduce into a paraconsistent logic without causing explo- siveness. Hewitt [3] proposed including the law of excluded middle and the proof by self-refutation rule (a very special case of proof by contradiction) but did not show whether the resulting logic would be explosive. In this paper, I show that for quasi-classical logic and its variant, the addition of either the law of excluded middle or the proof by self-refutation rule in fact leads to explosiveness.
  • Expressive Completeness

    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.
  • Application: Digital Logic Circuits

    Application: Digital Logic Circuits

    SECTION 2.4 Application: Digital Logic Circuits Copyright © Cengage Learning. All rights reserved. Application: Digital Logic Circuits Switches “in series” Switches “in parallel” Change closed and on are replaced by T, open and off are replaced by F? Application: Digital Logic Circuits • More complicated circuits correspond to more complicated logical expressions. • This correspondence has been used extensively in design and study of circuits. • Electrical engineers use language of logic when refer to values of signals produced by an electronic switch as being “true” or “false.” • Only that symbols 1 and 0 are used • symbols 0 and 1 are called bits, short for binary digits. • This terminology was introduced in 1946 by the statistician John Tukey. Black Boxes and Gates Black Boxes and Gates • Circuits: transform combinations of signal bits (1’s and 0’s) into other combinations of signal bits (1’s and 0’s). • Computer engineers and digital system designers treat basic circuits as black boxes. • Ignore inside of a black box (detailed implementation of circuit) • focused on the relation between the input and the output signals. • Operation of a black box is completely specified by constructing an input/output table that lists all its possible input signals together with their corresponding output signals. Black Boxes and Gates One possible correspondence of input to output signals is as follows: An Input/Output Table Black Boxes and Gates An efficient method for designing more complicated circuits is to build them by connecting less complicated black box circuits. Gates can be combined into circuits in a variety of ways. If the rules shown on the next page are obeyed, the result is a combinational circuit, one whose output at any time is determined entirely by its input at that time without regard to previous inputs.
  • Contradiction Or Non-Contradiction?  Hegel’S Dialectic Between Brandom and Priest

    Contradiction Or Non-Contradiction?  Hegel’S Dialectic Between Brandom and Priest

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Padua@research CONTRADICTION OR NON-CONTRADICTION? HEGEL’S DIALECTIC BETWEEN BRANDOM AND PRIEST by Michela Bordignon Abstract. The aim of the paper is to analyse Brandom’s account of Hegel’s conception of determinate negation and the role this structure plays in the dialectical process with respect to the problem of contradiction. After having shown both the merits and the limits of Brandom’s account, I will refer to Priest’s dialetheistic approach to contradiction as an alternative contemporary perspective from which it is possible to capture essential features of Hegel’s notion of contradiction, and I will test the equation of Hegel’s dialectic with Priest dialetheism. 1. Introduction According to Horstmann, «Hegel thinks of his new logic as being in part incompatible with traditional logic»1. The strongest expression of this new conception of logic is the first thesis of the work Hegel wrote in 1801 in order to earn his teaching habilitation: «contradictio est regula veri, non contradictio falsi»2. Hegel seems to claim that contradictions are true. The Hegelian thesis of the truth of contradiction is highly problematic. This is shown by Popper’s critique based on the principle of ex falso quodlibet: «if a theory contains a contradiction, then it entails everything, and therefore, indeed, nothing […]. A theory which involves a contradiction is therefore entirely useless I thank Graham Wetherall for kindly correcting a previous English translation of this paper and for his suggestions and helpful remarks. Of course, all remaining errors are mine.
  • Shalack V.I. Semiotic Foundations of Logic

    Shalack V.I. Semiotic Foundations of Logic

    Semiotic foundations of logic Vladimir I. Shalack abstract. The article offers a look at the combinatorial logic as the logic of signs operating in the most general sense. For this it is proposed slightly reformulate it in terms of introducing and replacement of the definitions. Keywords: combinatory logic, semiotics, definition, logic founda- tions 1 Language selection Let’s imagine for a moment what would be like the classical logic, if we had not studied it in the language of negation, conjunction, dis- junction and implication, but in the language of the Sheffer stroke. I recall that it can be defined with help of negation and conjunction as follows A j B =Df :(A ^ B): In turn, all connectives can be defined with help of the Sheffer stroke in following manner :A =Df (A j A) A ^ B =Df (A j B) j (A j B) A _ B =Df (A j A) j (B j B) A ⊃ B =Df A j (B j B): Modus ponens rule takes the following form A; A j (B j B) . B 226 Vladimir I. Shalack We can go further and following the ideas of M. Sch¨onfinkel to define two-argument infix quantifier ‘jx’ x A j B =Df 8x(A j B): Now we can use it to define Sheffer stroke and quantifiers. x A j B =Df A j B where the variable x is not free in the formulas A and B; y x y 8xA =Df (A j A) j (A j A) where the variable y is not free in the formula A; x y x 9xA =Df (A j A) j (A j A) where the variable y is not free in the formula A.
  • Notes on Proof Theory

