The significance of mathematical logic: words of Alfred Tarski

Reference: Introduction to Logic, and to the Methodology of Deductive Sciences by Alfred Tarski, Oxford University Press, New York; available in Amazon India.

Just my two cents worth: Of course, Euclid introduced the deductive method about 2500 years back. Prof Tarski says there is more to it 🙂

(below I just present his motivational explanation in the preface, 1937 A.D)

In the opinion of many laymen mathematics is today already a dead science; after having reached an unusually high degree of development/sophistication, it has become petrified in rigid perfection. This is an entirely erroneous view of the situation; there are but few domains of scientific research which are passing through a phase of such intensive development at present as mathematics. Moreover, this development is extraordinarily manifold: mathematics is expanding in all possible directions, it is growing in height, in width, and in depth. It is growing in height, since, on the soil of its old theories which look back upon hundreds if not thousands of years of development, new problems appear again and again, and ever more perfect results are being achieved. It is growing in width, since its methhods permeate other branches of sciences, while its domain of investigation embraces increasingly more comprehensive ranges of phenomena and ever new theories are being included in the large circle of mathematical disciplines. And finally it is growing in depth, since its foundations become more and more firmly established, its methods perfected, and its principles stabilized.

It has been my intention in this book to give those readers who are interested in contemporary mathematics, without being actively concerned with it, at least a very general idea of that third line mathematical development, that is, its growth in depth. My aim has been to acquaint the reader with the most important concepts of a discipline which is known as mathematical logic, and which has been created for the purpose of a firmer and more profound establishment of the foundations of mathematics; this discipline, in spite of its brief existence of barely a century, has already attained a high degree of perfection and plays today a role in the totality of our knowledge that far transcends its originally intended boundaries. It has been my intention to show that the concepts of logic permeate the whole of mathematics, that they comprehend all specifically mathematical concepts as special cases, and that logical laws are constantly applied — be it consciously or unconsciously — in mathematical reasonings.

Cheers,

Nalin Pithwa.

The book is quite accessible with some moderate concentration 🙂

Solutions 1: Introduction to Logic, Alfred Tarski

Reference: Previous Blog; Introduction to Logic and to the Methodology of Deductive Sciences, by Alfred Tarski, chapter 1.

Exercises:

Question 1: Which among the following are sentential functions and which are designatory functions?

1a) x is divisible by 3: sentential function

1b) the sum of the numbers x and 2: designatory function.

1c) y^{2}-x^{2}: designatory function

1d) y^{2}=x^{2}: sentential function

1e) x+2<y+3: sentential function

1f) (x+3)-(y+5): designatory function

1g) the mother of x and z: designatory function

1h) x is the mother of z? : sentential function.

Question 2: Give examples of sentential and designatory functions from the field of geometry.

Answer 2: Designatory function: two parallel lines

Answer 2: Sentential function: Area of parallelogram with sides x and y is xy\sin{\theta} where \theta is the angle between the sides.

Question 3: The sentential functions which are encountered in arithmetic and which contain only one variable (which may, however, occur at several different places in the given sentential function) can be divided into three categories: (a) functions satisfied by every number (b) functions not satisfied by any number; (c) functions satisfied by some numbers, and not satisfied by others.

To which of these categories do the following sentential functions belong:

(i) x+2=5+x; category b.

(ii) x^{2}=49; category c.

(iii) (y+2)(y-2)<y^{2}; category a.

(iv) y+24>36; category c.

(v) z=0 or z<0 or z>0; category a.

(vi) z+24>z+36? category b.

Question 4: Give examples of universal, absolutely existential and conditionally existential theorems from the fields of arithmetic and geometry.

Answer 4:

Universal existential theorem: arithmetic: commutative law: x+y=y+x

Universal existential theorem: geometry: Parallel lines do not meet.

Absolutely existential theorem: arithmetic: there are numbers x and y such that x<y+1

Absolutely existential theorem: geometry: there are three points which can form a triangle.

Conditionally existential theorem: arithmetic: For any numbers x and y, there is a number z such that x=y+z

Conditionally existential theorem: geometry: For any two lines which meet, there is an angle bisector.

Question 5: By writing quantifiers containing the variables “x” and “y” in front of the sentential function:

x>y

it is possible to obtain various sentences from it, for instance:

(i) for any numbers x and y, x>y; (not always true)

(ii) for any number x, there exists a number y such that x>y; (always true)

(iii) there is a number y such that, for any number x, x>y. (not always true)

Formulate them all (there are six altogether) and determine which of them are true.

The other three possibilities are as follows:

(iv) given x, there is no y such that x>y; false.

(v) given y, there is no x such that x>y; false.

(vi) there is no x or y such that x>y; false.

