OpenDocProject:LIS590ON/2010/08/24

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Reading: Forall(x) by PD Magnus

Chapter 1

Keep straight - false vs. true; and valid vs. invalid.

Basics: premise --> conclusion generally in logic we want our sentences to be statements that are true or false. Interrogative, Imperative and Exclamations are generally not logical sentences. 2 ways for argument to be weak: 1. One of the premises might be false 2. The premises do not support the conclusion

An argument that had no weakness of the second kind (both premises support the conclusion) would have perfect logical form. If its premises were true, then its conclusion would necessarily be true. We call such an argument `deductively valid'

An argument is deductively valid if and only if it is impossible for the premises to be true and the conclusion false.

A deductively valid argument does not need to have true premises or a true conclusion.

An "'inductive argument"' generalizes from many cases to a conclusion about all cases.

Truth Values= True or false is said to be the truth-value of a sentence.

contingent sentence = a sentence that is neither a tautology nor a contradiction.

tautology = a logically true sentence

"'contradiction = a logically false sentence

logically equivalent = two sentences have the same truth value

consistent = 2 sentences can be true at same time (does not mean that they are)

inconsistent = 2 sentences can't be true at the same time

'Set' A set can be inconsistent even when all of the sentences in it are either contingent or tautologous. When a single sentence is a contradiction, then that sentence alone cannot be true.

This book is concerned with two types of logic:

  1. Sentential Logic
  2. Quantified Logic


Sentential Logic

In SL, the smallest units are sentences themselves. Simple sentences are represented as letters and connected with logical connectives like 'and' and 'not' to make more complex sentences.

Quantified Logic = QL. In QL, the basic units are objects, properties of objects, and relations between objects.

Logical languages that make the assumption that there are only two truth-values (true and false) are called "'bivalent"' which means two-valued

Sentential Logic

Represent sentences with capital letters (A)

The sentences that can be symbolized with sentence letters are called atomic sentences, because they are the basic building blocks out of which more complex sentences can be built.

Negation

¬ wikimarkup = ¬

7. Elliott is happy.

8. Elliott is unhappy.


If we let H mean `Elliot is happy', then we can symbolize sentence 7 as H. However, it would be a mistake to symbolize sentence 8 as ¬H. If Elliott is unhappy, then he is not happy| but sentence 8 does not mean the same thing as `It is not the case that Elliott is happy.' It could be that he is not happy but that he is not unhappy either. Perhaps he is somewhere between the two. In order to symbolize sentence 8, we would need a new sentence letter.

For any sentence A: If A is true, then ¬A is false. If ¬A is true, then A is false. Using `T' for true and `F' for false, we can summarize this in a characteristic truth table for negation


Conjunction

& wikimarkup = "amp 'amp' ;"

A sentence can be symbolized as A & B if it can be paraphrased in English as `Both A, and B.' Each of the conjuncts must be a sentence.

Sentences that can be paraphrased `A, but B' or `Although A, B' are best symbolized using conjunction: A &B

Once we translate part of a sentence as B, any further structure is lost...so you cannot say for instnace

1. Barbara is athletic and energetic

Barbara is athletic = A

1 = A and energetic (this is incorrect)

1 = A and Barbara is energetic (this is correct)

Disjunction

∨ = wikimarkup "amp or ; "

D: Denison will play golf with me. E: Ellery will play golf with me. M: Denison will watch movies.

D∨M ... D and M are disjuncts

Like conjunction, disjunction is symmetrical. A∨B is logically equivalent to B∨A.

A sentence can be symbolized as A ∨B if it can be paraphrased in English as `Either A, or B.' Each of the disjuncts must be a sentence.

Inclusive OR = when an or statement means on, the other OR both (but always 1 of the three)

To negate an entire disjunction use () so `It is not the case that (S1 _S2).' is best represented as

¬(S1 _S2).

Conditional

⇒ (otherwise known as if then)

If you cut the red wire the bomb will explode = R → B The bomb will explode, only if you cut the red wire = B → R

In the first sentence above, R or the first clause is considered the antecedant and B is considered the Consequent. It is important to remember that the connective `!' says only that, if the antecedent is true, then the consequent is true. It says nothing about the causal connection between the two events.

The paraphrased sentence `A only if B' is logically equivalent to `If A, then B.'

Material Condition= This means that when A is false, the conditional A→B is automatically true, regardless of the truth value of B. If both A and B are true, then the conditional A→B is true.

In short, A→B is false if and only if A is true and B is false. We can summarize this with a characteristic truth table for the conditional

A| B| A→B
T  T   T
T  F    F
F  T    T
F  F    T

Biconditional

↔ (otherwise known as IF and Only IF)

Can also be represented as (T → S)&(S →T).

24. The figgure on the board is a triangle only if it has exactly three sides. 25. The figure on the board is a triangle if it has exactly three sides. 26. The figure on the board is a triangle if and only if it has exactly three sides.

