Business Mathematics Logic

BUSINESS MATHEMATICS LOGIC

MOHAMMED SHADAB

(Reg No: CC0018BK33AM25AAB)

GREAT EASTERN MANAGEMENT SCHOOL, BANGALORE

2009-2010

BUSINESS MATHEMATICS LOGIC

CERTIFICATE

This is to certify that the Project work “ LOGIC”

Is submitted to the college by the candidate MOHAMMED SHADAB bearing Reg No: CC0018BK33AM25AAB

Is the product of bonafide research carried out by the candidate

Under my supervision in BUSINESS MATHEMATICS.

(GUIDE)

DR.G.S.HEDGEBANGALORE Lecturer, Business Mathematics Great Eastern Management SchoolSEP 2009

Acknowledgment

The Project work was carried out under the remarkable guidance of

Lecturer, Great Eastern Management School. I am grateful for his guidance, valuable

I also express my sincere gratitude and thanks to all the subjects participated in the study.

CONTENTS

1. Statements and Logical Operators Exercises for Section 12. 2. Logical Equivalence, Tautologies and Contradictions Exercises

for Section 2

3. The Conditional and the Biconditional Exercises for Section 3

4. Tautological Implications and Tautological Equivalences Exercises

for Section 4

5. Rules of Inference Exercises for Section 5

6. Arguments and Proofs Exercises for Section 6

7. Predicate Calculus Exercises for Section 7

1 Nature of logic o 1.1 Logical form o 1.2 Deductive and inductive reasoning o 1.3 Consistency, soundness, and completeness o 1.4 Rival conceptions of logic

2 History of logic 3 Topics in logic

o 3.1 Syllogistic logic o 3.2 Sentential (propositional) logic o 3.3 Predicate logic o 3.4 Modal logic o 3.5 Informal reasoning o 3.6 Mathematical logic o 3.7 Philosophical logic o 3.8 Logic and computation

4 Controversies in logic o 4.1 Bivalence and the law of the excluded middle o 4.2 Is logic empirical? o 4.3 Implication: strict or material? o 4.4 Tolerating the impossible o 4.5 Rejection of logical truth

INTRODUCTION

Logic, from the Greek λογική (logiké) is defined by the Penguin Encyclopedia to be "The formal systematic study of the principles of valid inference and correct reasoning". As a discipline, logic dates back to Aristotle, who established its fundamental place in philosophy. It became part of the classical trivium, a fundamental part of a classical education, and is now an integral part of disciplines such as mathematics, computer science, and linguistics.

Logic concerns the structure of statements and arguments, in formal systems of inference and natural language. Topics include validity, fallacies and paradoxes, reasoning using probability and arguments involving causality and time. Logic is also commonly used today in argumentation theory.

You have been assigned the job of evaluating the attempts of mortals to prove the existence of God. And many attempts there have been. Three in particular have caught your attention: they are known as the cosmological argument, the teleological argument, and the ontological argument.

Cosmological Argument (St. Thomas Aquinas): No effect can cause itself, but requires another cause. If there were no first cause, there would be an infinite sequence of preceding causes. Clearly there cannot be an infinite sequence of causes, therefore there is a first cause, and this is God.

Teleological Argument (St. Thomas Aquinas): All things in the world act towards an end. They could not do this without their being an intelligence that directs them. This intelligence is God.

Ontological Argument (St. Anselm): God is a being than which none greater can be thought. A being thought of as existing is greater than one thought of as not existing. Therefore, one cannot think of God as not existing, so God must exist.

Are these arguments valid?

Logic is the underpinning of all reasoned argument. The Greeks recognized its role in mathematics and philosophy, and studied it extensively. Aristotle, in his Organon, wrote the first systematic treatise on logic. His work in particular had a heavy influence on philosophy, science and religion through the Middle Ages. But Aristotle's logic was logic expressed in ordinary language, so was still subject to the ambiguities of natural languages. Philosophers began to want to express logic more formally and symbolically, in the way that mathematics is written (Leibniz, in the 17th century, was probably the first to envision and call for such a formalism). It was with the publication in 1847 of G. Boole's The Mathematical Analysis of Logic and A. DeMorgan's Formal Logic that symbolic logic came into being, and logic became recognized as part of mathematics. This also marked the recognition that mathematics is not just about numbers (arithmetic) and shapes (geometry), but encompasses any subject that can be expressed symbolically with precise rules of manipulation of those symbols. It is symbolic logic that we shall study in this chapter.

Since Boole and DeMorgan, logic and mathematics have been inextricably intertwined. Logic is part of mathematics, but at the same time it is the language of mathematics. In the late 19th and early 20th century it was believed that all of mathematics could be reduced to symbolic logic and made purely formal. This belief, though still held in modified form today, was shaken by K. Gödel in the 1930's, when he showed that there would always remain truths that could not be derived in any such formal system. We'll mention more about this as we go along.

The study of symbolic logic is usually broken into several parts. The first and most fundamental is the propositional calculus, and this is the subject of most of this web text. Built on top of this is the predicate calculus, which

is the language of mathematics. We shall study the propositional calculus in the first six sections and look at the predicate calculus briefly in the last two.

1. Statements and Logical Operators

This on-line text is, for the most part, devoted to the study of so-called Propositional Calculus. Contrary to what the name suggests, this has nothing to do with the subject most people associate with the word "calculus." Actually, the term "calculus" is a generic name for any area of mathematics that concerns itself with calculating. For example, arithmetic could be called the calculus of numbers. Propositional Calculus is then the calculus of propositions. A proposition, or statement, is any declarative sentence, which is either true (T) or false (F). We refer to T or F as the truth-value of the statement.

Example 1 Propositions

The sentence "2+2 = 4" is a statement, since it can be either true or false. Since it happens to be a true statement, its truth value is T.

The sentence "1 = 0" is also a statement, but its truth value is F.

"It will rain tomorrow" is a proposition. For its truth value we shall have to wait for tomorrow.

The following statement might well be uttered by a Zen Master to a puzzled disciple: "If I am Buddha, then I am not Buddha." This is a statement which, we shall see later on, really amounts to the simpler statement "I am not Buddha." As long as the speaker is not Buddha, this is true.

"Solve the following equation for x" is not a statement, as it cannot be assigned any truth value whatsoever. (It is an imperative, or command, rather than a declarative sentence.)

"The number 5" is not a proposition, since it is not even a complete sentence.

"Mars is not a planet"

is a proposition with truth value T

is a proposition with truth value F

is not a proposition

.

"Ode to Spring"

is a proposition with truth value T

is a proposition with truth value F

is not a proposition

.

