Symbolic Language and Basic Operators

1. Symbolic Language and Basic Operators Kareem Khalifa Department of Philosophy Middlebury College

2. Overview • Why this matters • Artificial versus natural languages • Conjunction • Negation • Disjunction • Punctuation • Sample Exercises

3. Why this matters • Symbolic language allows us to abstract away the complexities of natural languages like English so that we can focus exclusively on ascertaining the validity of arguments • Judging the validity of arguments is an important skill, so symbolic languages allow us to focus and hone this skill. • Symbolic language encourages precision. This precision can be reintroduced into natural language.

4. More on why this matters • You are learning the conditions under which a whole host of statements are true and false. • This is crucial for criticizing arguments. • It is a good critical practice to think of conditions whereby a claim would be false.

5. Artificial versus natural languages • Symbolic language (logical syntax) is an artificial language • It was designed to be as unambiguous as possible. • English (French, Chinese, Russian, etc.) are natural languages • They weren’t really designed in any strong sense at all. They emerge and evolve through very “organic” and (often) unreflective cultural processes. • As a result, they have all sorts of ambiguities. • The tradeoff is between clarity and expressive richness. Both are desirable, but they’re hard to combine.

6. Propositions as letters • Logical syntax represents individual propositions as letters. • When we don’t care what the proposition actually stands for, we represent it with a lowercase letter, typically beginning with p. • When we have a fixed interpretation of a proposition, we represent it with a capital letter, typically beginning with P. • Ex. Let P = “It’s raining.” • Sometimes, letters are subscripted. Each subscripted letter should be interpreted as a different proposition.

7. Dispensable translation manuals • Often, the letters are given an interpretation, i.e., they are mapped onto specific sentences in English. • Ex. Let P be “It is raining;” Q be “The streets are wet,” etc. • However, this is not necessary. The validity of an argument doesn’t hinge on the interpretation. • If p then q p  q

8. Logical connectives: some basics • A logical connective is a piece of logical syntax that: • Operates upon propositions; and • Forms a larger (compound) proposition out of the propositions it operates upon, such that the truth of the compound proposition is a function of the truth of its component propositions. • Today, we’ll look at AND, NOT, and OR. • Khalifa is cunning and cute. • Khalifa is notcunning. • Either Khalifa is cunning orhe is foolish.

9. Conjunction • AND-statements • Middlebury has a philosophy department AND it has a neuroscience program. • Represented either as “” or as “&” • I recommend “&,” since it’s just SHIFT+7 • “p & q” will be true when p is true and q is true; false otherwise.

10. Truth-tables • Examine all of the combinations of component propositions, and define the truth of the compound proposition. T F F F

11. Subtleties in translating English conjunctions into symbolic notation • The “and” does not always appear in between two propositions. • Khalifa is handsome and modest. • Khalifa and Grasswick teach logic. • Khalifa teaches logic and plays bass.

12. More subtleties • Sometimes “and” in English means “and subsequently.” • The truth-conditions for this are the same as “&”, but the meaning of the English expression is not fully captured by the formal language. • Many English words have the same truth-conditions as “&” but have additional meanings. • Ex. “but,”“yet,”“still,”“although,”“however,”“moreover,”“nevertheless” • General lesson: The meaning of a proposition is not (easily) identifiable with the truth-functions that define it in logical notation.

13. Negation • Represented by a “~” • In English, “not,”“it’s not the case that,”“it’s false that,”“it’s absurd to think that,” etc. F T

14. Disjunction • Represented in English by “or.” • However, there are two senses of “or” in English: • Inclusive: when p AND q are true, p OR q is true • Exclusive: when p AND q is true, p OR q is false • Logical disjunction (represented as “v”) is an inclusive “or.”

15. Which is inclusive and which is exclusive? • You can take Intro to Logic in the fall or the spring. • Exclusive. You can’t take the same course twice! • You can take Intro to Logic or Calculus I. • Inclusive. You’d then be learned in logic and in math!

16. Truth table for disjunction T T T F

17. Punctuation • We can daisy-chain logical connectives together. • Either Polly and Quinn or Rita and Sam will not win the game show. • If we have no way of grouping propositions together, it becomes ambiguous • ~P & Q v R & S • Logic follows the same conventions as math { [ ( ) ] }, though some logicians prefer to use only ( ( ( ) ) ). • [(~P&~Q) v (~R&~S)]

18. A few quirks • A negation symbol applies to the smallest statement that the punctuation permits. • Ex. “~p & q” is equivalent to “(~p) & q” • It is NOT equivalent to “~(p & q)” • This reduces the number of ( ) • We can also drop the outermost brackets of any expression. • Ex. “[p & (q v r)]” is equivalent to “p & (q v r)”

19. Lessons about punctuation from logic • Make sure, in English, that you phrase things so that there is no ambiguity • Commas are very useful here • When reading, be especially sensitive to small subtleties about logical structure that would change the meaning of a passage.

20. Sample Exercise A3 (327) • ~London is the capital of England & ~Stockholm is the capital of Norway • ~T & ~F • F & T • F

21. Sample Exercise A4 (327) • ~(Rome is the capital of Spain v Paris the capital of France) • ~(F v T) • ~(T) • F

22. Sample Exercise A9 (327) • (London is the capital of England v Stockholm is the capital of Norway) & (~Rome is the capital of Italy & ~Stockholm is the capital of Norway) • (T v F) & (~T & ~F) • (T) & (F & T) • (T) & (F) • F

23. Sample Exercise C3 (329) • Q v ~X • Q v ~F • Q v T • T

24. Exercise C12 (329) • (P & Q) & (~P v ~Q) • The first conjunct (P&Q), can only be true if P = Q = T • However, this would make the whole conjunction false. Here’s the ‘proof’: • (T&T) & (~T v ~T) • (T) & (F v F) • (T) & (F) • F

25. Exercise D9 (330) • It is not the case that Egypt’s food shortage worsens, and Jordan requests more U.S. aid. • ~E & J