discreet math

8/9/2019 Discreet Math

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I. PROPOSITIONAL LOGIC

Propositional logic, also known as sentential logic and statement logic, isthe branch of logic that studies ways of joining and/or modifying entirepropositions, statements or sentences to form more complicatedpropositions, statements or sentences, as well as the logical relationshipsand properties that are derived from these methods of combining oraltering statements. In propositional logic, the simplest statements areconsidered as indivisible units, and hence, propositional logic does notstudy those logical properties and relations that depend upon parts ofstatements that are not themselves statements on their own, such as thesubject and predicate of a statement. The most thoroughly researchedbranch of propositional logic is classical truth-functional propositionallogic, which studies logical operators and connectives that are used toproduce complex statements whose truth-value depends entirely on thetruth-values of the simpler statements making them up, and in which it isassumed that every statement is either true or false and not both.

However, there are other forms of propositional logic in which other truth-values are considered, or in which there is consideration of connectivesthat are used to produce statements whose truth-values depend notsimply on the truth-values of the parts, but additional things such as theirnecessity, possibility or relatedness to one another.

The Facts

A statement or proposition is defined as a declarative sentence, or part ofa sentence, that is either true or false (this ability to be either true or falseis defined as having a truth-value), according to IEP. An example of astatement is "Washington, D.C. is the capitol of the United States." As aresult, propositional logic is a study of how certain statements can becombined or altered.

Features

Propositions can generally be combined in two ways. Statements can becombined using the word "or" and "and." The symbol for joiningstatements using "and" is an upside down V and the symbol for joiningstatements using "or" is V. A statement is logically valid when joined by an"and" when both components of the statement are true. For example, thefollowing argument is logically valid "Washington, D.C. is the capitol of theUnited States and Washington, D.C. has a population of over one million."Statements joined by "or" are logically valid if either one of the

components of the statement is true or if both are true.

Function

The truth value of a statement is defined as either its truth or falsity. Inparticular, all meaningful statement have truth values whether or not theyare simple statements or compound statements (those joined by "and" or"or").


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conclusion, the above argument could be symbolized in language PL asfollows:

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C & PP

In addition to statement letters like C and P and the operators, the onlyother signs that sometimes appear in the language PL are parentheseswhich are used in forming even more complex statements. Consider theEnglish compound sentence, Paris is the most important city in France ifand only if Paris is the capital of France and Paris has a population of overtwo million. If we use the letter M in language PL to mean that Paris isthe most important city in France, this sentence would be translated intoPL as follows:

I (C & P)

The parentheses are used to group together the statements C and Pand differentiate the above statement from the one that would be writtenas follows:

(I C) & P

This latter statement asserts that Paris is the most important city inFrance if and only if it is the capital of France, and (separate from this),Paris has a population of over two million. The difference between the two

is subtle, but important logically.

It is important to describe the syntax and make-up of statements in thelanguage PL in a precise manner, and give some definitions that will beused later on. Before doing this, it is worthwhile to make a distinctionbetween the language in which we will be discussing PL, namely, English,from PL itself. Whenever one language is used to discuss another, thelanguage in which the discussion takes place is called the metalanguage,and language under discussion is called the object language. In thiscontext, the object language is the language PL, and the metalanguage isEnglish, or to be more precise, English supplemented with certain special

devices that are used to talk about language PL. It is possible in English totalk about words and sentences in other languages, and when we do, weplace the words or sentences we wish to talk about in quotation marks.Therefore, using ordinary English, I can say that parler is a French verb,and I & C is a statement of PL. The following expression is part of PL, notEnglish:

(I C) & P


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However, the following expression is a part of English; in particular, it isthe English name of a PL sentence:

(I C) & P

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This point may seem rather trivial, but it is easy to become confused ifone is not careful.

In our metalanguage, we shall also be using certain variables that areused to stand for arbitrary expressions built from the basic symbols of PL.In what follows, the Greek letters , , and so on, are used for anyobject language (PL) expression of a certain designated form. Forexample, later on, we shall say that, if is a statement of PL, then so is

. Notice that itself is not a symbol that appears in PL; it is a symbolused in English to speak about symbols of PL. We will also be making useof so-called Quine corners, written and , which are a specialmetalinguistic device used to speak about object language expressionsconstructed in a certain way. Suppose is the statement (I C) and isthe statement (P & C); then v is the complex statement (I C) v (P & C).

Let us now proceed to giving certain definitions used in the metalanguagewhen speaking of the language PL.

Definition: A statement letterof PL is defined as any uppercase letterwritten with or without a numerical subscript.

Note: According to this definition, A, B, B2, C3, and P14 are examplesof statement letters. The numerical subscripts are used just in case weneed to deal with more than 26 simple statements: in that case, we canuse P1 to mean something different than P2, and so forth.

Definition: A connective or operatorof PL is any of the signs , &, v,, and .

Definition: A well-formed formula (hereafter abbrevated as wff) of PL isdefined recursively as follows:

1. Any statement letter is a well-formed formula.2. If is a well-formed formula, then so is .3. If and are well-formed formulas, then so is ( & ) .4. If and are well-formed formulas, then so is ( v ) .5. If and are well-formed formulas, then so is ( ) .


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6. If and are well-formed formulas, then so is ( ) .7. Nothing that cannot be constructed by successive steps of (1)-(6) is a well-formed formula.