"60 = 1"

is a proposition with truth value T

is a proposition with truth value F

is not a proposition

Example 1B Self-Referential Sentences

"This statement is false" gets us into a bind: If it were true, then, since it is declaring itself to be false, it must be false. On the other hand, if it were false, then it’s declaring itself false is a lie, so it is true! In other words, if it is true, then it is false, and if it is false, then it is true, and we go around in

circles. We get out of this bind by refusing to accord it the privileges of statement hood. In other words, it is not a statement. An equivalent pseudo-statement is: "I am lying," so we call this liar's paradox.

"This statement is true" may seem like a statement, but there is no way that its truth value can ever be determined: is it true, or is it false? We thus disqualify it as well. (In fact, it is the negation of the liar's paradox; see below for a discussion of negation.)

Sentences such as these are called self-referential sentences, since they refer to themselves.

Here are some rather amusing (and slightly disturbing) examples of self-referential sentences, the first two being taken from Douglas R. Hofstadter's Met magical Themas:

"This sentences no verb."

"This sentence was in the past tense."

"This sentence asserts absolutely nothing."

"While the last sentence had nothing to say, this sentence says a lot."

"This sentence has more to say than the last two sentences combined, if you count the number of words."

We shall use the letters p, q, r, s and so on to stand for propositions. Thus, for example, we might decide that p should stand for the proposition "the moon is round." We shall write

P: "the moon is round"

To express this. We read this as

P is the statment "the moon is round."

We can form new propositions from old ones in several different ways. For example, starting with p: "I am an Anchovian," we can form the negation of p: "It is not the case that I am an Anchovian" or simply "I am not an Anchovian." We denote the negation of p by ~p, read "not p." What we mean by this is that, if p is true, then ~p is false, and vice-versa. We can show this in the form of a truth table:

p ~p

T F

F T

On the left are the two possible truth values of p, with the corresponding truth values of ~p on the right. The symbol ~ is our first example of a logical operator.

Following is a more formal definition.

Negation

The negation of p is the statement ~p, which we read "not p." Its truth value is defined by the following truth table.

p ~p

T F

F T

The negation symbol "~" is an example of a unary logical operator (the term "unary" indicates that the operator acts on a single proposition).

Example 2 Negating Statements

Find the negations of the following propositions. (a) p: "2+2 = 4"(b) q: "1 = 0"(c) r: "Diamonds are a pearl's best friend."(d) s: "All the politicians in this town are crooks."

Solution

(a) ~p is the statement "it is not true that 2+2 = 4," or more simply, ~p: "2+2 4."

(b) ~q: "1 0."

(c) ~r: "Diamonds are not a pearl's best friend."

(d) ~s: " Not all the politicians in this town are crooks."

Notice that ~p is false, because p is true. However, ~q is true, because q is false. A statement of the form ~q can very well be true; it is a common mistake to think it must be false.

To say that diamonds are not a pearl's best friends is not to say that diamonds are a pearl's worst enemy. The negation is not the polar opposite, but whatever would deny the truth of the original statement. Similarly, saying that not all politicians are crooks is not the same as saying that no politicians are crooks, but is the same as saying that some politicians are not crooks. Negations of statements involving the quantifiers "all" or "some" are tricky. We'll study quantifiers in more depth when we discuss the predicate calculus.

Example 3 Conjunction

If p: "This galaxy will ultimately wind up in a black hole" and q: "2+2 = 4," what is p q?

Solution

p q: "This galaxy will ultimately disappear into a black hole and 2+2=4," or the more astonishing statement: "Not only will this galaxy ultimately disappear into a black hole, but 2+2 = 4!"

q is true, so if p is true then the whole statement p q will be true. On the other hand, if p is false, then the whole statement p q will be false.

Example 4 Combining Conjunction with Negation

With p and q as in Example 3, what does the statement p (~q) say?

Solution

p (~q) says: "This galaxy will ultimately disappear into a black hole and 2+2 4," or "Contrary to your hopes and aspirations, this galaxy is doomed to eventually disappear into a black hole; moreover, two plus two is decidedly different from four!"

Since ~q is false, the whole statement p (~q) is false (regardless of whether p is true or not).

Example 5 Combining Three Statements

Let p: "This topic is boring," q: "This whole web site is boring" and r: "Life is boring." Express the statement "Not only is this topic boring, but this whole web site is boring, and in fact life is boring (so there!)" in logical form.

Solution

The statement is asserting that all three statements p, q and r are true. (Note that "but" is simply an emphatic form of "and.") Now we can combine them all in two steps: Firstly, we can combine p and q to get p q, meaning "This topic is boring and this web site is boring." We can then conjoin this with r to get: (p q) r. This says: "This topic is boring, this web site is boring and life is boring." On the other hand, we could equally well have done it the other way around: conjoining q and r gives "This web site is boring and life is boring." We then conjoin p to get p (q r), which again says: "This topic is boring, this web site is boring and life is boring." We shall soon see that

(p q) r is logically the same as p (q r),a fact called the associative law for conjunction. Thus both answers (p q) r and p (q r) are equally valid. This is like saying that (1+2)+3 is the same as 1+(2+3). As with addition, we sometimes drop the parentheses and write p q r.

As we've just seen, there are many ways of expressing a conjunction in English. For example, if

p: "Waner drives a fast car" and

q: "Costenoble drives a slow car," the following are all ways of saying p q: Waner drives a fast car and Costenoble drives a slow car. Waner drives a fast car but Costenoble drives a slow car. Waner drives a fast car yet Costenoble drives a slow car. Although Waner drives a fast car, Costenoble drives a slow car.

Waner drives a fast car even though Costenoble drives a slow car. While Waner drives a fast car, Costenoble drives a slow car.

Any sentence that suggests that two things are both true is a conjunction. The use of symbolic logic strips away the elements of surprise or judgement that can also be expressed in an English sentence.

We now introduce a third logical operator. Starting once again with p: "I am clever," and q: "You are strong," we can form the statement "I am clever or you are strong," which we write symbolically as p q, read "p or q." Now in English the word "or" has several possible meanings, so we have to agree on which one we want here. Mathematicians have settled on the inclusive or: pq means p is true or q is true or both are true.  

With p and q as above, p q stands for "I am clever or you are atrong, or both." We shall sometimes include the phrase "or both" for emphasis, but even if we do not that is what we mean. We call p q the disjunction of p and q.

Disjunction

The disjunction of p and q is the statement p q, which we read "p or q." Its truth value is defined by the following truth table.

p q p q

T T T

T F T

F T T

F F F

This is the inclusive or, so p q is true when p is true or q is true or both are true.

Notice that the only way for the whole statement to be false is for both p and q to be false. For this reason we can say that p q also means "p and q are not both false." We'll say more about this in the next section.

The disjunction symbol " " is our second example of a binary

2. Logical Equivalence, Tautologies, and Contradictions

Example 1 Constructing a Truth Table

Construct the truth table for ~(p q).

Solution

Whenever we encounter a complex formula like this, we work from the inside out, just as we might do if we had to evaluate an algebraic expression, like -(a+b). Thus, we start with the p and q columns, then construct the p q column, and finally, the ~(p q) column:

p q p q ~(p q)

T T T F

T F F T

F T F T

F F F T

Notice how we get the ~(p q) column from the p q column: we reverse all its the truth values, since that is what negation means.

