Exercises 1: Alfred Tarski, Introduction to Logic

I. Which among the following expressions are sentential functions, and which are designatory functions:

a) x is divisible by 3

b) the sum of the numbers x and 2

c) x^{2}-z^{2}

d) y^{2}=z^{2}

e) x+2< y+3

f) (x+3) - (y+5)

g) the mother of x and z

h) x is the mother of z?

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

Problem 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 : (1) functions satisfied by every number; (ii) functions not satisfied by any number; (iii) functions satisfied by some numbers, and not satisfied by others.

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

(a) x+2=5+x

(b) x^{2}=49

(c) (y+2).(y-2)<y^{2}

(d) y+24>36

(e) z=0 or z<0 or z>0

(f) z+24>z+36?

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

Problem 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:

for any numbers x and y, x>y;

for any number x, there exists a number y such that x>y;

there is a number y such that, for any number x, x>y.

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

Problem 6: Do the same as in problem 5 for the following sentential 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.)

Problem 7: State a sentence of every day language that has the meaning as:

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

And, your sentence must not contain any quantifier or variables.

Problem 8:

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

Problem 9:

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

(a) x is divisible by y.

(b) for any x, x-y = x +(-y)

(c) if x<y, then there is a number z such that <y and y<z;

(d) for any number y, if y>0, then there is a number z such that x=y.z

(e) if x=y^{2} and y>0, then for any number z, x>-z^{2};

(f) if there exists a number y such that x>y^{2}, then, for any number z, x>-z^{2}.

Formulate the above expressions by replacing the quantifiers by the symbols introduced in Section 4.

Problem 10*: If, in the sentential function, (e) of the preceding exercise, we replace the variable “z” in both places by “y”, we obtain an expression in which “y” occurs in some places as a free and in others as a bound variable; in what places and y?

(In view of some difficulties in operating with expressions in which the same variable occurs both bound and free, some logicians prefer to avoid the use of such expressions altogether and not to treat them as sentential functions.)

Problem 11*: Try to state quite generally under which conditions a variable occurs at a certain place of a given sentential function as a free or as a bound variable.

Problem 12: Which numbers satisfy the sentential function: there is a number y such that x=y^{2}, and which satisfy: there is a number y such that x.y=1?


Nalin Pithwa

If you are pursuing higher math

Experience seems to show that the student usually finds a new theory difficult to grasp at a first reading. He needs to return to it several times before he becomes really familiar with it and can distinguish for himself which are the essential ideas and which results are of minor importance, and only then will he be able to apply it intelligently.

— quoted by Jean Dieudonne, in his preface to Foundations of Modern Analysis. (Academic Press, NY and London, 1969).

Set Theory…the beginning context…Cantor vs. others

“I protest…against the use of infinite magnitude as if it were something finished; this use is not something admissible. The infinite is only a facon de parler, one has in mind limits approached by certain ratios as closely as desirable while other ratios may increase indefinitely.” Carl Friedrich Gauss, presumably the foremost mathematician of the 19th century, expressed this view in 1831 in reply to an idea of Schumacher and in the process uttered a horror infiniti which up to almost the end of the century was the common attitude of mathematicians and seemed unassailable considering the authority of Gauss. Mathematics should deal with finite magnitudes and finite numbers only while the treatment of the actual infinity, whether infinitely great or infinitely small, might be left to philosophy.

It was the mathematician Georg Cantor (1845-1918) who dared to fight this attitude and in the opinion of the majority of 20th century mathematicians, has succeeded in the task of bestowing legitimacy upon infinitely great magnitude. Besides the creative intuition and the artistic power of production, which guided Cantor in his work, an enormous amount of energy and perseverance was required to carry through the new ideas, which for two decades were rejected by his contemporaries with the arguments that they were meaningless or false or “brought into the world of mathematics a hundred years too early”. Not only Gauss and other outstanding mathematicians were quoted in evidence against the actual infinity but also leading philosophers such as Aristotle, Descartes, Spinoza and modern logicians. Set theory was even charged with violating the principles of religion, an accusation rejected by Cantor with particular vigour and minuteness. Only in the last years of the nineteenth century, when Cantor had ceased engaging in mathematical research, did set theory begin to infiltrate many branches of mathematics.

Cantor has shown how definite and distinctly great magnitudes can be handled — another evidence of the free creation which is characteristic of mathematics to a higher extent than that of the other sciences. It is no mere accident that at the birth of the set theory, the slogan was coined: the very essence of mathematics is its freedom.

Ref: Abstract Set Theory by Abraham A. Fraenkel.

Shared by Nalin Pithwa …(a piece that I enjoyed in the beginning of the book)