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SECTION 1: C LASS NOTES
This section contains the class notes for the course.
1.1. Notes for Chapter 1: Introduction
1.1.1. Introduction (1.1-1.3 of the Text)
1. What is logic? 1.1 Arguments (1) Some examples of arguments
Mary will marry John only if John loves her. John loves Mary.
Therefore, Mary will marry John.
All human beings are mortal. Socrates is a human being.
Therefore, Socrates is mortal.
If you can win the game, I would be the uncle of monkey. ......
(Therefore, you will not win the game.)
I will die if I am killed. I am not killed.
Therefore, I will not die.
All the students in the room are logic students. Some logic students are really boring.
Some students in the room are boring.
Swan a is white. Swan b is white. ...... Swan n is white.
Therefore, all swans are white.
(2) Components of arguments
Definition: An argument is a group of statements, one or more of which (the premises) are claimed to provide support for, or reasons to believe, one of the others (the conclusion).
The structure of an argument:
Premise 1 Premise 2 support Conclusion
Premises provide some grounds (not necessarily guarantee) for the truths of the conclusion. There is an inferential relationship between premises and conclusion.
(3) Deductive vs. Inductive Arguments
1.2 Definition:
Logic is a subject (an art?) of the study of the methods and principles used to distinguish good / cogent from bad /fallacious argument.
2. How to evaluate (deductive) arguments: validity and soundness 2.1 Two basic criteria of evaluation
Validity--the inferential relationship between Ps and C: Whether An Argument Ps support C and to what extent?
Soundness—the status of premises: whether Ps are true or acceptable?
A good argument: (a) All Ps are acceptable (true) and (b) Ps support C to the extent that if all Ps are true, then it is impossible for C to be false.
2.2 Validity Definitions:
- An Argument is valid if and only if it is logically impossible for the conclusion to be false if all the premises to be true.
- An argument is valid iff the truths of the premises guarantee the truth of the conclusion.
A few feature of validity:
- Truth-preserving: from the truth of the premises to the truth of the conclusion.
Truth premises conclusion
- Hypothetical situation: Suppose / assume that all the premises are true, not that all the premises are actually true. For example, the following argument is valid although all the premises are actually false:
All cats are sea creatures. (False) All sea creatures are clod-blooded killers. (False)
All cats are cold-blooded killers. (False)
Argument in English Argument in Logical notions
Mary will marry John only if John loves her. Marry (Mary, John) → Love (John, Mary) John loves Mary. Love (John Mary)
Therefore, Mary will marry John. Marry (Mary, John)
M → L L
M
All the students in the room are logic students. ∀x [(S (x) ∧ I (x)) → L (x)] Some logic students are really boring. ∃x [L (x) ∧ B (x)]
Some students in the room are boring. ∃x [(S (x) ∧ I (x)) ∧ B (x) ]
Step II— Formal proof : using some formal methods to determine the validity of the argument in logical notion.
truth-tree method Formal methods truth-table method natural derivation
4. The function of artificial / formal language (the language of first-order logic or FOL) in symbolic sciences (see 1.2)
- in philosophy
- in computer science
- in math.
- In AI (thinking = mental presentation + computation)
- In linguistics 5. Logic in rational inquiry (see the text 1.1) 6. Play with Tarski’s World
1.2. Notes for Chapter 2: Atomic Sentences
1.2.1. The Basic Structure of Atomic Sentences
(2.1, 2.2, 2.3, and 2.5 of the Text)
1. Comparison between simple English sentences and atomic sentences
Simple English Sentences Atomic sentences (FOL) (subject-predicate sentences)
John is a freshman Freshman (John) John swims. Swim (John) John loves Jenny. Love (John, Jenny) John prefers Jenny to Amy. Prefer (John, Jenny, Amy) John’s mother loves Jenny. Love (mother (John), Jenny) The father of Jenny is angry. Angry (father (Jenny)) John is the brother of Jenny. John = brother (Jenny) [relational identity]
2. Names
Definition: Names are individual constants that refer to some fixed individual objects or other.
(1) The rule of naming (p. 10)
- No empty name.
- No multiple references (do not use one name to refer to different objects).
- Multiple names: you can name one object by different names.
(2) General terms / names: using a predicate, instead of a constant, to represent a general term. For example, John is a student Student (John) [correct] John = student [ wrong !!!]