Example 2 Constructing a Truth Table

Construct the truth table for p (p q).

Solution

Since there are two variables, p and q, we again start with the p and q columns. Working from inside the parentheses, we then evaluate p q, and finally take the disjunction of the result with p:

p q p q p (p q)

T T T T

T F F T

F T F F

F F F F

Before we go on...

How did we get the last column from the others? Since we are "or-ing" p with p q, we must look at the values in the p and p q columns and "or" those together, according to the instructions for "or." Thus, for example, in the second row, we get T F = T, and in the third row, we get F F = F. (If you look at the second row of the truth table for "or" you will see T | F | T, and in the last row you will see F | F | F )

Example 3 Three Variables

Construct the truth table for ~(p q) (~r).

Solution

Here, there are three variables: p, q and r. Thus we start with three initial columns showing all eight possibilities:

p q r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

We now add columns for p q, ~(p q) and ~r, and finally ~(p q) (~r) according to the instructions for these logical operators. Here is how the table would grow as you construct it:

p q r p q

T T T T

T T F T

T F T F

T F F F

F T T F

F T F F

F F T F

F F F F

p q r p q ~(p q) ~r

T T T T F F

T T F T F T

T F T F T F

T F F F T T

F T T F T F

F T F F T T

F F T F T F

F F F F T T

and finally,

p q r p q ~(p q) ~r ~(p q) (~r)

T T T T F F F

T T F T F T F

T F T F T F F

T F F F T T T

F T T F T F F

F T F F T T T

F F T F T F F

F F F F T T T

3. The Conditional and the Biconditional

The Conditional

Consider the following statement: "If you earn an A in logic, then I'll buy you a Yellow Mustang." It seems to be made up out of two simpler statements:

p: "You earn an A in logic," and

q: "I will buy you a Yellow Mustang."

What the original statement is then saying is this: if p is true, then q is true, or, more simply, if p, then q. We can also phrase this as p implies q, and we write p q.

Now let us suppose for the sake of argument that the original statement: "If you earn an A in logic, then I'll buy you a Yellow Mustang," is true. This does not mean that you will earn an A in logic; all it says is that if you do so, then I will buy you that Yellow Mustang. Thinking of this as a promise, the only way that it can be broken is if you do earn an A and I do not buy you a Yellow Mustang. In general, we ue this idea to define the statement p q.

Conditional

The conditional p q, which we read "if p, then q" or "p implies q," is defined by the following truth table.

p q p q

T T T

T F F

F T T

F F T

The arrow " " is the conditional operator, and in p q the statement p is classed the antecedent, or hypothesis, and q is called the consequent, or conclusion.

Notice that the conditional is a new example of a binary logical operator -- it assigns to each pair of statments p and q the new statement p q.

Notes

1. The only way that p q can be false is if p is true and q is false-this is the case of the "broken promise."

2. If you look at the truth table again, you see that we say that "p q" is true when p is false, no matter what the truth value of q. This again makes sense in the context of the promise — if you don't get that A, then whether or not I buy you a Corvette, I have not broken my promise. However, it goes against the grain if you think of "if p then q" as saying that p causes q. The problem is that there are really many ways in which the English phrase "if ... then ..." is used. Logicians have simply agreed that the meaning given by the truth table above is the most useful for mathematics, and so that is the meaning we shall always use. Shortly we'll talk about other English phrases that we interpret as meaning the same thing.

Here are some examples that will help to explain each line in the truth table.

Example 1 (True Implies True) is True

If p and q are both true, then p q is true. For instance: If 1+1 = 2 then the sun rises in the east.

Here p: "1+1 = 2" and q: "the sun rises in the east."

Notice that the statements need not have anything to do with one another. We are not saying that the sun rises in the east because 1+1 = 2, simply that the whole statement is logically true.

Example 2 True Can't Imply False

If p is true and q is false, then p q is false. For instance: When it rains, I carry an umbrella.

Here p: "It is raining," and q: "I am carrying an umbrella." In other words, we can rephrase the sentence as: "If it is raining then I am carrying an umbrella." Now there are lots of days when it rains (p is true) and I forget to bring my umbrella (q is false). On any of those days the statement p q is clearly false.

Notice that we interpreted "When p, q" as "If p then q."

Example 3 False Implies Anything

If p is false, then p q is true, no matter whether q is true or not. For instance: If the moon is made of green cheese, then I am the King of England.

Here p: "the moon is made of green cheese," which is false, and q: "I am the King of England." The statement p q is true, whether or not the speaker happens to be the King of England (or whether, for that matter, there even is a King of England).

"If I had a million dollars I'd be on Easy Street." "Yeah, and if my grandmother had wheels she'd be a bus." The point of the retort is that, if the hypothesis is false, the whole implication is true.

4. Tautological Implications and Tautological Equivalences

Tautological Implications

In this section we enlarge our list of "standard" tautologies by adding ones involving the conditional and the biconditional. From now on, we use small letters like p and q to denote atomic statements only, and uppercase letters like A and B to denote statements of all types, compound or atomic.

We first look at some tautological implications, tautologies of the form A B. You should check the truth table of each of the statements we give to see that they are, indeed, tautologies.

Modus Ponens or Direct Reasoning

[(p q) p] q.

In words: If p implies q, and if p is true, then q must be true.

Example Letting p: "I love math" and q: "I will pass this course," we get

If my loving math implies that I will pass this course, and if I indeed love math, then I will pass this course.

Another way of setting this up is in the following argument form:  

If I love math, then I will pass this course.

I love math.

Therefore, I will pass this course.

In symbols:

p q

p

q

Notice that we draw a line in the argument form to separate what we are given from the conclusion that we draw. This tautology represents the most direct form of everyday reasoning, hence its name "direct reasoning." Another bit of terminology: We say that p q and p together logically imply q.

To check that it is a tautology, we use a truth table.

p q p q (p q) p [(p q) p] q

T T T T T

T F F F T

F T T F T

F F T F T

Once more, modus ponens says that, if we know that p implies q, and we know that p is indeed true, then we can conclude that q is also true. This is sometimes known as affirming the hypothesis. You should not confuse this with a fallacious argument like: "If I were an Olympic athlete then I would drink Boors. I do drink Boors, therefore I am an Olympic athlete." (Do you see why this is nonsense?) This is known as the fallacy of affirming the consequent. There is, however, a correct argument in which we deny the consequent:

Modus Tollens or Indirect Reasoning

[(p q) ~q] ~p

In words, if p implies q, and q is false, then so is p.

Example If we once again take p: "I love math" and q: "I will pass this course," we get

If I love math then I will pass this course; but I know that I will fail it. Therefore, I must not love math.

