English For Mathematics

Eleventh Lecture

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Mathematical logic• Mathematical logic is a subfield of mathematics exploring the

applications of formal logic to mathematics. It bears close connectionsto metamathematics, the foundations of mathematics, and theoreticalcomputer science. The unifying themes in mathematical logic includethe study of the expressive power of formal systems and the deductivepower of formal proof systems.

• Mathematical logic is often divided into the fields of :1. set theory,2. model theory,3. recursion theory, and4. proof theory.

• These areas share basic results on logic, particularly first-order logic,and definability. In computer science (particularly in the ACMClassification) mathematical logic encompasses additional topics notdetailed in this article; see Logic in computer science for those.

Set theory

• Set theory is the branch of mathematicallogic that studies sets, which informally arecollections of objects. Although any type ofobject can be collected into a set, set theoryis applied most often to objects that arerelevant to mathematics. The language ofset theory can be used in the definitions ofnearly all mathematical objects.

Basic concepts and notation Set Theory• Set theory begins with a fundamental binary relation between

an object o and a set A. If o is a member (or element) of A, writeo ∈ A. Since sets are objects, the membership relation can relatesets as well.

• A derived binary relation between two sets is the subsetrelation, also called set inclusion. If all the members of set A arealso members of set B, then A is a subset of B, denoted A ⊆ B.For example, {1,2} is a subset of {1,2,3} , and so is {2} but {1,4} isnot. From this definition, it is clear that a set is a subset of itself;for cases where one wishes to rule this out, the term propersubset is defined. A is called a proper subset of B if and only if Ais a subset of B, but B is not a subset of A. Note also that 1 and 2and 3 are members (elements) of set {1,2,3} , but are notsubsets, and the subsets in turn are not as such members of theset.

Basic concepts and notation Set Theory• Just as arithmetic features binary operations on numbers, set theory features

binary operations on sets. The:• Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a

member of A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2,3, 4} .

• Intersection of the sets A and B, denoted A ∩ B, is the set of all objects thatare members of both A and B. The intersection of {1, 2, 3} and {2, 3, 4} is theset {2, 3} .

• Set difference of U and A, denoted U \ A, is the set of all members of U thatare not members of A. The set difference {1,2,3} \ {2,3,4} is {1} , while,conversely, the set difference {2,3,4} \ {1,2,3} is {4} . When A is a subset of U,the set difference U \ A is also called the complement of A in U. In this case, ifthe choice of U is clear from the context, the notation Ac is sometimes usedinstead of U \ A, particularly if U is a universal set as in the study of Venndiagrams.

• Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is the set ofall objects that are a

Basic concepts and notation Set Theory• Just as arithmetic features binary operations on numbers, set theory features

binary operations on sets. The:

• Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is the set ofall objects that are a member of exactly one of A and B (elements which arein one of the sets, but not in both). For instance, for the sets {1,2,3} and{2,3,4} , the symmetric difference set is {1,4} . It is the set difference of theunion and the intersection, (A ∪ B) \ (A ∩ B) or (A \ B) ∪ (B \ A).

• Cartesian product of A and B, denoted A × B, is the set whose members areall possible ordered pairs (a,b) where a is a member of A and b is a memberof B. The cartesian product of {1, 2} and {red, white} is {(1, red), (1, white), (2,red), (2, white)}.

• Power set of a set A is the set whose members are all possible subsets of A.For example, the power set of {1, 2} is { {}, {1}, {2}, {1,2} } .

• Some basic sets of central importance are the empty set (the unique setcontaining no elements), the set of natural numbers, and the set of realnumbers.

Types of Sentences• A mathematical sentence is one in which a fact or complete idea

is expressed. Because a mathematical sentence states a fact,many of them can be judged to be "true" or "false". Questionsand phrases are not mathematical sentences since they cannotbe judged to be true or false.

1. "An isosceles triangle has two congruent sides." is a truemathematical sentence.

2. "10 + 4 = 15" is a false mathematical sentence.

3. "Did you get that one right?" is NOT a mathematical sentence -it is a question.

4. "All triangles" is NOT a mathematical sentence - it is a phrase.

There are two types of mathematical sentences:An open sentence is a sentence which contains a variable.• "x + 2 = 8" is an open sentence -- the variable is "x."• "It is my favorite color." is an open sentence-- the variable is "It."• The truth value of theses sentences depends upon the value

replacing the variable.A closed sentence, or statement, is a mathematical sentence whichcan be judged to be true or false. A closed sentence, or statement,has no variables.• "Garfield is a cartoon character." is a true closed sentence, or

statement.• "A pentagon has exactly 4 sides." is a false closed sentence, or

statement.

A compound sentenceA compound sentence is formed when two or more thoughts areconnected in one sentence. Words such as and, or, if...then and ifand only if allow for the formation of compound sentences, orstatements. Notice that more than one truth value is involved inworking with a compound sentence.• "Today is a vacation day and I sleep late."

• "You can call me at 10 o'clock or you can call me at 2 o'clock."

• "If you are going to the beach, then you should take your sunscreen."

• "A triangle is isosceles if and only if it has two congruent sides."

