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

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

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.

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.

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.

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.

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

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 not cunning. Either Khalifa is cunning or he is foolish.

Conjunction AND-statements Represented either as “” or as “&” 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.

Truth-tables Examine all of the combinations of component propositions, and define the truth of the compound proposition. p q p & q T F T F F F

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.

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.

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

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.”

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!

Truth table for disjunction p q p v q T F T T T F

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)]

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)”

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.

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

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

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

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

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

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