In argument form:

If I love math, then I will pass this course.

I will fail the course.

Therefore, I do not love math.

In symbols:

p q

~q

~p

As you can see, this argument is not quite so direct as that in the first example; it seems to contain a little twist: "If p were true then q would also be true. However, q is false. Therefore p must also be false (else q would be true.)" That is why we refer to it as indirect reasoning.

We'll leave the truth table for the exercises. Note that there is again a similar, but fallacious argument form to avoid: "If I were an Olympic athlete then I would drink Boors. However, I am not an Olympic athlete. Therefore I can't drink Boors." This is a mistake Boors sincerely hopes you do not make!

More tautoligical implications:

Simplification (p q) p

and(p q) q

In words, the first says: If both p and q are true, then, in particular, p is true.

Example If the sky is blue and the moon is round, then (in particular) the sky is blue.

Argument Form

The sky is blue and the moon is round.

Therefore, the sky is blue.

In symbols:

p q

p

The other simplification, (p q) q is similar.

Addition p (p q)

In words, the first says: If p is true, then we know that either p or q is true.

Example If the sky is blue, then either the sky is blue of some ducks are kangaroos.

Argument Form

The sky is blue.

Therefore, the sky is blue or some ducks are kangaroos.

In symbols:

p

p q

Notice that it doesn't matter what we use as q, nor does it matter whether it is true or false. The reason is that the disjunction p q is true if at least one of p or q is true. Since we start out knowing that p is true, the truth value of q doesn't matter.

WarningThe following are not tautologies:

(p q) p;

p (p q).

5. Rules of Inference

In the last section, we wrote out all our tautologies in what we called "argument form." For instance, Modus Ponens [(p q) p] q was represented as

p q p

q

We think of the statements above the line, the premises, as statements given to us as true, and the statement below the line, the conclusion, as a statement that must then also be true.

Our convention has been that small letters like p stand for atomic statements. But, there is no reason to restrict Modus Ponens to such statements. For example, we would like to be able to make the following argument:

If roses are red and violets are blue, then sugar is sweet and so are you.Roses are red and violets are blue.

Therefore, sugar is sweet and so are you.

In symbols, this is

(p q) (r s)p q

r s

So, we really should write Modus Ponens in the following more general and hence usable form:

A B A

B

where, as our convention has it, A and B can be any statements, atomic or compound.

In this form, Modus Ponens is our first rule of inference. We shall use rules of inference to assemble lists of true statements, called proofs. A proof is a way of showing how a conclusion follows from a collection of premises. Modus Ponens, in particular, allows us to say that, if A B and A both appear as statements in a proof, then we are justified in adding B as another statement in the proof.

Example 1 Applying Modus Ponens

Apply Modus Ponens to statements 1 and 3 in the following list of premises (that is, statements that we take to be true). 1. (p q) (r ~s) 2. ~r s 3. p q

Solution

Notice that all the statements are compound statements, and that they have the following patterns: 1. A B 2. C 3. A.

Statement A appears twice; in lines (1) and (3). Looking at Modus Ponens, we see that we can deduce B = r ~s from these lines. (Line (2) is not going to be used at all; it just goes along for the ride.) Thus, we can enlarge our list as follows:

1. (p q) (r ~s) Premise2. ~r s Premise3. p q Premise4. r (~s) 1,3 Modus Ponens

On the right we have given the justification for each line: lines (1) through (3) were given as premises, and line (4) follows by an application of Modus Ponens to lines (1) and (3); hence the justification "1,3 Modus Ponens."

The above list of four statements consitutes a proof that Statement 4 follows from premises 1-3, and we refer to it as a proof of the argument

(p q) (r ~s) Premise~r s Premisep q Premise

r (~s) Conclusion

Example 2 Using T1

Apply Modus Tollens to the following premises: 1. (p q) (r ~s) 2. ~(r ~s) 3. (p q) p

Solution

Looking at the given premises, we see the pattern: 1. A B 2. ~B 3. A C

As a rule of inference, Modus Tollens has the following form:

A B ~B

~A

(In words, if A B appears on the list, and if ~B also appears on the list, we can add ~A to the list of true statements.)

This matches the first two premises, so we can apply Modus Tollens to get the following.

1. (p q) (r ~s) Premise 2. ~(r ~s) Premise

3. (p q) p Premise 4. ~(p q) 1,2 Modus Tollens

We used A C to represent the statement (p q) p, although we could just as well have represented it by D. Since we're not using this statement at all, it doesn't matter how we represent it. On the other hand, in order to be able to use Modus Tollens on lines (1) and (2), it was imperative that we represented line (1) by A B, and not by the single letter A. If you look at the argument form of Modus Tollens, you will see that it requires a statement of the form A B (as well as ~B, of course). Part of learning to apply the rules of inference is learning how to analyze the structure of statements at the right level of detail.

6. Arguments and Proofs

In Example 5 in the preceding section we saw the following argument.

a q b q

(a b) q

Precisely, an argument is a list of statements called premises followed by a statement called the conclusion. (We allow the list of premises to be empty, as in Example 3 in the preceding section.) We say that an argument is valid if the conjunction of its premises implies its conclusion. In other words, validity means that if all the premises are true, then so is the conclusion. Validity of an argument does not guarantee the truth of its premises, so does not guarantee the truth of its conclusion. It only guarantees that the conclusion will be true if the premises are.

Arguments and Validity

An argument is a list of statements called premises followed by a statement called the conclusion.

P1

P2

P3

. . . . .

Pr

C

The argument is said to be valid if the statement

(P1 P2 . . . Pr) C

is a tautology. In other words, validity means that if all the premises are true, then the conclusion must be true.

Question

To show the validity of an argument like

a q b q

(a b) qwhat we need to do is check that the statement [(a q)Š(b q)] [(a b)q] is a tautology. So to show that an argument is valid we need to construct a truth table, right?

Answer

Well, that would work, but there are a couple of problems. First, the truth table can get quite large. The truth table for [(a q)Š(b q)][(aæb) q] has eight rows and nine columns. It gets worse quickly, since each extra variable doubles the number of rows.

Second, checking the validity of an argument mechanically by constructing a truth table is almost completely unenlightening; it gives you no good idea why an argument is valid. We'll concentrate on an alternative way of showing that an argument is valid, called a proof, that is far more interesting and tells you much more about what is going on in the argument.

Lastly, while truth tables suffice to check the validity of statements in the propositional calculus, they do not work for the predicate calculus we will begin to discuss in the following section. Hence, they do not work for real mathematical arguments. One of our ulterior motives is to show you what mathematicians really do: They create proofs.

Question

OK, so what is a proof?

Answer

Informally, a proof is a way of convincing you that the conclusion follows from the premises, or that the conclusion must be true if the premises are. Formally, a proof is a list of statements, usually beginning with the premises, in which each statement that is not a premise must be true if the statements preceding it are true. In particular, the truth of the last statement, the conclusion, must follow from the truth of the first statements, the premises. How do we know that each statement follows from the preceding ones? We cite a rule of inference that guarantees that it is so.