3. Predicates
Definition: Predicates are symbols used to denote some property of objects or some relationship between objects.
(1) Arity of predicates
- Unary predicates--property
- Binary predicates Relations
- Ternary predicates
(2) The predicates used in Tarski’s World : see p. 11.
(3) Two rules of predicates: see p.12.
Translation Manual
Components English FOL
Names John John Jenny Jenny Amy Amy
Predicates x is a freshman Freshman (x) x is angry Angry (x) x swims Swim (x) x loves y Love (x, y) x is y x = y x prefer y to z Prefer (x, y, z)
Functions x’s mother mother (x) the father of x father (x) the brother of x brother (x)
(2) The procedure of translation:
Simple English sentences atomic sentences in FOL (the sentences on the left of (the sentences on the left hand side of 1 ) hand side of 1 )
Translation manual
7. A classification of terms
Definition: A term is a noun / noun phrase or expression used to refer to an individual objects (either fixed or unfixed) object.
Simple terms: names / constants such as John. Fixed terms Complex terms (name-like terms): functions (function symbols + terms) such as the father of John Terms
Unfixed terms: variables such as x.
8. Class exercises
Problem 9 (p. 19). Translation between the relational language and the functional language. First of all, make a list of all available symbols in each language:
the relational language the functional language
Names Claire, Melanie, Jon Claire, Melanie, Jon
Predicates TallerThan(x, y) TallerThan(x, y) FatherOf (x, y): x is the father of y. x = y x = y
Functions father (x): the father of x
Second, translate from the relational language to the functional language:
- FatherOf (Jon, Claire) Jon = father (Claire)
Jon is the father of Claire.
- FatherOf (Jon, Melanie) Jon = father (Melanie)
- TallerThan (Melanie, Claire) TallerThan (Melanie, Claire)
Third, only sentence 1 can be translated into atomic sentence of the relational language.
- father (Melanie) = Jon FatherOf (Jon, Melanie)
- “father (Melanie) = father (Claire)” means “Melanie’s father is Claire’s father” or “Melanie and Claire have the same father.”
- “TallenThan (father (Claire), father (Jon))” means “Claire’s father is taller than Jon’s father.”
Problem 10 (p. 20)
First of all, make a translation manual as follows:
English FOL
Names Carl, Sam, Mary the same
Predicates x is the same as y x = y x is greater than y x > y
Functions the height of x height (x)
Then, translate sentences into logical notations, for example:
Problem 17 (p. 24) (1) One suggested translation manual.
English FOL
Names Claire, John, Jon, Nancy, Max, Mary Ellen France, Spain, Portugal, AIDS, influenza Company, Misery
Predicates x is between y and z in size. BetweenInSize (x, y, z) x is less contagious than y. LessContagious (x, y) x loves y Love (x, y) x shook y Shook (x, y) x is younger than y. YoungerThan (x, y)
Functions the father of x father (x) the hand of x hand (x) the eldest child of x and y eldestChild (x, y)
(2) Translation based on the above manual:
- LessContagious (AIDS, Influenza)
- Between (Spain, France, Portugal)
- Loves (Misery, Company)
- Shook (Max, hand (father (Claire)))
- YoungerThan (eldest (John, Nancy), eldest (Jon, Marry Ellen))
1.2.3. Methods of Proof (2.8 of the Text)
1. Logical Consequence
The conclusion is a logical consequence of its premises iff the argument is valid. A statement C is a logical consequence of a set of statements {P} iff if{P}are true, then C must be true.
{P1, P2, ..., Pn} ╞ C
For example,
{if A, then B; A} ╞ B If A, then B A
B
2. A Proof 2.1 Definition
A proof is a step-by-step demonstration that a given conclusion (say C) follows from some premises {P1, P2, and P3} in any circumstance.
If you can give a proof that C follows from {P1, P2, P3), then C is the logical consequence of {P1, P2, P3}. Accordingly, the argument P1, P2, P3 / C is valid.
For example,
(1) If I study hard enough for a course, then I will pass it. (1)&(2) I will pass symbolic logic. (2) I study hard enough for symbolic logic course. (3) I will graduate this fall if I pass symbolic logic course.
(4) I will graduate this fall.
