Day 8: Truth-Functional Connectives and Symbolization

1. Day 8: Truth-Functional Connectives and Symbolization Copyright 2003 Julian C. Cole Revised by Makoto Suzuki The Figure in the Picture: George Boole (1815-1864) Introducer of a fully symbolic logic. portrait source:  George Boole An Investigation of the Laws of Thought.Dover Edition (1958, NY).

2. Aims for Day 8 • Define connectives, simple and compound sentences, and truth-functional connectives. • Introduce several truth-functional connectives. • Negation • Conjunction • Disjunction, inclusive and exclusive • Material Conditional • Material Biconditional • (Neither A nor B) • Discuss elementary symbolization.

3. Connectives, Simple Sentences and Compound Sentences • A connective is something that creates a compound sentence. • Ex.: if, it is not the case that, or, and, if and only if, after, because and so on. • A compound sentence is one that contains another sentence as a proper part. • Ex.: Mary is upset because her partner has gotten an STD. • The sentences that are not compound are called simple or atomic. • In the above example, “Mary is upset” and ”Mary’s partner has gotten an STD” are simple sentences.

4. Truth Values and Two Principles • Declarative sentences have truth values. • Hereafter I mean declarative sentence(s) by “sentence(s)”. • For the purpose of this course there are (only) two truth values; true and false. • We assume two principles: • Principle of the Excluded Middle: • Every sentence is either true or false; • Principle of Contradiction: • No sentence is both true and false.

5. Truth-Functional Connectives: Definition • A connective is truth-functional if and only if: the truth value of the compound sentence is entirely determined by the truth values of its component sentences connected by the connective. • Not all connectives are truth-functional. • Ex.: “On a diet, you lose your patience before you lose your weight.” • Suppose both component sentences “you lose your patience” and “you lose your weight” are true. Is the truth value of the whole statement then determined? • No, this compound sentence can still be true or false, depending on which happens first, patience loss or weight loss. • So its truth value is not entirely determined by the truth values of its component sentences.

6. Truth-Functional Connectives • Logic involving non-truth-functional connectives is too complicated for an introductory course like ours. • Thus, we focus on the logic involving only truth-functional connectives. • There are five truth-functional connectives that we will be interested in for the purposes of this class: • the negation (not) • the conjunction (and) • the disjunction (or) • the conditional (if) • the (material) biconditional (if and only if) • We will see them in this order. In addition, we will see how to symbolize sentences involving “neither A nor B”.

7. 1. The Negation (Not) • Negating/negation is an operation that changes a true sentence into a false one, and a false sentence into a true one. • The negation is a truth-functional connective usually expressed by the English term “not”. • The negation’s truth-functional properties are summarized by the following truth-table: A Not A T F F T

8. Ways of Expressing Negation • There are numerous ways in which negation can be expressed in English: A Not A • She will do it. She will not do it. • Harry is normal. Harry is abnormal. • Jack loves Jill. It is not the case that Jack loves Jill. • Ice is present. Ice is absent. • ‘Not A’ is symbolized “~A” by Logicians. (Try 9 on p.38.)

9. 2. The Conjunction (And) • A conjunction is usually expressed with the English word “and”, i.e., as “A and B”. • The conjunction is a truth-functional connective. • The conjunction is also expressed by the English terms “also”, “but”, “furthermore”, “moreover”, and “while”. • Conjunction is symbolized “A  B”. (Try Q4, 12, 1, 11 and 15.)

10. The Conjunction (And) • Example: A pessimist forgets to laugh, but an optimist laughs to forget. • This conjunction has two conjuncts: “A pessimist forgets to laugh” and “An optimist laughs to forget”. • The conjuncts of a conjunction are the simpler sentences that the connective “and” joins together.

11. The Conjunction (And): Truth-Table • The truth-functional properties of the conjunction are summarized by the following truth-table: A B A  B T T T T F F F T F F F F • Summary: a conjunction is true if and only if both conjuncts are true.

12. 3. The disjunction (Either … or …) • The disjunction is expressed in English using the word “or” or sometimes the phrase “either … or…”. • The two simpler sentences on either side of “or” are called disjuncts. • Logicians generally think there are two types of the disjunction, exclusive and inclusive.