Proofs

A proof of an argument is a list of statements, each of which is obtained from the preceding statements using one of the rules of inference T1, T2, S, C, or P. The last statement in the proof must be the conclusion of the argument.

Example As an example, we have the following proof of the argument given above, which we considered in the preceding section:

1. a q Premise 2. b q Premise 3. ~a q 1, Switcheroo 4. ~b q 2, Switcheroo 5. (~a q) (~b q) 3,4 Rule C 6. (~a ~b) q 5, Distributive Law 7. ~(a b) q 6, De Morgan 8. (a b) q 7, Switcheroo

Question

I'm convinced that proofs may be a good thing, but I'm still a little skeptical. What does a proof actually have to do with the validity of an argument?

Answer

On the one hand, a proof establishes the validity of an argument. The reason is that, in a proof, every line must be true if the preceding lines are true. In particular, the truth of the first lines, the premises, implies the truth of the last line, the conclusion. Hence a proof does show that an argument is valid. Much less obvious, but reassuring, is the fact that every valid argument in propositional calculus has a proof.   In other words, an argument is valid if and only if there is a proof of it.

The only way to learn to find proofs is by looking at lots of examples and doing lots of practice. In the following examples we'll try to give you some tips as we go along.

7. Predicate Calculus

The Limits of Propositional Calculus

One of the most famous arguments in logic goes as follows. All men are mortal. Socrates is a man. Therefore, Socrates is mortal. There is really no good way to express this argument using propositional calculus.

Question

What are you talking about? These are just three ordinary statements in the propositional calculus: p: All men are mortal. q: Socrates is a man. r: Socrates is mortal.

Answer

But then the above argument has the form pq

rand is therefore not a valid argument in the propositional calculus.

Question

OK. That was a tricky one. I now see that we cannot take those statments as atomic statements, but should write them as compound statements. Now I get it! It is just the transitive rule: Something is a man It is mortal Something is Socrates It is a man

Something is Socrates It is mortal

Answer

This looks more convincing, but there is another catch: "Something is a man", and "It is a man", while a perfectly good sentences, are not propositions (what, after all, are their truth values?). The same goes for the other "statements" in the argument No matter how we try to rephrase the argument as a valid argument in propositional calculus, we are doomed to run into some or other technical difficulty.

Universal Quantifier

We need to go beyond the propositional calculus to the predicate calculus, which allows us to manipulate statements about all or some things, suggested by the above attempt at formulating the argument about Socrates.

We begin by rewording the statment "All men are mortal" a little more slickly than we did above:

"For all x, if x is a man then x is mortal." The sentence "x is a man" is not a statement in propositional calculus, since it involve an unknown thing x and we can't assign a truth value without knowing what x we're talking about. This sentence can be broken down into its subject, x, and a predicate, "is a man." We say that the sentence is a statement form, since it becomes a statement once we fill in x. Here is how we shall write it symbolically: The subject is already represented by the symbol x, called a term here, and we use the symbol P for the predicate "is a man." We then write Px for the statement form. (It is traditional to write the predicate before the term; this is related to the convention of writing function names before variables in other parts of mathematics.) Similarly, if we use Q to represent the predicate "is mortal" then Qx stands for "x is mortal." We can then write the statement "If x is a man then x is mortal" as Px Qx. To write our whole statement, "For all x, if x is a man then x is mortal" symbolically, we need symbols for "For all x." We use the symbol "" to stand for the words "for all" or "for every." Thus, we can write our

complete statement as x[Px Qx].

The symbol " " is called a quantifier because it describes the number of things we are talking about: all of them. Specifically, it is the universal quantifier because it makes a claim that something happens universally.

Question

What are those square brackets doing around Px Qx?

Answer

They define what is called the scope of the quantifier x. That is, they surround what it is we are claiming is true for all x.

Nature of logic

The concept of logical form is central to logic; it being held that the validity of an argument is determined by its logical form, not by its content. Traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logics.

Informal logic is the study of natural language arguments. The study of fallacies is an especially important branch of informal logic. The dialogues of Plato are a good example of informal logic.

Formal logic is the study of inference with purely formal content, where that content is made explicit. (An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The works of Aristotle contain the earliest known formal study of logic, which were incorporated in the late nineteenth century into modern formal logic. In many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuance of natural language.)

Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is often divided into two branches, propositional logic and predicate logic.

Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of model theory, proof theory, set theory, and recursion theory.

These families generally give logic a similar structure: to establish the relation of the sentences in topic of interest to their representation in logic through the analysis of logical form and semantics, and to present an account of inference relating these formal propositions.

Logical form

Logic is generally accepted to be formal, in that it aims to analyses and represent the form (or logical form) of any valid argument type. The form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. If one considers the notion of form to be too philosophically loaded, one could say that formalizing is nothing else than translating English sentences in the language of logic.

This is known as showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a considerable variety of form and complexity that makes their use in inference impractical. It requires, first, ignoring those grammatical features which are irrelevant to logic (such as gender and declension if the argument is in Latin), replacing conjunctions which are not relevant to logic (such as 'but') with logical conjunctions like 'and' and replacing ambiguous or alternative logical expressions ('any', 'every', etc.) with expressions of a standard type (such as 'all', or the universal quantifier).

Second, certain parts of the sentence must be replaced with schematic letters. Thus, for example, the expression 'all As are Bs' shows the logical form which is common to the sentences 'all men are mortals', 'all cats are carnivores', 'all Greeks are philosophers' and so on.

That the concept of form is fundamental to logic was already recognized in ancient times. Aristotle uses variable letters to represent valid inferences the Prior Analytics, leading Jan Łukasiewicz to say that the introduction of variables was 'one of Aristotle's greatest inventions'. According to the followers of Aristotle (such as Ammonius), only the logical principles stated in schematic terms belong to logic, and not those given in concrete terms. The concrete terms 'man', 'mortal', etc., are analogous to the substitution values of the schematic placeholders 'A', 'B', 'C', which were called the 'matter' (Greek 'hyle') of the inference.

The fundamental difference between modern formal logic and traditional or Aristotelian logic lies in their differing analysis of the logical form of the sentences they treat.

In the traditional view, the form of the sentence consists of (1) a subject (e.g. 'man') plus a sign of quantity ('all' or 'some' or 'no'); (2)

the copula which is of the form 'is' or 'is not'; (3) a predicate (e.g. 'mortal'). Thus: all men are mortal. The logical constants such as 'all', 'no' and so on, plus sentential connectives such as 'and' and 'or' were called 'syncategorematic' terms (from the Greek 'kategorei' – to predicate, and 'syn' – together with). This is a fixed scheme, where each judgement has an identified quantity and copula, determining the logical form of the sentence.