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Note: According to part (1) of this definition, the statement letters C, Pand M are wffs. Because C and P are wffs, by part (3), (C & P) is awff. Because it is a wff, and M is also a wff, by part (6), (M (C & P)) isa wff. It is conventional to regard the outermost parentheses on a wff asoptional, so that M (C & P) is treated as an abbreviated form of (M (C & P)). However, whenever a shorter wff is used in constructing a morecomplicated wff, the parentheses on the shorter wff are necessary.

The notion of a well-formed formula should be understood ascorresponding to the notion of a grammatically correct or properlyconstructed statement of language PL. This definition tells us, forexample, that (Q v R) is grammatical for PL because it is a well-formed formula, whereas the string of symbols, )Qv(P&, whileconsisting entirely of symbols used in PL, is not grammatical because it isnot well-formed.

Truth Table Propositional Forms

So far we have in effect described the grammarof language PL. Whensetting up a language fully, however, it is necessary not only to establishrules of grammar, but also describe the meanings of the symbols used inthe language. We have already suggested that uppercase letters are usedas complete simple statements. Because truth-functional propositionallogic does not analyze the parts of simple statements, and only considersthose ways of combining them to form more complicated statements thatmake the truth or falsity of the whole dependent entirely on the truth orfalsity of the parts, in effect, it does not matter what meaning we assignto the individual statement letters like P, Q and R, etc., provided that

each is taken as either true or false (and not both).

However, more must be said about the meaning or semantics of thelogical operators &, v, , , and . As mentioned above, these areused in place of the English words, and, or, if then, if and only if,and not, respectively. However, the correspondence is really only rough,because the operators of PL are considered to be entirelytruth-functional,


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whereas their English counterparts are not always used truth-functionally.Consider, for example, the following statements:

1. If Bob Dole is president of the United States in 2004, then thepresident of the United States in 2004 is a member of the Republican

party.2. If Al Gore is president of the United States in 2004, then thepresident of the United States in 2004 is a member of the Republicanparty.

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For those familiar with American politics, it is tempting to regard theEnglish sentence (1) as true, but to regard (2) as false, since Dole is aRepublican but Gore is not. But notice that in both cases, the simple

statement in the if part of the if then statement is false, and thesimple statement in the then part of the statement is true. This showsthat the English operator if then is not fully truth-functional.However, all the operators of language PL are entirely truth-functional, sothe sign , though similar in many ways to the English if then isnot in all ways the same. More is said about this operator below.

Since our study is limited to the ways in which the truth-values of complexstatements depend on the truth-values of the parts, for each operator, theonly aspect of its meaning relevant in this context is its associated truth-function. The truth-function for an operator can be represented as a table,

each line of which expresses a possible combination of truth-values for thesimpler statements to which the operator applies, along with the resultingtruth-value for the complex statement formed using the operator.

The signs &, v, , , and , correspond, respectively, to the truth-functions ofconjunction, disjunction, material implication, materialequivalence, and negation. We shall consider these individually.

Conjunction: The conjunction of two statements and , written in PLas ( & ) , is true if both and are true, and is false if either is falseor is false or both are false. In effect, the meaning of the operator &

can be displayed according to the following chart, which shows the truth-value of the conjunction depending on the four possibilities of the truth-values of the parts:

( & )

TTF

TFT

TFF


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is false, and is true if either is false or is true (or both). This truth-function generates the following chart:

( )

T

TFF

T

FTF

T

FTT

Because the truth of a statement of the form ( ) rules out thepossibility of being true and being false, there is some similaritybetween the operator and the English phrase, if then, which isalso used to rule out the possibility of one statement being true andanother false; however, is used entirely truth-functionally, and so, forreasons discussed earlier, it is not entirely analogous with if then inEnglish.

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If is false, then ( ) is regarded as true, whether or not there is anyconnection between the falsity of and the truth-value of . In astatement of the form, ( ) , we call theantecedent, and we call the consequent, and the whole statement ( ) is sometimes alsocalled a(material) conditional.

The sign is sometimes used instead of for material implication.

Material Equivalence: This truth-function is represented in language PLwith the sign . A statement of the form ( ) is regarded as true if and are either both true or both false, and is regarded as false if theyhave different truth-values. Hence, we have the following chart:

( )

TTFF

TFTF

TFFT

Since the truth of a statement of the form ( ) requires and tohave the same truth-value, this operator is often likened to the Englishphrase if and only if. Again, however, they are not in all ways alike,because is used entirely truth-functionally. Regardless of what and are, and what relation (if any) they have to one another, if both are false,( ) is considered to be true. However, we would not normally regardthe statement Al Gore is the President of the United States in 2004 if andonly if Bob Dole is the President of the United States in 2004 as true


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simply because both simpler statements happen to be false. A statementof the form ( ) is also sometimes referred to as a (material)binconditional.

The sign is sometimes used instead of for material equivalence.

Negation: The negation of statement , simply written in languagePL, is regarded as true if is false, and false if is true. Unlike the otheroperators we have considered, negation is applied to a single statement.The corresponding chart can therefore be drawn more simply as follows:

TF

FT

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The negation sign bears obvious similarities to the word not used inEnglish, as well as similar phrases used to change a statement fromaffirmative to negative or vice-versa. In logical languages, the signs ~ or are sometimes used in place of .

The five charts together provide the rules needed to determine the truth-

value of a given wff in language PL when given the truth-values of theindependent statement letters making it up. These rules are very easy toapply in the case of a very simple wff such as (P & Q). Suppose that Pis true, and Q is false; according to the second row of the chart given forthe operator, &, we can see that this statement is false.