Rule used: If A, then B A “Modus Ponens” or “Conditional Elimination” B
2.2 Informal vs. Formal Proofs How can we prove that {Cube (a), a = b} ╞ Cube (b)?
Informal proof:
Suppose that Cube (a) and a = b. According to the principle of indiscernibility of identicals, since a = b, we can replace a in Cube (a) by b. We come up with Cube (b), as desired.
- Used by mathematicians;
- Stated in natural language;
- More free-wheeling style;
- Not always include every step.
Formal proof:
- Cube (a)
- a = b
- Cube (b) Ind Id: 1, 2
- Used by logicians;
- Stated in FOL;
- Follow stylized method of presentation, such as “Fitch-style” system used by our book.
Problem 20 (p. 30) We assume that only one person can own a disk at any given time.
- (3) does follow from (1)m and (2).
- No! For example, let Folly belong to Claire at 2 pm, and let Silly belong to Mary at 2 pm. So, under this situation, (1) is true, (3) is True, but (2) is false.
- No! For example, let Folly and Silly belong to Max at 2 pm. Then (2) is true and (3) is true. But (1) is false in this case.
1.2.4. Formal Proofs (2.9 of the Text)
1. The structure of a formal proof in “Fitch-Style” system 1. P 2. P2 premises 3. P
.
. intermediate conclusions .
#n C final conclusion
2. Derivation Rules
Rule 1: Reflexivity of Identity (Refl=)
a = a no justification needed
Rule 2: Indiscernibility of Identicals (Ind Id)
#m P(a) : : #n a = b : : P(b) Ind Id: #m, #n
the # of Identity
Examples:
- Cube (a) 1. a = b 1. a = b
- a = b 2. b = c
- a = a Refl=
- Cube (b) Ind Id: 1, 2 3. b = a Ind Id: 2, 1 3. a = c Ind Id: 1, 2
Be careful here: (1) You can substitute some or all of the occurrences of one name. : : #m a = b : #n P(a, a, c) Ind Id: #n, #m P(b, a, c) P(a, b, c) P(b, b, c) :
(2) Only a name on the RIGHT hand side of an identity can be used for substitution.
: : a = b #m a = b : : P(b) WRONG!!! #n P(b) : : P(a) #l a = a Refl= #h b = a Ind Id: #l, #m P(a) Ind Id: #n, #h
Rule 3: Reiteration (Reit)
: P : P Reit
1.3. Notes for Chapter 3:
Conjunctions, Disjunctions, and Negations
1.3.1. Introduction to Conjunctions, Disjunctions, and Negations
(3.1, 3.2, 3.3, and 3.4 of the Text)
1. Introduction 1.1 Moving from atomic sentences to compound sentences
one single predicate Simple sentences: no truth-functional connectives. Sentences More than one predicate Compound sentences Two or more simple sentences connected by some truth- functional connectives.
For example:
John is a student and Joe is a teacher.
John or Joe is a student = John is a student or Joe is a student.
If John is a student, then Joe is a teacher.
John is not a student.
1.2 Truth-functional vs. Non-truth-functional sentences.
Definition: a compound sentence is truth-functional iff the truth-value of the sentence is fully determined by the truth-value of its component simple sentences.
Then the connectives connected component sentences of a truth-functional sentence is truth- functional connectives. They are: conjunction, disjunction, negation, and conditional.
For example, Truth-functional sentences:
John is a student and Joe is a teacher. (False) True False
If John is a student, then Joe is a teacher. (False) True False
Non-truth-functional sentences:
John loves Kathy because he kisses her. (True or False) True
I believe that Pat is on the mat (propositional attitude). (True or False) True
2. Syntax of truth-functional connectives
In English In FOL
Conjunction and, but, however, although, ∧ / & nevertheless, moreover, in additions, ,
Disjunction or, either...or..., at least one of two..., unless ∨
Negation not, be hardly, un happy, im possible, in complete ¬
Attention: two senses of disjunction
In exclusive sense: exactly one alternative (at least one and at most one alternative) Disjunction In inclusive sense: at least one alternative (and could be both)
For example:
Waitress: “You can have ice cream or a cake as desert” (but not both). Alice and Katy’s father: “John, you can marry either Alice or Kathy” (but not both). Professor: “Joe, you can take either ethics or human nature course to fulfill your philosophy requirement” (sure you can take both if you like).