13. The Disjunction:Exclusive & Inclusive • The exclusive disjunction means “one or the other, but not both”. • Example: When a menu states: “Soup or salad is included in the price of the meal”. • The inclusive disjunction means “one or the other, or possibly both”. • Example: When a road sign says: “This bridge is open to automobile or truck traffic.” • Read all disjunctions inclusively in the following slides without a special notice.

14. The Disjunction (Either … or …): Symbolization and Truth-Table • Inclusive disjunction is symbolized “A  B”. (Try Q6 and 13.) • The truth-functional properties of the (inclusive) disjunction are given by the following truth-table: A B A  B T T T T F T F T T F F F • Summary: a(n inclusive) disjunction is false if and only if both disjuncts are false.

15. Symbolizing the Exclusive Disjunction • Occasionally it is clear that the intended meaning of an “either … or …” statement is an exclusive disjunction. • In this case the disjunctive statement should be symbolized “(A  B) (A  B)”, which, when read, says: “A or B, and not, A and B”.

16. 4. Conditional Sentences: Definition • Conditional sentences (AKA conditionals) are used when we want to say that the truth of one claim is a sufficient condition for the truth of another claim. • The classic way to express a conditional sentence is to use the phrase structure “if A (then) B”. • Logicians symbolize this “A  B”. (Try Q5 and 14.)

17. Conditional Sentences: A  B • Conditional sentences are made up of two sentences: • one that states a sufficient condition (labeled A in the above), and • one that expresses a necessary condition (labeled B in the above) • The sufficient condition part is called the antecedent of the conditional • The other part is called the consequent of the conditional

18. Material Conditionals • Material conditionals are the conditionals in which the relationship between the antecedent and consequent is truth- functional. • Example 1: If the sun is cold, then the Pope isn’t a Catholic. • Example 2: If I don’t die today, then I will die eventually.

19. The Material Conditional as A Truth-Functional Connective • A material conditional is true unless the antecedent is true and the consequent is false. • That is, the only situation in which a material conditional is false is that in which the antecedent is true and the consequent is false. • For simplicity, we will take all conditionals as material conditionals, though actually not all conditionals are material. (See p. 33 of our course packet.)

20. The Material Conditional • The properties of the material conditional are summarized by the following truth-table: • A B A  B • T T T • T F F • F T T • F F T

21. Is This Truth Table Correct? • The second line in the last page is obvious; it is the others we might doubt. • In particular, why are material conditionals always TRUE if the antecedent is false? • For one, this interpretation gets things right. • For illustration, consider this sentence: “If x is less than 5, then x is less than 20.” • We think this sentence cannot be false: that is, it is necessarily true. The truth table of the last slide makes it so. p: x is less than 5, and q: x is less than 20; • x = 3 p: true q: true p  q: true • x = 7 p: false q: true p  q: true! • x = 24 p: false q: false p  q: true!

22. Continued • In contrast, suppose the truth table of material conditionals are changed so that they are false when their antecedents are false. Then, the sentence “If x is less than 5, then x is less than 20” can be false. That is: p: x is less than 5, and q: x is less than 20; • x = 3 p: true q: true p  q: true • x = 7 p: false q: true p  q: false! • x = 24 p: false q: false p  q: false! • We think “If x is less than 5, then x is less than 20” is necessarily true, i.e., cannot be false. Thus, this result is unacceptable.

23. Continued • There is another reason for not making material conditionals false when their antecedents are false. • The reason is that it makes material conditionals equivalent to conjunctions. Remember the truth table of conjunctions: A B A  B T T T T F F F T F F F F

24. Conditional Sentences:Ordinary Ways of Expression • There are many ways to express conditionals: • “B provided A” / “B, provided A” • “B whenever A” / “Whenever A, B” • “B if A” / “If A (then) B”. • All of the above are symbolized “A  B”. • This is because the expressions like “if”, “provided” and “whenever” indicate that the immediately following sentence is a sufficient condition for the truth of the other sentence. • “B unless A” (or “Unless A, B”) means “B if not A,” so it should be symbolized “~A  B”. (Try Q2.)