However, the charts also provide the rules necessary for determining thetruth-value of more complicated statements. We have just seen that (P &Q) is false if P is true and Q is false. Consider a more complicatedstatement that contains this statement as a part, e.g., ((P & Q) R),and suppose once again that P is true, and Q is false, and furthersuppose that R is also false. To determine the truth-value of thiscomplicated statement, we begin by determining the truth-value of theinternal parts. The statement (P & Q), as we have seen, is false. Theother substatement, R, is true, because R is false, and reversesthe truth-value of that to which it is applied. Now we can determine thetruth-value of the whole wff, ((P & Q) R), by consulting the chartgiven above for . Here, the wff (P & Q) is our , and R is our ,and since their truth-values are F and T, respectively, we consult the third


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row of the chart, and we see that the complex statement ((P & Q) R)is true.

We have so far been considering the case in which P is true and Q andR are both false. There are, however, a number of other possibilities with

regard to the possible truth-values of the statement letters, P, Q andR. There are eight possibilities altogether, as shown by the following list:

P Q R

TTTTFFFF

TTFFTTFF

TFTFTFTF

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Strictly speaking, each of the eight possibilities above represents adifferent truth-value assignment, which can be defined as a possibleassignment of truth-values T or F to the different statement letters makingup a wff or series of wffs. If a wff has n distinct statement letters makingup, the number of possible truth-value assignments is 2n. With the wff,((P & Q) R), there are three statement letters, P, Q and R, and sothere are 8 truth-value assignments.

It then becomes possible to draw a chart showing how the truth-value of agiven wff would be resolved for each possible truth-value assignment. Webegin with a chart showing all the possible truth-value assignments for thewff, such as the one given above. Next, we write out the wff itself on thetop right of our chart, with spaces between the signs. Then, for each,truth-value assignment, we repeat the appropriate truth-value, T, or F,underneath the statement letters as they appear in the wff. Then, as the

truth-values of those wffs that are parts of the complete wff aredetermined, we write their truth-values underneath the logical sign that isused to form them. The final column filled in shows the truth-value of theentire statement for each truth-value assignment. Given the importanceof this column, we highlight it in some way. Here, we highlight it in yellow.

P Q R | ((P & Q) R)


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TTTTF

FFF

TTFFT

TFF

TFTFT

FTF

TTTTF

FFF

TTFFF

FFF

TTFFT

TFF

FTTTT

TTT

FTFTF

TFT

TFTFT

FTF

Charts such as the one given above are called truth tables. In classicaltruth-functional propositional logic, a truth table constructed for a givenwff in effects reveals everything logically important about that wff. Theabove chart tells us that the wff ((P & Q) R) can only be false if P,Q and R are all true, and is true otherwise.

II. PROVIDING TECHNIQUES IN PROPOSITIONAL LOGIC

A proof is a valid argument that establishes the truth of a mathematicalstatement. A proof can use the hypotheses of the theorem, if any, axiomsassumed to be true, and previously proven theorems.

11Using these ingredients and rules of inference, the final step of the proofestablishes the truth of the statement being proved.

Theoretically, a proof of a mathematical statement is no different than a

logically valid argument starting with some premises and ending with thestatement. However, in the real world such logically valid arguments canget so long and involved that they lose their "punch" and require toomuch time to verify.

In mathematics, the purpose of a proof is to convince the reader of theproof that there is a logically valid argument in the background. Both thewriter and the reader must be convinced that such an argument can beproduced if needed.

When you read proofs, you will often find the words "obviously" or"clearly." These words indicate that steps have been omitted that the

author expects the reader to be able to fill in. Unfortunately, thisassumption is often not warranted and readers are not at all sure how tofill in the gaps. We will assiduously try to avoid using these words and trynot to omit too many steps. However, if we included all steps in proofs,our proofs would often be excruciatingly long.

A. Methods of Proving


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We will survey the basic proof methods. In doing so, our examples toillustrate the techniques should not be very complicated, so we willrestrict them to fairly simple statements which do not need a great deal ofbackground to understand.

Direct Proof

A direct proofof a conditional statementp q is constructed when thefirst step is the assumption thatp is true; subsequent steps areconstructed using rules of inference, with the final step showing that qmust also be true. A direct proof shows that a conditional statementp qis true by showing that ifp is true, then q must also be true, so that thecombinationp true and q false never occurs. In a direct proof, we assumethatp is true and use axioms, definitions, and previously proventheorems, together with rules of inference, to show that q must also be

true. You will find that direct proofs of many results are quitestraightforward, with a fairly obvious sequence of steps leading from thehypothesis to the conclusion. However, directproofs sometimes requireparticular insights and can be quite tricky.

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The integer n is even if there exists an integer ksuch that n = 2k, and n isoddif there exists an integer ksuch that n = 2k+ 1. (Note that an integeris either even or odd, and no integer is both even and odd.)

In a directproofone starts with the premise (hypothesis) and proceeddirectly to the conclusion with a chain of implications. Most simple proofsare of this kind.

Proof by Contraposition

Direct proofs lead from the hypothesis of a theorem to the conclusion.They begin with the premises, continue with a sequence of deductions,

and end with the conclusion. However, we will see that attempts at directproofs often reach dead ends. We need other methods of provingtheorems of the form Vx(P(x) Q(x)). Proofs of theorems of this type thatare not direct proofs, that is, that do not start with the hypothesis and endwith the conclusion, are called indirect proofs.