S ↔ T; Sentence 26 says that T is true if and only if S is true; we can infer S from T, and we can infer T from S.

A↔B is true if and only if A and B have the same truth value. This is the characteristic truth table for the biconditional:

A | B | A↔B
T    T    T
T    F    F
F    T    F
F    F    T

Other Symbolization

If a sentence can be paraphrased as `Unless A, B,' then it can be symbolized as A∨B.


Object Language = SL, or any language we are using to model

Metalanguage = language we are using to speak about Object Language (such as English) we also use Metavariables, to talk about examples that are true for instances of an Object Language, such as AB

Expressions = Any string of an object language

WFF = Well Formed Forumlae

Recursion = Recursive definitions begin with some speci�able base elements and de�ne ways to inde�nitely compound the base elements.

Main Logical Operator = The connective you look to first when decomposing a sentence.

Truth Tables

Truth-functional connectiveness

The truth-value of the compound sentence depends only on the truth-value of the atomic sentences that comprise it. In order to know the truth-value of (D ↔ E), for instance, you only need to know the truth-value of D and the truth-value of E. Connectives that work in this way are called truth-functional.

  • All logical operators in SL are truth functional-- This is not true of modal logic, which also incorporates an operator for possibility


Logical equivalence- Two sentences are logically equivalent in English if they have the same truth value as a matter logic.


Consistency - A set of sentences in English is consistent if it is logically possible for them all to be true at once.


Validity - An argument in English is valid if it is logically impossible for the premises to be true and for the conclusion to be false at the same tim

Quantified Logic

Predicates are the basic unit of QL -- such as 'is a dog' It is neither true nor false. In order to be true or false, we need to specify something: Who or what is it that is a dog?

In QL we rep. Predicates with capital letters (e.g. D) and lower case letters for specific "things" (e.g. b)

Such that 'Db' might mean "Bettie is a dog"

‘∃’ will mean ‘There is some x’ So to say that there is a dog, we can write ∃xDx; that is: "There is some x such that x is a dog"

Singular Terms

In English -- a singular term is like a noun- except that it refers to some SPECIFIC person place or thing. A dog is not a singular term. There are millions of dogs. My dog Cesc is a specific dog (and a fictional dog), and it is singular term.

In general, a name is singular term because it picks out a specific individual instance of a person.

Definite Description - picks out an individual by means of a unique description. For example, ‘the tallest member of Monty Python’ and ‘the first emperor of China’ are definite descriptions.

Singular terms should be referenced with lowercase letters such as "a" - "w"

Singular terms are also called "constants" because they pick out specific instances of things. "x" "y" and "z" will be reserved for Variables.

Predicates

The simplest predicates are properties of individuals. They are things you can say about an object.

  • One-Place or Monadic-predicates are those which have only one space for the "owner" of the property..."The piano fell on ____" is a one place predicate.
  • Two-Place or Dyadic predicates use the form "____ is bigger than _____"
  • Three-place or Triadic predicate "___ borrowed ____ from _______"

Predicates with more than one place are called polyadic. Predicates with n places, for some number n, are called n-place or n-adic.

  • When we give a symbolization key for predicates, we will not use blanks; instead, we will use variables.


Quantifiers

‘∀’ symbol. This is called the universal quantifier.

∀xHx. Paraphrased in English, this means ‘For all x, x is happy.’ We call ∀x an x-quantifier. The formula that follows the quantifier is called the scope of the quantifier. We will give a formal definition of scope later, but intuitively it is the part of the sentence that the quantifier quantifies over. In ∀xHx, the scope of the universal quantifier is Hx.

Existential quantifier, ∃

∀xA is logically equivalent to ¬∃x¬A and vice versa

Someone - ∃ Everyone - ∀

  • A conditional will usually be the natural connective to use with a universal quantifier, but a conditional within the scope of an existential quantifier can do very strange things. As a general rule, do not put conditionals in the scope of existential quantifiers unless you are sure that you need one.

=Empty Predicates

A predicate need not apply to anything in the UD. A predicate that applies to nothing in the UD is called an empty predicate.


When we use quantifiers, followed by predicates in ( ) we are using the Universal domain to make statements about those predicates within that domain-- so to say ∀x(Rx→Mx) means that any member of the UD that is a refrigerator is a monkey. (or If RX is in the Domain ∀x then it is Mx)

Some Generalities:

 A UD must have at least one member.

 A predicate may apply to some, all, or no members of the UD.

 A constant must pick out exactly one member of the UD. A member of the UD may be picked out by one constant, many constants, or none at all.

More on Universe of Discourse

Universe of Discourse

In order to eliminate this ambiguity, we will need to specify a universe of discourse— abbreviated UD. The UD is the set of things that we are talking about. So if we want to talk about people in Chicago, we define the UD to be people in Chicago.

In QL, the UD must be non-empty; that is, it must include at least one thing.

Non Referring Terms

In QL, each constant must pick out exactly one member of the UD. A constant cannot refer to more than one thing- it is a singular term.