In FOL, we define “OR” in inclusive sense only.
3. Semantics of truth-functional connectives 3.1 Truth-table definitions : Suppose that P and Q here represent any sentence (either simple or compound sentences). We can define the truth-value of a compound sentence consisting of P and Q as follows:
P Q P ∧ Q P ∨ Q ¬ P
T T T T F
T F F T F
F T F T T
F F F F T
1.3.2. Logical Equivalency (3.5 of the Text)
1. Definition:
Two sentences are logically equivalent iff they have same truth value in exactly the same circumstances (under any possible interpretation / in any possible world).
Illustrations: 1.1 In the language of the Tarski’s World (with fixed interpretation)
a is to one side or other of cube b , but is in front of dodecahedron c.
Suppose that you have two different translations of the above English sentence as follows:
(a) [Cube(b) ∧ ( LeftOf(a, b) ∨ RightOf(a, b) ) ] ∧ [Dodec(c) ∧ FrontOf (a, c)]
(b) [ (Cube(b) ∧ LeftOf(a, b)) ∨ (Cube(b) ∧ RightOf(a, b)) ] ∧ [Dodec(c) ∧ FrontOf (a, c)]
Are sentence (a) and (b) logically equivalent? To determine this, you need to see whether they always have the same truth value in any Tarski’s world. If they always have the same truth value in any Tarski’s world, then they are logically equivalent (Question: How can you do this???). If they do not have the same truth value in one Tarski’s world, then they are not logically equivalent (an counterexample).
1.2 In any formal language
(a) ¬ (P(a) ∧ Q(a)) (b) ¬ P(a) ∧ ¬ Q(a)
Are the above sentences in FOL logically equivalent? To find out, let us give a possible interpretation to the predicates and names under considerations. Suppose:
P(x): x is a student. Q(x): x is a female. a: Sean
Under this interpretation, sentence (a) means that it is not the case that Sean is a female student or Sean is not a female student (but Sean may be a student). Sentence (b) says that Sean is not a female, and Sean is not a student either. Suppose further that Sean is a male student in a possible world. Then under the above interpretation and in the above circumstance, sentence (a) is true but sentence (b) is false. In conclusion, sentences (a) and (b) are not logically equivalent (since we have found one possible world in which they do not have the same truth value).
2. How to test logical equivalency? There are many different formal methods to test for logical equivalency. We will only introduce two of them at this stage.
2.1 Truth-table method Make a truth table for both sentences to be tested. If they have the same truth values at all rows, then they are logically equivalent. Otherwise they are not.
P Q ¬ (P ∧ Q) ¬ P ∨ ¬ Q
T T F F T F T T F T T T F F T T
Conclusion: “¬ (P ∧ Q)” and “¬ P ∨ ¬ Q” are logically equivalent. That is,
¬ (P ∧ Q) ⇔ ¬ P∨ ¬ Q
P Q ¬ (P ∧ Q) ¬ P ∧ ¬ Q
T T F F
T F T F
F T T F counterexamples F F T T
Conclusion: “¬ (P ∧ Q)” and “¬ P∧ ¬ Q” are not logically equivalent.
2.2 Rules of logical equivalency Rules: (1) Double negation: ¬¬ P ⇔ P
(2) DeMorgan Rules: ¬ (P ∧ Q) ⇔ ¬ P ∨ ¬ Q ¬ (P ∨ Q) ⇔ ¬ P ∧ ¬ Q
- Negation normal form: “¬” only applies to atomic sentences.
(3) Idempotence: P ∧ P ⇔ P P ∨ P ⇔ P
(4) Commutative rules: P ∧ Q ∧ R ⇔ R ∧ P ∧ Q P ∨ Q ∨ R ⇔ Q ∨ R ∨ P
(5) Association rules: (P ∧ Q) ∧ R ⇔ P ∧ (Q ∧ R) P ∨ (Q ∨ R) ⇔ (P ∨ Q) ∨ R (6) Distribution rules: P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R) P ∨ (Q ∧ R) ⇔ (P ∨ Q) ∧ (P ∨ R)
- Disjunctive normal form: disjunction of conjunction of literals.
- Conjunctive normal form: conjunction of disjunction of literals.