An extremely useful type of indirect proof is known as proof bycontraposition. Proofs by contraposition make use of the fact that the


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conditional statementp q is equivalent to its contrapositive, q p.This means that the conditional statementp q can be proved byshowing that its contrapositive, q p, is true. In a proof bycontraposition ofp q, we take -.q as a hypothesis, and using axioms,definitions, and previously proven theorems, together with rules of

inference, we show that p

must follow.

Example:

If n2 is an odd integer, then n is odd.

Pf:

Suppose n is an even integer.There exists an integer k so that n = 2k.n2 = (2k)2 = 4k2 = 2(2k2)

Since 2k2 is an integer, n2 is even.

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Proofs by Contradiction

This proof method is based on the Law of the Excluded Middle.Essentially, if you can show that a statement cannot be false, then it mustbe true. In practice, you assume that the statement you are trying toprove is false and then show that this leads to a contradiction (anycontradiction).

This method can be applied to any type of statement, not just conditionalstatements. There is no way to predict what the contradiction will be. Themethod is wide-spread and is often found in short segments of larger

proofs.

In proof by contradiction (also known as reductio ad absurdum, Latin for"by reduction toward the absurd"), it is shown that if some statementwere so, a logical contradiction occurs, hence the statement must be notso. This method is perhaps the most prevalent of mathematical proofs.


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A famous example of a proof by contradiction shows that isan Irrational number:

Suppose that is a rational number, so where a and b are

non-zero integers with no common factor (definition of a rational number).

Thus, . Squaring both sides yields 2b2 = a2. Since 2 divides theleft hand side, 2 must also divide the right hand side (as they are equaland both integers). So a2 is even, which implies that a must also be even.So we can write a = 2c, where c is also an integer. Substitution into theoriginal equation yields 2b2 = (2c)2 = 4c2. Dividing both sides by 2yields b2 = 2c2. But then, by the same argument as before, 2 divides b2,so b must be even. However, ifa and b are both even, they share a factor,namely 2. This contradicts our assumption, so we are forced to conclude

that is an irrational number.

Proof of Biconditionals

A proof of a P Q statement usually uses the tautology P Q (P Q) (Q P) That is, we prove an iff statement by separately proving the "if"part and the "only if" part.

Example:

Integer a is odd if and only if a+1 is even.

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PF:

(Sufficiency, if a is odd then a+1 is even)

Suppose a is an odd integer.There exists an integer k so that a = 2k + 1.a+1 = (2k+1) + 1 = 2k+2 = 2(k+1)

Since k+1 is an integer, a+1 is even.

Example:

Integer a is odd if and only if a+1 is even.

PF:


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(Necessity, if a+1 is even then a is odd)Suppose a+1 is an even integer.There exists an integer k so that a+1 = 2k.a = a + 1 1 = (2k) - 1 = (2(k-1) + 2) 1 = 2(k-1) + 1Since k-1 is an integer, a is odd.

Uniqueness Proof

Proofs of existentially quantified statements (x P(x)) can be constructive in which case you produce an x which makes P(x) true, or non-constructive when you use contradiction to show that ~(x P(x)) is false.

Definition: To say that there is one and only one x which makes the

predicate P(x) true, we write (!x P(x)) (there exists a unique x such thatP(x)).

To prove a (!x P(x)) statement, we first prove (x P(x)) and then showthat if P(x) and P(y) are both true, we must have x = y.

Definition: Let a and b be two positive integers. If n is a positive integerand a|n and b|n, then we call n a common multiple of a and b. If n is acommon multiple of a and b, and if for every other common multiple, m,of a and b we have that n|m, we say that n is a least common multipleof a and b. In this case, we write n = LCM(a,b).

Example:

For all positive integers a and b, LCM(a,b) is unique.

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PF:

(We shall omit the proof of the existence of the LCM and just show it'suniqueness, assuming that it exists.)

Let a and b be positive integers.

Suppose m1 and m2 are two LCM's for a and b.

Since m1 is an LCM and m2 is a common multiple, m1|m2, so m1 m2.

Since m2 is an LCM and m1 is a common multiple, m2|m1, so m2 m1.


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Therefore, m1 = m2.

B. Tautologies, Logical Equivalence and Validity

Truth-functional propositional logic concerns itself only with those ways ofcombining statements to form more complicated statements in which thetruth-values of the complicated statements depend entirely on the truth-values of the parts. Owing to this, all those features of a complexstatement that are studied in propositional logic derive from the way inwhich their truth-values are derived from those of their parts. Thesefeatures are therefore always represented in the truth table for a givenstatement.

Some complex statements have the interesting feature that they would betrue regardless of the truth-values of the simple statements making them

up. A simple example would be the wff P v P; i.e., P or not P. It isfairly easy to see that this statement is true regardless of whether P istrue or P is false. This is also shown by its truth table:

P | P v P

TF

TF

TT

FT

TF

There are, however, statements for which this is true but it is not soobvious. Consider the wff, R ((P Q) v (R Q)). This wff also comes

out as true regardless of the truth-values of P, Q and R.

P Q R | R ((P Q) V (R Q))

TTTTFFFF

TTFFTTFF

TFTFTFTF

TFTFTFTF

TTTTTTTT

TTTTFFFF

TTFFTTTT

TTFFTTFF

TTTFTTTT

FFTFFFTF

TFTFTFTF

TTFTTTFT

TTFFTTFF

Statements that have this interesting feature are called tautologies. Letdefine this notion precisely.

