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?

Cheers,

Nalin Pithwa

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