According to the modern view, the fundamental form of a simple sentence is given by a recursive schema, involving logical connectives, such as a quantifier with its bound variable, which are joined to by juxtaposition to other sentences, which in turn may have logical structure.

The modern view is more complex, since a single judgement of Aristotle's system will involve two or more logical connectives. For example, the sentence "All men are mortal" involves in term logic two non-logical terms "is a man" (here M) and "is mortal" (here D): the sentence is given by the judgement A (M, D). In predicate logic the sentence involves the same two non-logical concepts, here analyzed as m(x) and d(x), and the sentence is

given by , involving the logical connectives for universal quantification and implication.

But equally, the modern view is more powerful: medieval logicians recognized the problem of multiple generality, where Aristotelian logic is unable to satisfactorily render such sentences as "Some guys have all the luck", because both quantities "all" and "some" may be relevant in an inference, but the fixed scheme that Aristotle used allows only one to govern the inference. Just as linguists recognize recursive structure in natural languages, it appears that logic needs recursive structure.

Deductive and inductive reasoning

Deductive reasoning concerns what follows necessarily from given premises. However, inductive reasoning—the process of deriving a reliable generalization from observations—has sometimes been included in the study of logic. Correspondingly, we must distinguish between deductive validity and inductive validity (called "cogency"). An inference is deductively valid if and only if there is no possible situation in which all the premises are true and the conclusion false.

The notion of deductive validity can be rigorously stated for systems of formal logic in terms of the well-understood notions of semantics. Inductive validity on the other hand requires us to define a reliable generalization of some set of observations. The task of providing this definition may be approached in various ways, some less formal than others; some of these definitions may use mathematical models of probability. For the most part this discussion of logic deals only with deductive logic.

Consistency, soundness, and completeness

Among the important properties that logical systems can have:

Consistency, which means that no theorem of the system contradicts another.

Soundness, which means that the system's rules of proof will never allow a false inference from a true premise. If a system is sound and its axioms are true then its theorems are also guaranteed to be true.

Completeness, which means that there are no true sentences in the system that cannot, at least in principle, be proved in the system.

Some logical systems do not have all three properties. As an example, Kurt Gödel's incompleteness theorems show that no standard formal system of arithmetic can be consistent and complete. At the same time his theorems for first-order predicate logics not extended by specific axioms to be arithmetic formal systems with equality, show those to be complete and consistent.

Rival conceptions of logic

Logic arose (see below) from a concern with correctness of argumentation. Modern logicians usually wish to ensure that logic studies just those arguments that arise from appropriately general forms of inference. For example, Thomas Hofweber writes in the Stanford Encyclopedia of Philosophy that logic "does not, however, cover good reasoning as a whole. That is the job of the theory of rationality. Rather it deals with inferences whose validity can be traced back to the formal features of the representations that are involved in that inference, be they linguistic, mental, or other representations".

By contrast, Immanuel Kant argued that logic should be conceived as the science of judgment, an idea taken up in Gottlob Frege's logical and philosophical work, where thought (German: Gedanke) is substituted for judgement (German: Urteil). On this conception, the valid inferences of logic follow from the structural features of judgements or thoughts.

History of logic

The earliest sustained work on the subject of logic is that of Aristotle,[11] In contrast with other traditions, Aristotelian logic became widely accepted in science and mathematics, ultimately giving rise to the formally sophisticated systems of modern logic.

Several ancient civilizations have employed intricate systems of reasoning and asked questions about logic or propounded logical paradoxes. In India, the Nasadiya Sukta of the Rig-Veda (RV 10.129) contains ontological speculation in terms of various logical divisions that were later recast formally as the four circles of catu ṣ ko ṭ i : "A", "not A", "Neither A or not A", and "Both not A and not not A". The Chinese philosopher Gongsun Long (ca. 325–250 BC) proposed the paradox "One and one cannot become two, since neither becomes two." Also, the Chinese 'School of Names' is recorded as having examined logical puzzles such as "A White Horse is not a Horse" as early as the fifth century BCE. In China, the tradition of scholarly investigation into logic, however, was repressed by the Qin dynasty following the legalist philosophy of Han Feizi.

Logic in Islamic philosophy also contributed to the development of modern logic, which included the development of "Avicennian logic" as an alternative to Aristotelian logic. Avicenna's system of logic was responsible for the introduction of hypothetical syllogism,[16] temporal modal logic, and inductive logic. The rise of the Asharite school, however, limited original work on logic in Islamic philosophy, though it did continue into the 15th century and had a significant influence on European logic during the Renaissance.

In India, innovations in the scholastic school, called Nyaya, continued from ancient times into the early 18th century, though it did not survive long into the colonial period. In the 20th century, Western philosophers like Stanislaw Schayer and Klaus Glashoff have tried to explore certain aspects of the Indian tradition of logic.

During the later medieval period, major efforts were made to show that Aristotle's ideas were compatible with Christian faith. During the later period of the Middle Ages, logic became a main focus of philosophers, who would engage in critical logical analyses of philosophical arguments.

The syllogistic logic developed by Aristotle predominated until the mid-nineteenth century when interest in the foundations of mathematics stimulated the development of symbolic logic (now called mathematical logic). In 1854, George Boole published An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, introducing symbolic logic and the principles of what is now known as Boolean logic. In 1879 Frege published Begriffsschrift which inaugurated modern logic with the invention of quantifier notation. In 1903 Alfred North Whitehead and Bertrand Russell published Principia Mathematical on the foundations of mathematics, attempting to derive mathematical truths from axioms and inference rules in symbolic logic. In 1931 Gödel raised serious problems with the foundation list program and logic ceased to focus on such issues.

The development of logic since Frege, Russell and Wittgenstein had a profound influence on the practice of philosophy and the perceived nature of philosophical problems (see Analytic philosophy), and Philosophy of mathematics. Logic, especially sentential logic, is implemented in computer logic circuits and is fundamental to computer science. Logic is commonly taught by university philosophy departments often as a compulsory discipline.

Topics in logic

Syllogistic logic

The Organon was Aristotle's body of work on logic, with the Prior Analytics constituting the first explicit work in formal logic, introducing the syllogistic. The parts of syllogistic, also known by the name term logic, were the analysis of the judgements into propositions consisting of two terms that are related by one of a fixed number of relations, and the expression of inferences by means of syllogisms that consisted of two propositions sharing a common term as premise, and a conclusion which was a proposition involving the two unrelated terms from the premises.

Aristotle's work was regarded in classical times and from medieval times in Europe and the Middle East as the very picture of a fully worked out system. It was not alone: the Stoics proposed a system of propositional logic that was studied by medieval logicians; nor was the perfection of Aristotle's system undisputed; for example the problem of multiple generality was recognised in medieval times. Nonetheless, problems with syllogistic logic were not seen as being in need of revolutionary solutions.