Definition: a wff is a tautologyif and only if it is true for all possible truth-value assignments to the statement letters making it up.


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Tautologies are also sometimes called logical truths or truths oflogic because tautologies can be recognized as true solely in virtue of theprinciples of propositional logic, and without recourse to any additionalinformation.

On the other side of the spectrum from tautologies are statements thatcome out as false regardless of the truth-values of the simple statementsmaking them up. A simple example of such a statement would be the wffP & P; clearly such a statement cannot be true, as it contradicts itself.This is revealed by its truth table:

P | P & P

TF

TF

FF

FT

TF

To state this precisely:

Definition: a wff is a self-contradiction if and only if it is false for allpossible truth-value assignments to the statement letters making it up.

Another, more interesting, example of a self-contradiction is thestatement (P Q) & (Q P); this is not as obviously self-contradictory. However, we can see that it is when we consider its truthtable:

P Q | (P Q) & (Q P)

TTFF

TFTF

FTFF

TTFF

TFTT

TFTF

FFFF

FFTF

TFTF

TTFT

TTFF

A statement that is neither self-contradictory nor tautological is calleda contingentstatement. A contingent statement is true for some truth-value assignments to its statement letters and false for others.

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The truth table for a contingent statement reveals which truth-valueassignments make it come out as true, and which make it come out asfalse. Consider the truth table for the statement (P Q) & (P Q):

P Q | (P Q) & (P Q)

T T T T T F T F F T


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TFF

FTF

TFF

FTT

FTF

FTT

TFF

TTT

TFT

FTF

We can see that of the four possible truth-value assignments for this

statement, two make it come as true, and two make it come out as false.Specifically, the statement is true when P is false and Q is true, andwhen P is false and Q is false, and the statement is false when P istrue and Q is true and when P is true and Q is false.

Truth tables are also useful in studying logical relationships that holdbetween two or more statements. For example, two statements are saidto be consistentwhen it is possible for both to be true, and are said tobe inconsistentwhen it is not possible for both to be true. In propositionallogic, we can make this more precise as follows.

Definition: two wffs are consistentif and only if there is at least onepossible truth-value assignment to the statement letters making them upthat makes both wffs true.

Definition: two wffs are inconsistentif and only if there is no truth-valueassignment to the statement letters making them up that makes themboth true.

Whether or not two statements are consistent can be determined bymeans of a combined truth table for the two statements. For example, thetwo statements, P v Q and (P Q) are consistent:

P Q | P v Q (P Q)

TTFF

TFTF

TTFF

TTTF

TFTF

TFFT

TTFF

FTTF

FTFT

TFTF

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Here, we see that there is one truth-value assignment, that in which bothP and Q are true, that makes both P v Q and (P Q) true.However, the statements (P Q) & P and (Q v P) are inconsistent,because there is no truth-value assignment in which both come out astrue.

P Q | (P Q) & P (Q v P))


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TTFF

TFTF

TTFF

TFTT

TFTF

TFFF

TTFF

FTFF

TFTF

TFTT

FFTT

TTFF

Another relationship that can hold between two statements is that ofhaving the same truth-value regardless of the truth-values of the simplestatements making them up. Consider a combined truth table for the wffsP Q and (Q & P):

P Q | P Q (Q & P))

TTFF

TFTF

FFTT

TTFF

TTFT

FTFT

TFTF

TTFT

TFTF

FFTF

FFTT

TTFF

Here we see that these two statements necessarily have the same truth-value.

Definition: two statements are said to be logically equivalentif and onlyif all possible truth-value assignments to the statement letters makingthem up result in the same resulting truth-values for the wholestatements.

The above statements are logically equivalent. However, the truth tablegiven above for the statements P v Q and (P Q) show that they,on the other hand, are not logically equivalent, because they differ in

truth-value for two of the four possible truth-value assignments.

Finally, and perhaps most importantly, truth tables can be utilized todetermine whether or not an argument is logically valid. In general, anargument is said to be logically valid whenever it has a form that makes itimpossible for the conclusion to be false if the premises are true. Inclassical propositional logic, we can give this a more precisecharacterization.

Definition: a wff is said to be a logical consequence of a set of wffs 1,2, , n, if and only if there is no truth-value assignment to the statement

letters making up these wffs that makes all of 1, 2, , ntrue but doesnot make true.

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An argument is logically validif and only if its conclusion is a logicalconsequence of its premises. If an argument whose conclusion is and


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whose only premise is is logically valid, then is said to logicallyimply.

For example, consider the following argument:

P QQ PQ

We can test the validity of this argument by constructing a combined truthtable for all three statements.

P Q | P Q Q P Q

TTF

F

TFT

F

TTF

F

TFT

T

TFT

F

FTF

T

TFT

F

TTT

F

TTF

F

TFT

F

Here we see that both premises come out as true in the case in whichboth P and Q are true, and in which P is false but Q is true. However,in those cases, the conclusion is also true. It is possible for the conclusionto be false, but only if one of the premises is false as well. Hence, we cansee that the inference represented by this argument is truth-

preserving. Contrast this with the following example:

P QQ v P

Consider the truth-value assignment making both P and Q true. If wewere to fill in that row of the truth-value for these statements, we wouldsee that P Q comes out as true, but Q v P comes out as false.Even if P and Q are not actually both true, it ispossible for them to bothbe true, and so this form of reasoning is not truth-preserving. In otherwords, the argument is not logically valid, and its premise does notlogically imply its conclusion.

