Symbolic Language and Basic Operators Kareem Khalifa Department
Symbolic Language and Basic Operators Kareem Khalifa Department of Philosophy Middlebury College
Overview o o o o Why this matters Artificial versus natural languages Conjunction Negation Disjunction Punctuation Sample Exercises
Why this matters o 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 n Judging the validity of arguments is an important skill, so symbolic languages allow us to focus and hone this skill. o Symbolic language encourages precision. This precision can be reintroduced into natural language.
More on why this matters o You are learning the conditions under which a whole host of statements are true and false. n This is crucial for criticizing arguments. n It is a good critical practice to think of conditions whereby a claim would be false.
Artificial versus natural languages o Symbolic language (logical syntax) is an artificial language n It was designed to be as unambiguous as possible. o English (French, Chinese, Russian, etc. ) are natural languages n They weren’t really designed in any strong sense at all. They emerge and evolve through very “organic” and (often) unreflective cultural processes. n As a result, they have all sorts of ambiguities. o The tradeoff is between clarity and expressive richness. Both are desirable, but they’re hard to combine.
Propositions as letters o Logical syntax represents individual propositions as letters. n When we don’t care what the proposition actually stands for, we represent it with a lowercase letter, typically beginning with p. n When we have a fixed interpretation of a proposition, we represent it with a capital letter, typically beginning with P. o Ex. Let P = “It’s raining. ” o Sometimes, letters are subscripted. Each subscripted letter should be interpreted as a different proposition.
Dispensable translation manuals o Often, the letters are given an interpretation, i. e. , they are mapped onto specific sentences in English. n Ex. Let P be “It is raining; ” Q be “The streets are wet, ” etc. o However, this is not necessary. The validity of an argument doesn’t hinge on the interpretation. n If p then q p q
Logical connectives: some basics o A logical connective is a piece of logical syntax that: n Operates upon propositions; and n 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. o Today, we’ll look at AND, NOT, and OR. n Khalifa is cunning and cute. n Khalifa is not cunning. n Either Khalifa is cunning or he is foolish.
Conjunction o AND-statements n Middlebury has a philosophy department AND it has a neuroscience program. o Represented either as “ ” or as “&” n I recommend “&, ” since it’s just SHIFT+7 o “p & q” will be true when p is true and q is true; false otherwise.
Truth-tables o Examine all of the combinations of component propositions, and define the truth of the compound proposition. p q p&q T T F F F T F F
Subtleties in translating English conjunctions into symbolic notation o The “and” does not always appear in between two propositions. n Khalifa is handsome and modest. n Khalifa and Grasswick teach logic. n Khalifa teaches logic and plays bass.
More subtleties o Sometimes “and” in English means “and subsequently. ” n The truth-conditions for this are the same as “&”, but the meaning of the English expression is not fully captured by the formal language. o Many English words have the same truthconditions as “&” but have additional meanings. n Ex. “but, ” “yet, ” “still, ” “although, ” “however, ” “moreover, ” “nevertheless” o General lesson: The meaning of a proposition is not (easily) identifiable with the truth-functions that define it in logical notation.
Negation o Represented by a “~” o 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 o Represented in English by “or. ” o However, there are two senses of “or” in English: n Inclusive: when p AND q are true, p OR q is true n Exclusive: when p AND q is true, p OR q is false o Logical disjunction (represented as “v”) is an inclusive “or. ”
Which is inclusive and which is exclusive? o You can take Intro to Logic in the fall or the spring. n Exclusive. You can’t take the same course twice! o You can take Intro to Logic or Calculus I. n Inclusive. You’d then be learned in logic and in math!
Truth table for disjunction p q pvq T T F F F
Punctuation o We can daisy-chain logical connectives together. n Either Polly and Quinn or Rita and Sam will not win the game show. o If we have no way of grouping propositions together, it becomes ambiguous n ~P & Q v R & S o Logic follows the same conventions as math { [ ( ) ] }, though some logicians prefer to use only ( ( ( ) ) ). n [(~P&~Q) v (~R&~S)]
A few quirks o A negation symbol applies to the smallest statement that the punctuation permits. n Ex. “~p & q” is equivalent to “(~p) & q” o It is NOT equivalent to “~(p & q)” n This reduces the number of ( ) o We can also drop the outermost brackets of any expression. n Ex. “[p & (q v r)]” is equivalent to “p & (q v r)”
Lessons about punctuation from logic o Make sure, in English, that you phrase things so that there is no ambiguity n Commas are very useful here o When reading, be especially sensitive to small subtleties about logical structure that would change the meaning of a passage.
Sample Exercise A 3 (327) o ~London is the capital of England & ~Stockholm is the capital of Norway o ~T & ~F o. F&T o. F
Sample Exercise A 4 (327) o ~(Rome is the capital of Spain v Paris the capital of France) o ~(F v T) o ~(T) o. F
Sample Exercise A 9 (327) o (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) o (T v F) & (~T & ~F) o (T) & (F & T) o (T) & (F) o. F
Sample Exercise C 3 (329) o o Q v ~X Q v ~F Qv. T T
Exercise C 12 (329) o (P & Q) & (~P v ~Q) o The first conjunct (P&Q), can only be true if P = Q = T o However, this would make the whole conjunction false. Here’s the ‘proof’: n n (T&T) & (~T v ~T) (T) & (F v F) (T) & (F) F
Exercise D 9 (330) o It is not the case that Egypt’s food shortage worsens, and Jordan requests more U. S. aid. o ~E & J
- Slides: 25