Symbolic Logic

1. Symbolic Logic Joan Ridgway

2. Basic Concepts of Symbolic Logic The word “logic” derives from the Greek word logos meaning “word”. For a mathematician, logic deals with the conversion of word statements to symbolic form, and the use of that symbolic form to make deductions and create proofs. Mathematical logic deals with basic statements called propositions. A proposition can be true, T, or false, F, but might be indeterminate. Questions, exclamations and orders are not propositions. In the study of mathematical logic, we concentrate on propositions which have a well-defined truth value, that is, they are true or false.

3. Which of the following are propositions? For each proposition identified, discuss whether it is true, false or indeterminate. • All dogs have tails. • My little sister is cute. • Today it is snowing here. • (i) 32 = 9 (ii) 42 = 21 (iii)  is a rational number • How many people are there is the world? • Quadratic equations contain at most a second power of the variable. • y = 6 • For all xR, x2  0 Proposition. False Proposition. Indeterminate Proposition. Indeterminate Proposition. True Proposition. False Proposition. False Not a proposition. Proposition. True Not a proposition. (We are not told what y is). Proposition. True

4. To avoid writing all the words in a proposition all the time, we label propositions with letters. Usually we use p, q, r, s etc. In Maths Studies, the maximum number of propositions used at once is three, usually p, q, and r. e.g. p : The wind is blowing. q : I will lose my hat. In this example, there might be a connection between these propositions. Compound Statements All relations between two or more propositions are called compound statements or compound propositions. Symbols used to connect propositions are called connectives. We will look at the symbols and meaning of Implication, Equivalence, Negation, Conjunction, Disjunction and Exclusive disjunction.

5. Implication E.g. p : The wind is blowing. q : I will lose my hat. “If the wind is blowing then I will lose my hat”. This is called implication. The symbol for this is an arrow in the direction of implication. We write it as pq (or sometimes as q p ) p : This is a zebra. q : This is striped. “If this is a zebra then it is striped”. This could be phrased as “All zebras are striped”

6. Equivalence If an implication works in both directions, it is called an equivalence. E.g. r : Fiona is a good footballer s : Fiona scores lots of goals We could write r sThis is the same as writingr sandr  s It is also the same as saying that r is trueif and only if s is true. Negation If we deny a proposition, we are saying it is not true. The negation of p is usually written as p E.g. q : I will not lose my hat r : Fiona is not a good footballer.

7. Conjunction, Disjunction and Exclusive disjunction These are fancy names for the ideas of “and”, “or” and “or but not both”. Each has its own symbol: • Conjunction ( = and ) with the symbol  • Disjunction ( = or, possibly both ) with the symbol • Exclusive disjunction ( = or but not both ) with the symbol The use is straightforward: p : Mischa has a dog q : Yuri has a cat p  q Mischa has a dog and Yuri has a cat. p q Either Mischa has a dog or Yuri has a cat (and both could be true). pq Either Mischa has a dog or Yuri has a cat but not both are true.

8. Given w : I am wearing shorts. s : I am going to swim. r: I am going to run. • Write in words these compound propositions: • ws • wrs • sr •  sr • e. w  (sr) I am wearing shorts and I am going to swim. If I am wearing shorts then I am going to swim or run. I am going to run or swim, but not both. If I am not going to swim, then I am going to run, and if I am going to run, then I am not going to swim. I am wearing shorts and I am not going to run or to swim. (or I am not wearing shorts and I am going to neither swim nor run.)

9. Some compound propositions have the same meaning (are equivalent). For example: p qqp is the same as pq  (pq) is the same as pq  (pq) is the same as p q To understand why these propositions are equivalent, an example helps. Take p : The sun is shining. q : It is raining. Then pq becomes “the sun is not shining and it is not raining” while (pq) becomes “the sun is shining or it is raining (or both)” and hence  (pq) is “neither is the sun shining nor is it raining”, which is clearly the same as “the sun is not shining and it is not raining”. Warning: The use of brackets is important when a compound proposition could be ambiguous without their use. For example if I write pq, this is different from (pq). It is like the difference between– x + yand– (x + y)in algebra. If in doubt, use brackets!

11. r p q r  p q

12. r  q r  p q (q  r)   p

13. r   r  p q (q  r) p If (I visited Sarah’s Snackbar or I visited Pete’s Eats) then I did not visit Alan’s Diner If I visited either Sarah’s Snackbar or Pete’s Eats (or both), then I did not visit Alan’s Diner.

14. r  p q If I visited either Sarah’s Snackbar or Pete’s Eats, then I did not visit Alan’s Diner.

15. U t A B The connectives used in symbolic logic are closely related to ideas which you have met when working with sets. If we have: p : Today is a cold day. q : Today is a wet day then p q means that today is cold and wet. Suppose that A is the set of days which are cold, and B is the set of days which are wet. Then A B stands for the set of days which are cold and wet. So if ‘today’ is a member of the set AB, i.e. t  AB, it is cold and wet today. So t  A B gives the same information as p q being true.

16. U t B A A B The connectives used in symbolic logic are closely related to ideas which you have met when working with sets. If we have: p : Today is a cold day. q : Today is a wet day then p q means that today is cold or wet. Suppose that A is the set of days which are cold, and B is the set of days which are wet. A B stands for the set of days which are either cold or wet (or both). So if ‘today’ is a member of the set AB, i.e. t  AB, it is cold or wet today. So t  A B gives the same information as p q being true.

17. U t B A B The connectives used in symbolic logic are closely related to ideas which you have met when working with sets. If we have: p : Today is a cold day. q : Today is a wet day then  pmeans that today is not cold. Suppose that A is the set of days which are cold, and B is the set of days which are wet. A’stands for the set of days which are not cold. So if ‘today’ is a member of the set A’, i.e. t  A’, it is not cold today. So t  A’ gives the same information as  pbeing true.

18. The connectives used in symbolic logic are closely related to ideas which you have met when working with sets. If we have: p : Today is a cold day. q : Today is a wet day then p q means that if it is cold today, then it is wet today.. Suppose that A is the set of days which are cold, and B is the set of days which are wet. A  Bstands for all cold days are a subset of wet days, so all cold days are wet. So if ‘today’ is a member of the set A, it must also be a member of the set B. U B A t So tA and AB gives the same information as pq being true.

19. P Q R Draw a Venn diagram showing the truth sets for the propositions: p : n is an even natural number less than 17 q : n is an odd natural number less than 17 r : n is a positive multiple of 3 which is less than 20 Take U to be the set of integers from 1 to 20 inclusive. Using the Venn diagram, list the truth sets for the propositions: p q p r n is a positive multiple of 3 which is less than 20 but is not an even number less than 17  ( p  q ) U The set P Q, which is  1 14 2 13 4 3 11 The set P R, which is {6, 12} 10 6 8 5 15 12 16 9 7 19 18 17 20 {3, 9, 15, 18} The set (P Q)’, which is {17, 18, 19, 20 }