One of the most striking features of truth tables is that they providean effective procedure for determining the logical truth, or tautologyhoodof any single wff, and for determining the logical validity of any argumentwritten in the language PL. The procedure for constructing such tables ispurely rote, and while the size of the tables grows exponentially with thenumber of statement letters involved in the wff(s) under consideration,the number of rows is always finite and so it is in principle possible tofinish the table and determine a definite answer. In sum, classicalpropositional logic is decidable.

19


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Rules of Displacement/Inference Rules

a. Natural Deduction

Truth tables, as we have seen, can theoretically be used to solve anyquestion in classical truth-functional propositional logic. However, thismethod has its drawbacks. The size of the tables grows exponentially withthe number of distinct statement letters making up the statementsinvolved. Moreover, truth tables are alien to our normal reasoningpatterns. Another method for establishing the validity of an argumentexists that does not have these drawbacks: the method of naturaldeduction. In natural deduction an attempt is made to reduce thereasoning behind a valid argument to a series of steps each of which isintuitively justified by the premises of the argument or previous steps inthe series.

Consider the following argument stated in natural language:

Either cat fur or dog fur was found at the scene of the crime. If dog furwas found at the scene of the crime, officer Thompson had an allergyattack. If cat fur was found at the scene of the crime, then Macavity isresponsibile for the crime. But officer Thompson didnt have an allergyattack, and so therefore Macavity must be responsible for the crime.

The validity of this argument can be made more obvious by representingthe chain of reasoning leading from the premises to the conclusion:

1. Either cat fur was found at the scene of the crime, or dog fur wasfound at the scene of the crime.(Premise)2. If dog fur was found at the scene of the crime, then officerThompson had an allergy attack.(Premise)3. If cat fur was found at the scene of the crime, then Macavity isresponsible for the crime. (Premise)4. Officer Thompson did not have an allergy attack. (Premise)5. Dog fur was not found at the scene of the crime. (Follows from 2and 4.)6. Cat fur was found at the scene of the crime. (Follows from 1 and 5.)7. Macavity is responsible for the crime. (Conclusion. Follows from 3and 6.)

Above, we do not jump directly from the premises to the conclusion, butshow how intermediate inferences are used to ultimately justify theconclusion by a step-by-step chain. Each step in the chain represents asimple, obviously valid form of reasoning.


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In this example, the form of reasoning exemplified in line 4 iscalled modus tollens, which involves deducing the negation of theantecedent of a conditional from the conditional and the negation of its

consequent. The form of reasoning exemplified in step 5 iscalled disjunctive syllogism, and involves deducing one disjunct of adisjunction on the basis of the disjunction and the negation of the otherdisjunct. Lastly, the form of reasoning found at line 7 is called modus

ponens, which involves deducing the truth of the consequent of aconditional given truth of both the conditional and its antecedent. Modus

ponensis Latin for affirming mode, and modus tollensis Latinfor denying mode.

A system ofnatural deduction consists in the specification of a list ofintuitively valid rules of inference for the construction ofderivations orstep-by-step deductions. Many equivalent systems of deduction have

been given for classical truth-functional propositional logic. In whatfollows, we sketch one system, which is derived from the popular textbookby Irving Copi (1953). The system makes use of the language PL.

b. Rules of Inference

Here we give a list of intuitively valid rules of inference. The rules arestated in schematic form. Any inference in which any wff of language PL issubstituted unformly for the schematic letters in the forms belowconstitutes an instance of the rule.

Modus ponens (MP):

(Modus ponens is sometimes also called modus ponendo ponens,

detachment or a form of -elimination.)

Modus tollens (MT):

(Modus tollens is sometimes also called modus tollendo tollens or a

form of -elimination.)


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Disjunctive syllogism (DS): (two forms)

v

v

(Disjunctive syllogism is sometimes also called modus tollendo ponens

or v-elimination.)

Addition (DS): (two forms)

v

v

(Addition is sometimes also called disjunction introduction or v-introduction.)

Simplification (Simp): (two forms)

&

&

(Simplification is sometimes also called conjunction elimination or &-elimination.)

Conjunction (Conj):

&

(Conjunction is sometimes also called conjunction introduction, &-introduction or logical multiplication.)


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Hypothetical syllogism (HS):

(Hypothetical syllogism is sometimes also called chain reasoning or

chain deduction.)

Constructive dilemma (CD):

( ) & ( ) v v

Absorption (Abs):

( & )

c. Rules of Replacement

The nine rules of inference listed above represent ways of inferringsomething new from previous steps in a deduction. Many systems ofnatural deduction, including those initially designed by Gentzen, consist

entirely of rules similar to the above. If the language of a system involvessigns introduced by definition, it must also allow the substitution of adefined sign for the expression used to define it, or vice versa. Still othersystems, while not making use of defined signs, allow one to make certainsubstitutions of expressions of one form for expressions of another form incertain cases in which the expressions in question are logically equivalent.These are called rules of replacement, and Copis natural deductionsystem invokes such rules. Strictly speaking, rules of replacement differfrom inference rules, because, in a sense, when a rule of replacement isused, one is not inferring something new but merely stating what amountsto the same thing using a different combination of symbols. In somesystems, rules for replacement can be derived from the inference rules,

but in Copis system, they are taken as primitive.