25. “A only if B”/ ”Only if B, A” • “A only if B” (or “Only if B, A”) is equivalent to “If A then B.” So it is symbolized A  B. • To illustrate this point, consider this English sentence: “Lawyers are nice to you only if you have money.” This has the structure A only if B. This sentence can be translated as: “If you don’t have money, lawyers are not nice to you.” This sentence can be further translated as: “If lawyers are nice to you, you have money.” This sentence has the structure if A, then B. • Thus, we can symbolize the forms of English sentences “A only if B” and “Only if B, A” A  B. (Try Q3.)

26. Necessary Condition, Sufficient Condition, Once Again • In “If A, B” and “B if A”, Aexpresses a sufficient condition, and B a necessary condition. • In “Only if A, B” and “B only if A”, A expresses a necessary condition, and B a sufficient condition. • This is why these sentences are translated as “B A” • A  Bis NOT logically equivalent to B A. Therefore, don’t symbolize “p only if q” and “Only if q, p” q  p.

27. 5. The Material Biconditional(If and Only If) • The material biconditional is usually expressed by the English phrase “if and only if”. • It can also be expressed by such English phrases as “when and only when” or “just in case”. • These expresses the idea of a necessary and sufficient condition. • The material biconditional is a truth-functional connective. • A material biconditional is symbolized “A  B”. (Try Q7 and 10.) • “A  B” has the same meaning as “(A  B)  (B  A)”.

28. The Material Biconditional(If and Only If): Truth-Table • The truth-functional properties of the material biconditional are summarized by the following truth-table: A B A  B T T T T F F F T F F F T • Summary: a material biconditional is true if and only if both component sentences take the same truth value.

29. Two Cautions on Conditionals and Biconditionals • Don’t symbolize “A only if B” (or “Only if B, A”) “A  B”. • “A  B” is the symbolization of “A if and only if B”. • Don’t symbolize as “B → A” the sentences that should be symbolized “A → B”, for example, “B if A” and “A only if B”. • “A → B” is NOT equivalent to “B → A”.

30. 6. Neither A nor B • “Neither A nor B” can be symbolized in two ways; either “A  B” or “(A  B)”. (Try Q8.) • We will see later that these two sentences are logically equivalent. • (Logical equivalence roughly means the sameness of the meanings. We will see the more strict definition of logical equivalence later.)

31. Symbols for Truth-Functional Connectives: A Summary • Arrow  (if…then…) • Tilde  (not) • Dot  (and) • Wedge  (either…or…) • Double Arrow  (if and only if)

32. Dictionaries • In the symbolization of ordinary sentences, we need a dictionary. • A dictionary is an assignment of lower-case letters to simple sentences. • Example: the following chart is a dictionary: p: Makoto is a weirdo. q: Everyone is someone else’s weirdo. r: All philosophers are weirdoes.

33. Symbolization • Symbolization (in this course) is the process of using a dictionary and the truth-functional connectives to express a sentence, or a collection of sentences, symbolically. • Example: If we use the dictionary on the last page, the sentence “Makoto is a weirdo and everyone is someone else’s weirdo” can be symbolized as “p  q”.

34. Symbolization: Example • A Sample Argument: If all philosophers are weirdoes, then Makoto is a weirdo. All philosophers are weirdoes.  Makoto is a weirdo. • This argument can be symbolized: r  p r  p

35. Sample Symbolization 1 • Consider the following sentence: Hilary is more irritable if and only if Bill is more irritating. • Dictionary: (Try to make it!) • a: Hilary is more irritable. • b: Bill is more irritating. • Symbolization: • a  b

36. Sample Symbolization 2 • Consider the following sentence: I want to be a feminist unless my boyfriend stops me. • Dictionary: (Try to make it!) • f: I want to be a feminist. • s: My boyfriend stops me. • Symbolization: • s  f.

37. Sample Symbolization 3 • Consider the following sentence: Neither making a Japanese kid speak up nor making an American kid shut up is easy. • Dictionary: (Try to make it!) • j: Making a Japanese kid speak up is easy. • m: Making an American kid shut up is easy. • Symbolization: • (j  m), or j  m.