Question 6: Do the same as in Exercise 5 for the following existential functions:

x+y^{2}>1 and “x is the father of y”.

(assuming that the variables ‘x” and “y” in the latter stand for names of human beings.)

part 1: x+y^{2}>1

1a: for any numbers x and y, x+y^{2}>1 (not always true)

1b: for any number x, there exists a number y such that x+y^{2}>1 (always true)

1c: there is a number y such that, for any number x, x+y^{2}>1 (always true)

1d: given x, there is no y such that x+y^{2}>1 (always false)

1e: given y, there is no x such that x+y^{2}>1 (always false)

1f: there is no x or y such that x=y^{2}>1; always false.

Part 2: x is the father of y:

2a: for any x and y, x is the father of y (not always true)

2b: for any x, there exists a y such that x is the father of y. (not always true)

2c: there is a y such that for any x, x is the father of y. (not always true).

2d: given x, there is no y such that “x is the father of y” (not always true).

2e: given y, there is no x such that “x is the father of y” (false).

2f: there is no x or y such that “x is the father of y” (false).

Question 7: State a question of every day language that has the same meaning as:

for every x, if x is a dog, then x has a good sense of smell.

Answer 7: Every dog has a good sense of smell.

Question 8: Replace the sentence: “some snakes are poisonous” by one which has the same meaning but is formulated with the help of quantifiers and variables.

Answer 8; There exist some snakes which are poisonous.

Question 9: Differentiate, in the following expressions, between the free and bound variables:

9a: x is divisible by y: y is free, x is bound.

9b: for any x, x-y=x+(-y) ; both x and y are free.

9c: if x<y, then there is a number z such that x<y and y<z; x and y are free, z is bound.

9d: for any number y, if y>0. then, for any number z such that x=y.z; z is bound; x is bound, y is free.

9e: if x=y^{2} and y>0, then, for any number z, x>-z^{2}; x and y are bound; z is not bound.

PS: I am skipping two, three questions in the exercise.

Your comments are welcome!

I hope to continue this logic blog albeit quite slow.

Cheers,

Nalin Pithwa.

Tutorial Problems I: Topology: Hocking and Young

Reference: Topoology by Hocking and Young, Dover Publications, Inc., NY. Available in Amazon India.

Exercises 1-1:

Show that if S is a set with the discrete topology and f: S \rightarrow T is any transformation of S into a topologized set T, then f is continuous.

Solution 1-1:

Definition A: The set S has a topology (or is topologized) provided that, for every point p in S and every subset X of S, the question : “is p a limit point of X?” can be answered.

Definition B: A topology is said to be a discrete topology when we assume that for no point p in S, and every subset X of S: the answer to the question: “is p a limit point of X?” is NO.

Definition C: A transformation f: S \rightarrow T is continuous provided that if p is a limit point of a subset X of S, then f(p) is a limit point or a point of f(X).

So, the claim is vacuously true. QED.

Exercises 1-2:

A real-valued function y=f(x) defined on an interval [a,b] is continuous provided that if a \leq x_{0} \leq b and \epsilon >0, then there is a number \delta>0 such that if $|x-x_{0}|<\delta$, where x \in [a,b], then |f(x)-f(x_{0})|<\epsilon. Show that this is equivalent to our definition, using definition 1-1.

Solution 1-2:

Definition 1-1: The real number p is a limit point of a set X of real numbers provided that for every positive number \epsilon, there is an element x of the set X such that $0<|p-x|<\epsilon$.

Definition C: A transformation f: S \rightarrow T is continuous provided that if p is a limit point of a subset X of S, then f(p) is a limit point or a point of f(X).

Part 1: Let us assume that given function f is continuous as per definition given just above.

Then, as p is a limit point of X: it means: For any \delta>0, there exists a real number p such that there is an element x \in X such that |p-x|<\delta..

So, also, by definition C, f(p) is a limit point or a point of f(X); this means the following: if f(p) is a point of f(X), there exists some x_{0} \in X such that $f(x_{0} \in f(X)$, and so quite clearly in this case p=x_{0} so that |p-x_{0}|=|x_{0}-x_{0}|<\delta, as \delta is positive.

On the other hand, if f(p) is a limit point of f(X), as per the above definition of continuity, then also for any \epsilon>0, there exists a point y \in f(X) such that |y-f(p)|<\epsilon. So, in this case also the claim is true.

We have proved Part 1. QED.

Now, part II: We assume the definition of continuity given in the problem statement is true. From here, we got to prove definition C as the basic definition given by the authors.

But this is quite obvious as in this case p=x_{0}.

We have proved Part II. QED.

Thus, the two definitions are equivalent.

Cheers,

Nalin Pithwa