Rules of replacement also differ from inference rules in other ways.Inference rules only apply when the main operators match the patternsgiven and only apply to entire statements. Inference rules are also strictlyunidirectional: one must infer what is below the horizontal line from whatis above and not vice-versa.


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However, replacement rules can be applied to portions of statementsand not only to entire statements; moreover, they can be implemented ineither direction.

The rules of replacement used by Copi are the following:

Double negation (DN):

is interreplaceable with

(Double negation is also called -elimination.)

Commutativity (Com): (two forms)

& is interreplaceable with &

v is interreplaceable with v

Associativity (Assoc): (two forms)

( & ) & is interreplaceable with & ( & )( v ) v is interreplaceable with v ( v )

Tautology (Taut): (two forms)

is interreplaceable with & is interreplaceable with v

DeMorgans Laws (DM): (two forms)

( & ) is interreplaceable with v ( v ) is interreplaceable with &

Transposition (Trans):

is interreplaceable with

(Transposition is also sometimes called contraposition.)

Material Implication (Impl):

is interreplaceable with v

Exportation (Exp):

( ) is interreplaceable with ( & )


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Distribution (Dist): (two forms)

& ( v ) is interreplaceable with ( & ) v ( & )

v ( & ) is interreplaceable with ( v ) & ( v )

Material Equivalence (Equiv): (two forms)

is interreplaceable with ( ) & ( ) is interreplaceable with ( & ) v ( & )

(Material equivalence is sometimes also called biconditional

introduction/elimination or -introduction/elimination.)

d. Direct Deductions

A direct deduction of a conclusion from a set of premises consists of anordered sequence of wffs such that each member of the sequence iseither (1) a premise, (2) derived from previous members of the sequenceby one of the inference rules, (3) derived from a previous member of thesequence by the replacement of a logically equivalent part according tothe rules of replacement, and such that the conclusion is the final step ofthe sequence.

To be even more precise, a direct deduction is defined as an orderedsequence of wffs, 1, 2, , n, such that for each step i where i is

between 1 and n inclusive, either (1) i is a premise, (2) i matches theform given below the horizontal line for one of the 9 inference rules, andthere are wffs in the sequence prior to i matching the forms given abovethe horizontal line, (3) there is a previous step in the sequencej wherej < i and j differs from i at most by matching or containing apart that matches one of the forms given for one of the 10 replacementrules in the same place in whcih i contains the wff of the correspondingform, and such that the conclusion of the argument is n.

Using line numbers and the abbreviations for the rules of the system toannotate, the chain of reasoning given above in English, when transcribedinto language PL and organized as a direct deduction, would appear as

follows:

1. C v D Premise2. C O Premise3. D M Premise4. O Premise5. C 2,4 MT6. D 1,5 DS


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7. M 2,6 MP

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There is no unique derivation for a given conclusion from a given set ofpremises. Here is a distinct derivation for the same conclusion from thesame premises:

1. C v D Premise2. C O Premise3. D M Premise4. O Premise5. (C O) & (D M) 2,3 Conj6. O v M 1,5 CD7. M 4,6 DS

Consider next the argument:

P Q(S v T) QP v (T & R)T U

This argument has six distinct statement letters, and hence constructing atruth table for it would require 64 rows. The table would have 22 columns,thereby requiring 1,408 distinct T/F calculations. Happily, the derivation ofthe conclusion of the premises using our inference and replacement rules,while far from simple, is relatively less exhausting:

1. P Q Premise2. (S v T) Q Premise3. P v (T & R) Premise4. (P Q) & (Q P) 1 Equiv5. Q P 4 Simp6. (S v T) P 2,5 HS7. P (T & R) 3 Impl8. (S v T) (T & R) 6,7 HS9. (S v T) v (T & R) 8 Impl10. (S & T) v (T & R) 9 DM11. [(S & T) v T] & [(S & T) v R] 10 Dist

12. (S & T) v T 11 Simp13. T v (S & T) 12 Com14. (T v S) & (T v T) 13 Dist15. T v T 14 Simp16. T 15 Taut17. T v U 16 Add18. T U 17 Impl


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III.PREDICATE LOGIC AND QUANTIFIERS

Predicates

Consider the following syllogism.

Premise: All human beings are mortal.

Premise: Socrates is a human being.

Conclusion: Socrates is mortal.

The last statement seems an irrefutable conclusion of the premises, yetthe validity of this type of argument lies beyond the rules of sentential

logic. The key of the argument is the quantifier "all" that precedes the firstpremise.

Before we deal with quantifiers let's consider the arithmetic sentence "x+1 = 2". Here the letterxis called a variable since the symbol "x" apartfrom its position in the alphabet has no standard interpretation as adefinite object. In contrast, the symbols "1", "=", and "2" have specificmeanings. The first thing we need to specify for a variable isits Domain or Universe which is the collection of objects (i.e., a set) fromwhich a given variable takes its particular values. In "x+ 1 = 2" the mostreasonable domain is some set of numbers (more on these in Section V).In the sentence "zwas president of the United States in 1955" the domain

forzwould be a set of human beings.