    Notes on Proof Theory

    Notes on Proof Theory Master 1 “Informatique”, Univ. Paris 13 Master 2 “Logique Mathématique et Fondements de l’Informatique”, Univ. Paris 7 Damiano Mazza November 2016 1Last edit: March 29, 2021 Contents 1 Propositional Classical Logic 5 1.1 Formulas and truth semantics . 5 1.2 Atomic negation . 8 2 Sequent Calculus 10 2.1 Two-sided formulation . 10 2.2 One-sided formulation . 13 3 First-order Quantification 16 3.1 Formulas and truth semantics . 16 3.2 Sequent calculus . 19 3.3 Ultrafilters . 21 4 Completeness 24 4.1 Exhaustive search . 25 4.2 The completeness proof . 30 5 Undecidability and Incompleteness 33 5.1 Informal computability . 33 5.2 Incompleteness: a road map . 35 5.3 Logical theories . 38 5.4 Arithmetical theories . 40 5.5 The incompleteness theorems . 44 6 Cut Elimination 47 7 Intuitionistic Logic 53 7.1 Sequent calculus . 55 7.2 The relationship between intuitionistic and classical logic . 60 7.3 Minimal logic . 65 8 Natural Deduction 67 8.1 Sequent presentation . 68 8.2 Natural deduction and sequent calculus . 70 8.3 Proof tree presentation . 73 8.3.1 Minimal natural deduction . 73 8.3.2 Intuitionistic natural deduction . 75 1 8.3.3 Classical natural deduction . 75 8.4 Normalization (cut-elimination in natural deduction) . 76 9 The Curry-Howard Correspondence 80 9.1 The simply typed l-calculus . 80 9.2 Product and sum types . 81 10 System F 83 10.1 Intuitionistic second-order propositional logic . 83 10.2 Polymorphic types . 84 10.3 Programming in system F ...................... 85 10.3.1 Free structures .
  • Boolean Algebra

    Boolean Algebra

    Boolean Algebra Definition: A Boolean Algebra is a math construct (B,+, . , ‘, 0,1) where B is a non-empty set, + and . are binary operations in B, ‘ is a unary operation in B, 0 and 1 are special elements of B, such that: a) + and . are communative: for all x and y in B, x+y=y+x, and x.y=y.x b) + and . are associative: for all x, y and z in B, x+(y+z)=(x+y)+z, and x.(y.z)=(x.y).z c) + and . are distributive over one another: x.(y+z)=xy+xz, and x+(y.z)=(x+y).(x+z) d) Identity laws: 1.x=x.1=x and 0+x=x+0=x for all x in B e) Complementation laws: x+x’=1 and x.x’=0 for all x in B Examples: • (B=set of all propositions, or, and, not, T, F) • (B=2A, U, ∩, c, Φ,A) Theorem 1: Let (B,+, . , ‘, 0,1) be a Boolean Algebra. Then the following hold: a) x+x=x and x.x=x for all x in B b) x+1=1 and 0.x=0 for all x in B c) x+(xy)=x and x.(x+y)=x for all x and y in B Proof: a) x = x+0 Identity laws = x+xx’ Complementation laws = (x+x).(x+x’) because + is distributive over . = (x+x).1 Complementation laws = x+x Identity laws x = x.1 Identity laws = x.(x+x’) Complementation laws = x.x +x.x’ because + is distributive over .
  • CS 173: Discrete Structures, Spring 2012 Homework 1 Solutions

    CS 173: Discrete Structures, Spring 2012 Homework 1 Solutions

    CS 173: Discrete Structures, Spring 2012 Homework 1 Solutions This homework contains 3 problems worth a total of 25 points. 1. [1 point] Finding information on Piazza How many pet rats does Margaret’s family have? (Hint: see Piazza.) Solution: Margaret’s family has six pet rats. 2. [4 points] Demographic Survey 3. [20 points] Logic operators The late 19th century philosopher Charles Peirce (rhymes with ‘hearse,’ not ‘fierce’) wrote about a set of logically dual operators and, in his writings, coined the term ‘Ampheck’ to describe them. The two most common Ampheck operators, the Peirce arrow (written ↓ or ⊥ or ∨ by different people) and the Sheffer stroke (written ↑ or | or ∧ by different people), are defined by the following truth table: p q p ↑ q p ↓ q T T F F T F T F F T T F F F T T (a) (4 points) The set of operators {∧, ∨, ¬} is functionally complete, which means that every logical statement can be expressed using only these three operators. Is the smaller set of operators {∨, ¬} also functionally complete? Explain why or why not. Solution: By De Morgan’s laws, ¬(¬p∨¬q) ≡ ¬¬p∧¬¬q ≡ p∧q So every logical statement using the ∧ operator can be rewritten in terms of the ∨ and ¬ operators. Since every logical statement can be expressed in terms of the ∧, ∨, and ¬ operators, this implies that every logical statement can be expressed in terms of the ∨ and ¬ operators, and so {∨, ¬} is functionally complete. (b) (4 points) Express ¬p using only the Sheffer stroke operation ↑. Solution: Note that ¬p is true if and only if p is false.
  • Boolean Logic

    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.
  • Logic, Proofs

    Logic, Proofs

    CHAPTER 1 Logic, Proofs 1.1. Propositions A proposition is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: “Paris is in France” (true), “London is in Denmark” (false), “2 < 4” (true), “4 = 7 (false)”. However the following are not propositions: “what is your name?” (this is a question), “do your homework” (this is a command), “this sentence is false” (neither true nor false), “x is an even number” (it depends on what x represents), “Socrates” (it is not even a sentence). The truth or falsehood of a proposition is called its truth value. 1.1.1. Connectives, Truth Tables. Connectives are used for making compound propositions. The main ones are the following (p and q represent given propositions): Name Represented Meaning Negation p “not p” Conjunction p¬ q “p and q” Disjunction p ∧ q “p or q (or both)” Exclusive Or p ∨ q “either p or q, but not both” Implication p ⊕ q “if p then q” Biconditional p → q “p if and only if q” ↔ The truth value of a compound proposition depends only on the value of its components. Writing F for “false” and T for “true”, we can summarize the meaning of the connectives in the following way: 6 1.1. PROPOSITIONS 7 p q p p q p q p q p q p q T T ¬F T∧ T∨ ⊕F →T ↔T T F F F T T F F F T T F T T T F F F T F F F T T Note that represents a non-exclusive or, i.e., p q is true when any of p, q is true∨ and also when both are true.