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.

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