In instances like the above two examples the variablesxandzare said tobe "free", in that any member of the domain is allowed to be substitutedinto the sentence. The sentence "x+ 1 = 2" could be represented by thesymbol S(x) . The sentence S(3) is the false proposition "3+ 1 = 2", whilethe sentence S(1) is the true proposition "1+ 1 = 2". A statement like S(x)with free variables is called a predicate or open sentence. Note: theresemblance to function notation is deliberate. The idea is that the truthof the proposition S(x) depends on or is a function of the variablex. Thus,some authors refer to predicates as propositional functions. If you wereasked to determine the truth value ofS(x), the question would be

meaningless. The statement is sometimes true (whenxis replaced by 1)and sometimes false (whenxis not replaced by 1). The truth ofS(x) is anopen question until a value forxis specified. Similarly, let P(z) be thepredicate "zwas president of the United States in 1955". Then P(DwightDavid Eisenhower) is true, P(John F. Kennedy) is false, and P(z) is open.


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Universal and Existential Quantifiers

When we introduce quantifiers like all, every, some, there exist, etc., infront of a predicate, the variables in the sentence are bound by thequantifier. The two quantifiers we will use are the Universal

Quantifier, , commonly read as "for all" or "for every", and

the Existential Quantifier, , commonly read as "there exists such

that" or "for some". For example, is interpreted to mean thesentence "there existszsuch thatzwas President of the United States in1955". This sentence is considered closed, not open. In fact, it is certainlya true statement since Dwight David Eisenhower is just such an individual!If we wanted to be more specific as to the domain of this statement, we

could introduce the additional predicate H(x) "xis a human being" into theclosed sentence . Again this is a true statement since

substituting Dwight David Eisenhower forzmakes true.

Note: the predicate H(x) was defined using the variablexrather thanz.The choice of variable names in the definition of a predicate or in aquantified closed sentence is usually arbitrary. Such variables are oftencalled "dummy" variables, since the choice of the name used is immaterialto the interpretation of the sentence. Thus, the

statements and are equivalent.

The closed sentence is interpreted as "everything is botha human being and president of the United States in 1955". Clearly, forany domain having more members than Dwight David Eisenhower, this is

a false statement. Using a similar analysis, is true sincex= 1 does

the job, while is false since onlyx= 1 does the job.

The use of quantifiers and predicates is summarized as follows.

1. The closed sentence is true if and only ifQ(x) is true for everyvalue in the domain ofx.

2. The closed sentence is true if and only ifQ(x) is true for at leastone value in the domain ofx.

Obviously, , but in general the converse is false.


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Negation of Quantified Predicates

Let C(x) be the predicate "xis a citizen of the United States" and let T(x)be the predicate "xpays taxes". Then we probably suspect that the closed

sentence is false. Not every US citizen pays taxes. This

means of course is true. What is the correctinterpretation of this negation? It is certainly not thestatement

, whichcan be rendered as "everyone is both a US citizen and doesn't pay

taxes"! is false if and only if we can find at least one US

citizen who doesn't pay taxes. Hence, is true if andonly if we can find at least one US citizen who doesn't pay taxes. Thus, weare lead to the following equivalent statements.

It's not the case that every US citizen pays taxes if and only if there is atleast one US citizen who doesn't pay taxes.

In general, and by similar

reasoning .

Multiple Quantifiers

Often we need predicates with more than one variable. Consider theclosed sentence "Every team wants to win all of its games". Let T(x) bethe predicate "xis a team", G(x,y) the predicate "yis a game played byx"and W(x,y) the predicate "xwants to winy". The above sentence can be

symbolized as . Note: The

sentence , which could be rendered as

"In every game played by any team, the team wants to win", is identical in

meaning to . In general, for any

predicate Q(x,y), .


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As a second example,

let S(x,y) be the predicate "the sum ofxandyis zero". Let the

domains of bothxandybe the set of integers .Now consider the following four closed sentences.

1. "The sum of every two integers iszero."

2. "There are two integers whose sum iszero."

3. "There is an integer whose sum withany integer is zero."

4. "Every integer has an opposite."

Statement 2 is true. For example, is such a pair.

Statement 4 is true. For any .

Statement 1 is clearly false. When .

Statement 3 is also false. There is no integer that acts as the opposite forall the integers. Note: The corresponding statement for multiplication,

, is true, since is just such anx.

Thus, we see that for any predicate,

and .

However, is not equivalent to . In

fact, , but the converse need not be true.


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For example, Consider the predicate M(x,y) "xis a human being, andyisa human being, andyis the mother ofx". Then the closed

sentence could be rendered as "Everyone has a biologicalmother". While this statement might not be true due to advances incloning technology, it is at least plausible. The closed

sentence is the absurdity "There is a person who iseveryone's biological mother".

The rules of negation for multiple quantifiers follow from repeatedapplication of the rules for negating a single quantifier. The results arestated and paraphrased below.

It's not true that allx,ypairs make Q(x,y) true if and only if you can findat least onex,ypair that makes Q(x,y) false.

It's not true that for everyxyou can find aythat makes Q(x,y) true if andonly if there is at least onexvalue for which allyvalues make Q(x,y)false.

It's not true that you can find anxvalue that for everyyvaluemakes Q(x,y) true if and only if for everyxvalue you can find ayvaluethat makes Q(x,y) false.

It's not true that you can find anx,ypair that makes Q(x,y) true if andonly if everyx,ypair makes Q(x,y) false.

Unique Existence

A third quantifier often used is unique existence . The

sentence is read as "There exists a uniquexsuch that Q(x)". Thismeans that there is one and only one value in the domain ofxthat makesthe predicate Q(x) true. Using the notion of equality, i.e.,x= yif and onlyifxandyare the same object, one has the following equivalence.


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discreet math

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