Today, some academics claim that Aristotle's system is generally seen as having little more than historical value (though there is some current interest in extending term logics), regarded as made obsolete by the advent of propositional logic and the predicate calculus. Others use Aristotle in argumentation theory to help develop and critically question argumentation schemes that are used in artificial intelligence and legal arguments.

Sentential (propositional) logic

A propositional calculus or logic (also a sentential calculus) is a formal system in which formulae representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows certain formula to be established as "theorems".

Predicate logic

Predicate logic is the generic term for symbolic formal systems such as first-order logic, second-order logic, many-sorted logic, and infinitary logic.

Predicate logic provides an account of quantifiers general enough to express a wide set of arguments occurring in natural language. Aristotelian syllogistic logic specifies a small number of forms that the relevant part of the involved judgements may take. Predicate logic allows sentences to be analyzed into subject and argument in several additional ways, thus allowing predicate logic to solve the problem of multiple generality that had perplexed medieval logicians.

The development of predicate logic is usually attributed to Gottlob Frege, who is also credited as one of the founders of analytical philosophy, but the formulation of predicate logic most often used today is the first-order logic presented in Principles of Mathematical Logic by David Hilbert and Wilhelm Ackermann in 1928. The analytical generality of predicate logic allowed the formalisation of mathematics, drove the investigation of set theory, and allowed the development of Alfred Tarski's approach to model theory. It provides the foundation of modern mathematical logic.

Frege's original system of predicate logic was second-order, rather than first-order. Second-order logic is most prominently defended (against the criticism of Willard Van Orman Quine and others) by George Boolos and Stewart Shapiro.

Modal logic

In languages, modality deals with the phenomenon that sub-parts of a sentence may have their semantics modified by special verbs or modal particles. For example, "We go to the games" can be modified to give "We should go to the games” and "We can go to the games"" and perhaps "We will go to the games". More abstractly, we might say that modality affects the circumstances in which we take an assertion to be satisfied.

The logical study of modality dates back to Aristotle, who was concerned with the alethic modalities of necessity and possibility, which he observed to be dual in the sense of De Morgan duality.[citation needed] While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of Clarence Irving Lewis in 1918, who formulated a family of rival axiomatizations of the alethic modalities. His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to include deontic logic and epistemic logic. The seminal work of Arthur Prior applied the same formal language to treat temporal logic and paved the way for the marriage of the two subjects. Saul Kripke discovered (contemporaneously with rivals) his theory of frame semantics which revolutionized the formal technology available to modal logicians and gave a new graph-theoretic way of looking at modality that has driven many applications in computational linguistics and computer science, such as dynamic logic.

Informal reasoning

The motivation for the study of logic in ancient times was clear: it is so that one may learn to distinguish good from bad arguments, and so become more effective in argument and oratory, and perhaps also to become a better person. Half of the works of Aristotle's Organon treat inference as it occurs in an informal setting, side by side with the development of the syllogistic, and in the Aristotelian school, these informal works on logic were seen as complementary to Aristotle's treatment of rhetoric.

This ancient motivation is still alive, although it no longer takes centre stage in the picture of logic; typically dialectical logic will form the heart of a course in critical thinking, a compulsory course at many universities.

Argumentation theory is the study and research of informal logic, fallacies, and critical questions as they relate to every day and practical situations. Specific types of dialogue can be analyzed and questioned to reveal premises, conclusions, and fallacies. Argumentation theory is now applied in artificial intelligence and law.

Mathematical logic

Mathematical logic really refers to two distinct areas of research: the first is the application of the techniques of formal logic to mathematics and mathematical reasoning, and the second, in the other direction, the application of mathematical techniques to the representation and analysis of formal logic.

The earliest use of mathematics and geometry in relation to logic and philosophy goes back to the ancient Greeks such as Euclid, Plato, and Aristotle. Many other ancient and medieval philosophers applied mathematical ideas and methods to their philosophical claims.

The boldest attempt to apply logic to mathematics was undoubtedly the logicism pioneered by philosopher-logicians such as Gottlob Frege and Bertrand Russell: the idea was that mathematical theories were logical tautologies, and the programmer was to show this by means to a reduction of mathematics to logic. The various attempts to carry this out met with a series of failures, from the crippling of Frege's project in his Grundgesetze by Russell's paradox, to the defeat of Hilbert's program by Gödel's incompleteness theorems.

Both the statement of Hilbert's program and its refutation by Gödel depended upon their work establishing the second area of mathematical logic, the application of mathematics to logic in the form of proof theory. Despite the negative nature of the incompleteness theorems, Gödel's completeness theorem, a result in model theory and another application of mathematics to logic, can be understood as showing how close logicism came to being true: every rigorously defined mathematical theory can be exactly captured by a first-order logical theory; Frege's proof calculus is enough to describe the whole of mathematics, though not equivalent to it. Thus we see how complementary the two areas of mathematical logic have been.

If proof theory and model theory have been the foundation of mathematical logic, they have been but two of the four pillars of the subject. Set theory originated in the study of the infinite by Georg Cantor, and it has been the source of many of the most challenging and important issues in mathematical logic, from Cantor's theorem, through the status of the Axiom of Choice and the question of the independence of the continuum hypothesis, to the modern debate on large cardinal axioms.

Recursion theory captures the idea of computation in logical and arithmetic terms; its most classical achievements are the undecidability of the Entscheidungsproblem by Alan Turing, and his presentation of the Church-Turing thesis. Today recursion theory is mostly concerned with the more refined problem of complexity classes — when is a problem efficiently solvable? — and the classification of degrees of unsolvability.

Philosophical logic

Philosophical logic deals with formal descriptions of natural language. Most philosophers assume that the bulk of "normal" proper reasoning can be captured by logic, if one can find the right method for translating ordinary language into that logic. Philosophical logic is essentially a continuation of the traditional discipline that was called "Logic" before the invention of mathematical logic. Philosophical logic has a much greater concern with the connection between natural language and logic. As a result, philosophical logicians have contributed a great deal to the development of non-standard logics (e.g., free logics, tense logics) as well as various extensions of classical logic (e.g., modal logics), and non-standard semantics for such logics (e.g., Kripke's technique of super valuations in the semantics of logic).

Logic and the philosophy of language are closely related. Philosophy of language has to do with the study of how our language engages and interacts with our thinking. Logic has an immediate impact on other areas of study. Studying logic and the relationship between logic and ordinary speech can help a person better structure their own arguments and critique the arguments of others. Many popular arguments are filled with errors because so many people are untrained in logic and unaware of how to correctly formulate an argument.

Logic and computation

Logic cut to the heart of computer science as it emerged as a discipline: Alan Turing's work on the Entscheidungsproblem followed from Kurt Gödel's work on the incompleteness theorems, and the notion of general purpose computers that came from this work was of fundamental importance to the designers of the computer machinery in the 1940s.