Negation• In logic, negation, also called logical complement, is an operation that takes a

proposition p to another proposition "not p", written ¬p, which is interpretedintuitively as being true when p is false and false when p is true. Negation isthus a unary (single-argument) logical connective. It may be applied as anoperation on propositions, truth values, or semantic values more generally. Inclassical logic, negation is normally identified with the truth function thattakes truth to falsity and vice versa. In intuitionistic logic, according to theBrouwer–Heyting–Kolmogorov interpretation, the negation of a proposition pis the proposition whose proofs are the refutations of p.

• Classical negation is an operation on one logical value, typically the value of aproposition, that produces a value of true when its operand is false and avalue of false when its operand is true. So, if statement A is true, then ¬A(pronounced "not A") would therefore be false; and conversely, if ¬A is true,then A would be false.

• The truth table of ¬p is as follows: 𝒑 ∼ 𝒑

True False

False True

Logical conjunction• In logic and mathematics, and is the truth-functional operator of logical

conjunction; the and of a set of operands is true if and only if all of itsoperands are true. The logical connective that represents this operator istypically written as ∧ or ⋅ .

• "A and B" is true only if A is true and B is true.

• Venn diagram of 𝐴 ∧ 𝐵

• Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true.

Logical Form• Logic is generally considered formal when it analyzes and represents the form of any

valid argument type. The form of an argument is displayed by representing itssentences in the formal grammar and symbolism of a logical language to make itscontent usable in formal inference. If one considers the notion of form toophilosophically loaded, one could say that formalizing simply means translatingEnglish sentences into the language of logic.

• This is called showing the logical form of the argument. It is necessary becauseindicative sentences of ordinary language show a considerable variety of form andcomplexity that makes their use in inference impractical. It requires, first, ignoringthose grammatical features irrelevant to logic (such as gender and declension, if theargument is in Latin), replacing conjunctions irrelevant to logic (such as "but") withlogical conjunctions like "and" and replacing ambiguous, or alternative logicalexpressions ("any", "every", etc.) with expressions of a standard type (such as "all", orthe universal quantifier ∀).

• Second, certain parts of the sentence must be replaced with schematic letters. Thus,for example, the expression "all As are Bs" shows the logical form common to thesentences "all men are mortals", "all cats are carnivores", "all Greeks arephilosophers", and so on.

Syllogism• A syllogism (Greek: συλλογισμός syllogismos, "conclusion, inference") is a

kind of logical argument that applies deductive reasoning to arrive at aconclusion based on two or more propositions that are asserted or assumedto be true

• In its earliest form, defined by Aristotle, from the combination of a generalstatement (the major premise) and a specific statement (the minor premise),a conclusion is deduced. For example, knowing that all men are mortal (majorpremise) and that Socrates is a man (minor premise), we may validlyconclude that Socrates is mortal. Syllogistic arguments are usuallyrepresented in a three-line form (without sentence-terminating periods)

Basic structure

A categorical syllogism consists of three parts:

a. Major premise

b. Minor premise

c. Conclusion

Syllogism• Each part is a categorical proposition, and each categorical proposition contains two

categorical terms. In Aristotle, each of the premises is in the form "All A are B," "SomeA are B", "No A are B" or "Some A are not B", where "A" is one term and "B" isanother. "All A are B," and "No A are B" are termed universal propositions; "Some Aare B" and "Some A are not B" are termed particular propositions. More modernlogicians allow some variation. Each of the premises has one term in common withthe conclusion: in a major premise, this is the major term (i.e., the predicate of theconclusion); in a minor premise, it is the minor term (the subject) of the conclusion.For example:

1) Major premise: All men are mortal

2) Minor premise: Socrates is a man

3) Conclusion: Therefore, Socrates is mortalThere are infinitely many possible syllogisms, but only 256 logically distinct types and only 24 valid types (enumerated below). A syllogism takes the form:

1. Major premise: All M are P.

2. Minor premise: All S are M.

3. Conclusion: All S are P.

(Note: M – Middle, S – subject, P – predicate. See below for more detailed explanation.)

Relationships between thefour types of propositions inthe square of opposition

(Black areas are empty,red areas are nonempty.)

Syllogism• The premises and conclusion of a syllogism can be any of four types, which

are labeled by letters[9] as follows. The meaning of the letters is given by thetable:

• In Analytics, Aristotle mostly uses the letters A, B and C (actually, the Greekletters alpha, beta and gamma) as term place holders, rather than givingconcrete examples, an innovation at the time. It is traditional to use is ratherthan are as the copula, hence All A is B rather than All As are Bs.

Syllogism• On the other hand, in modern mathematical logic, however, statements

containing words "all", "some" and "no", can stated in terms of set theory. Ifthe set of all A's is labeled as s(A) and the set of all B's as s(B), then:

• By definition, the empty set is a subset of all sets. From this it follows that,according to this mathematical convention, if there are no A's, then thestatements "All A is B" and "No A is B" are always true whereas thestatements "Some A is B" and "Some A is not B" are always false. This,however, implies that AaB does not entail AiB, and some of the syllogismsmentioned above are not valid when there are no A's.