In the 1950s and 1960s, researchers predicted that when human knowledge could be expressed using logic with mathematical notation, it would be possible to create a machine that reasons, or artificial intelligence. This turned out to be more difficult than expected because of the complexity of

human reasoning. In logic programming, a program consists of a set of axioms and rules. Logic programming systems such as Prolog compute the consequences of the axioms and rules in order to answer a query.

Today, logic is extensively applied in the fields of artificial intelligence, and computer science, and these fields provide a rich source of problems in formal and informal logic. Argumentation theory is one good example of how logic is being applied to artificial intelligence. The ACM Computing Classification System in particular regards:

Section F.3 on Logics and meanings of programs and F. 4 on Mathematical logic and formal languages as part of the theory of computer science: this work covers formal semantics of programming languages, as well as work of formal methods such as Hoare logic

Boolean logic as fundamental to computer hardware: particularly, the system's section B.2 on Arithmetic and logic structures, relating to operatives AND, NOT, and OR;

Many fundamental logical formalisms are essential to section I.2 on artificial intelligence, for example modal logic and default logic in Knowledge representation formalisms and methods, Horn clauses in logic programming, and description logic.

Furthermore, computers can be used as tools for logicians. For example, in symbolic logic and mathematical logic, proofs by humans can be computer-assisted. Using automated theorem proving the machines can find and check proofs, as well as work with proofs too lengthy to be written out by hand.

Controversies in logic

Just as we have seen there is disagreement over what logic is about, so there is disagreement about what logical truths there are.

Bivalence and the law of the excluded middle

The logics discussed above are all "bivalent" or "two-valued"; that is, they are most naturally understood as dividing propositions into true and false propositions. Non-classical logics are those systems which reject bivalence.

Hegel developed his own dialectic logic that extended Kant's transcendental logic but also brought it back to ground by assuring us that "neither in heaven nor in earth, neither in the world of mind nor of nature, is there anywhere such an abstract 'either–or' as the understanding maintains. Whatever exists is concrete, with difference and opposition in itself".

In 1910 Nicolai A. Vasiliev rejected the law of excluded middle and the law of contradiction and proposed the law of excluded fourth and logic tolerant to contradiction. In the early 20th century Jan Łukasiewicz investigated the extension of the traditional true/false values to include a third value, "possible", so inventing ternary logic, the first multi-valued logic.

Logics such as fuzzy logic have since been devised with an infinite number of "degrees of truth", represented by a real number between 0 and 1.

Intuitionistic logic was proposed by L.E.J. Brouwer as the correct logic for reasoning about mathematics, based upon his rejection of the law of the excluded middle as part of his intuitionism. Brouwer rejected formalization in mathematics, but his student Arend Heyting studied intuitionistic logic formally, as did Gerhard Gentzen. Intuitionistic logic has come to be of great interest to computer scientists, as it is a constructive logic, and is hence a logic of what computers can do.

Modal logic is not truth conditional, and so it has often been proposed as a non-classical logic. However, modal logic is normally formalized with the principle of the excluded middle, and its relational semantics is bivalent, so this inclusion is disputable.

Is logic empirical?

What is the epistemological status of the laws of logic? What sort of argument is appropriate for criticising purported principles of logic? In an influential paper entitled "Is logic empirical?" Hilary Putnam, building on a suggestion of W.V. Quine, argued that in general the facts of propositional logic have a similar epistemological status as facts about the physical universe, for example as the laws of mechanics or of general relativity, and in particular that what physicists have learned about quantum mechanics provides a compelling case for abandoning certain familiar principles of classical logic: if we want to be realists about the physical phenomena described by quantum theory, then we should abandon the principle of distributivity, substituting for classical logic the quantum logic proposed by Garrett Birkhoff and John von Neumann.

Another paper by the same name by Sir Michael Dummett argues that Putnam's desire for realism mandates the law of distributivity. Distributivity of logic is essential for the realist's understanding of how propositions are true of the world in just the same way as he has argued the principle of bivalence is. In this way, the question, "Is logic empirical?" can be seen to lead naturally into the fundamental controversy in metaphysics on realism versus anti-realism.

Implication: strict or material?

It is obvious that the notion of implication formalized in classical logic does not comfortably translate into natural language by means of "if… then…", due to a number of problems called the paradoxes of material implication.

The first class of paradoxes involves counterfactuals, such as "If the moon is made of green cheese, then 2+2=5", which are puzzling because natural language does not support the principle of explosion. Eliminating this class of paradoxes was the reason for C. I. Lewis's formulation of strict

implication, which eventually led to more radically revisionist logics such as relevance logic.

The second class of paradoxes involves redundant premises, falsely suggesting that we know the succedent because of the antecedent: thus "if that man gets elected, granny will die" is materially true if granny happens to be in the last stages of a terminal illness, regardless of the man's election prospects. Such sentences violate the Gricean maxim of relevance, and can be modelled by logics that reject the principle of monotonicity of entailment, such as relevance logic.

Tolerating the impossible

Hegel was deeply critical of any simplified notion of the Law of Non-Contradiction. It was based on Leibniz's idea that this law of logic also requires a sufficient ground in order to specify from what point of view (or time) one says that something cannot contradict itself, a building for example both moves and does not move, the ground for the first is our solar system for the second the earth. In Hegelian dialectic the law of non-contradiction, of identity, itself relies upon difference and so is not independently assert able.

Closely related to questions arising from the paradoxes of implication comes the suggestion that logic ought to tolerate inconsistency. Relevance logic and paraconsistent logic are the most important approaches here, though the concerns are different: a key consequence of classical logic and some of its rivals, such as intuitionistic logic, is that they respect the principle of explosion, which means that the logic collapses if it is capable of deriving a contradiction. Graham Priest, the main proponent of dialetheism, has argued for paraconsistency on the grounds that there are in fact, true contradictions.

Rejection of logical truth

The philosophical vein of various kinds of skepticism contains many kinds of doubt and rejection of the various bases upon which logic rests, such as the idea of logical form, correct inference, or meaning, typically leading to the conclusion that there are no logical truths. Observe that this is opposite to the usual views in philosophical skepticism, where logic directs skeptical enquiry to doubt received wisdoms, as in the work of Sextus Empiricus.

Friedrich Nietzsche provides a strong example of the rejection of the usual basis of logic: his radical rejection of idealization led him to reject truth as a mobile army of metaphors, metonyms, and anthropomorphisms—in short ... metaphors which are worn out and without sensuous power; coins which have lost their pictures and now matter only as metal, no longer as coins. His rejection of truth did not lead him to reject the idea of either inference or logic completely, but rather suggested that logic [came] into existence in man's head [out] of illogic, whose realm originally must have been immense. Innumerable beings who made inferences in a way different from ours perished. Thus there is the idea that logical inference has a use as a tool for human survival, but that its existence does not support the existence of truth, nor does it have a reality beyond the instrumental: Logic, too, also rests on assumptions that do not correspond to anything in the real world.