SCHAUM’S OUTLINES: LOGIC

John Nolt

Outline

12/6/2011

CHAPTER 1: ARGUMENT STRUCTURE

1.1 What Is an Argument?

Definitions

Logic: the study of arguments.

Argument: a sequence of statements of which one is intended as a conclusion and the others, the premises, are intended to prove or at least provide some evidence for the conclusion

They don’t need to actually prove the conclusion

The premises and conclusion of an argument are always statements or propositions

Statements: assertion that is either true or false

Nonstatements: questions, commands, or exclamations, are neither true nor false

Standard form:

All humans are mortal.

Socrates is human.

:. Socrates is mortal.

1.2 Identifying Arguments

Inference indicators: words or phrases used to signal the presence of an argument

Two types: conclusion indicators and premise indicators

Inference indicators often have other functions in other contexts; no hard and fast rule

Conclusion Indicators

Thus

Hence

For this reason

Accordingly

Consequently

This being so

It follows that

The moral is

Which proves that

Which means that

From which we can infer that

As a result

In conclusion

Premise Indicators

For

Since

Because

Assuming that

Seeing that

Granted that

This is true because

The reason is that

For the reason that

In view of the fact that

It is a fact that

As shown by the fact that

Given that

Inasmuch as

One cannot doubt that

1.3 Complex Arguments

Conclusion of one argument becomes the premise of another

Nonbasic premises (intermediate conclusions): premises which are intended as conclusions from previous premises

Basic premises: Those which are not conclusions from previous premises

1.4 Argument Diagrams

Circle inference indicators, bracket and number statements

Plus sign: together with

Arrow: intended as evidence for

Bracketing: bracket statements in a way that best reveals the argument structure

Compound sentence: sentence that contains two or more statements

It’s helpful to break apart some compound statements, such as those that include “and”

Unbreakable compounds: should not be split into separate statements and treated as complete statements (e.g., either-or, if-then, which are not inference indicators)

1.5 Convergent Arguments

Convergent argument: an argument contains several steps of reasoning which all support the same (final or intermediate) conclusion

Several statements have their own arrows that independently point to (or converge on) the same conclusion

1.6 Implicit Statements

Implicit statements: hidden statements in arguments that are incompletely expressed

e.g., the author may intend for the reader to draw the conclusion

Principle of charity: give the arguer the benefit of the doubt, and make the argument as strong as possible while remaining faithful to the arguer's thought

Goal is to minimize misinterpretation

1.7 Use and Mention

Place single quotes around words when referring to the word itself

Use: Socrates was a Greek philosopher (about the person; using the word)

Mention: ‘Socrates’ is a name containing eight letters (about the word; mentioning the word)

1.8 Formal vs. Informal Logic

Formal logic is the study of argument forms, abstract patterns common to many different arguments. (Chapters 3, 4, 5, 6, 9, and 10)

Informal logic is the study of particular arguments in natural language and the contexts in which they occur. (Chapters 1, 2, 7, and 8)

Tips for Chapter 1

Valid argument forms

It helps to first become familiar with a few valid argument forms (modus ponens, modus tollens, disjunctive syllogism, hypothetical syllogism in chapter 4.4); this makes it easier to identify premises and conclusions

Steps for diagraming arguments

Bracket each statement, and circle clue words (and, but, therefore)

Number each statement sequentially

Identify conclusion; if conclusion is implied, supply it and number it

Work backwards from conclusion to premises;

Determine if any premise leads independently to a conclusion

Determine if there are any intermediate conclusions

Determine which premises need to be taken together

CHAPTER 2: ARGUMENT EVALUATION

2.1 Evaluative Criteria

Main purpose of an argument: to demonstrate that a conclusion is true or at least likely to be true. Arguments may be judged better or worse based on this

Four criteria for judging the truth of a conclusion

1. Truth: whether all the premises are true;

2. Inductive probability: whether the conclusion is at least probable, given the truth of the premises;

3. Relevance: whether the premises are relevant to the conclusion;

4. Total evidence: whether the conclusion is vulnerable to new evidence.

2.2 Truth of Premises

Truth by itself not adequate for argument evaluation, but it provides a good start

Truth or falsity of one or more premises may be unknown, so that the argument fails to establish its conclusion so far as we know

2.3 Validity and Inductive Probability

Deductive argument: an argument whose conclusion follows necessarily from its basic premises.

An argument is deductive if it is impossible for its conclusion to be false while its basic premises are all true.

Inductive argument: an argument whose conclusion is not necessary relative to the premises

Inductive probability: the probability of a conclusion, given a set of premises

The inductive probability of a deductive argument is maximal, i.e., equal to 1 (on a scale from 0 to 1)

Inductive probability of inductive arguments is typically less than 1

Validity:

Valid deductive arguments are those which are genuinely deductive in the sense defined above

Invalid deductive arguments are arguments which purport to be deductive but in fact are not.

Soundness: a valid deductive argument with all true premises

Weak and strong induction

Inductively strong: the inductive probability of an argument is high

Inductively weak: the inductive probability of the argument is weak

No sharp line between strong and weak inductive reasoning

2.4 Relevance

Fallacy of relevance: argument which is irrelevant for demonstrating the truth of its conclusion, and is thereby useless

Relevance is a matter of degree

Sometime occurs with conclusions that are logically necessary statements

Logically necessary statement: a statement whose very conception or meaning requires its truth; its falsehood, in other words, is logically impossible (e.g., either something exists or nothing at all exists)

2.5 The Requirement of Total Evidence

An inductive argument can be strengthened or weakened with the addition of new evidence (deductive arguments remain deductive even with new premises added)

If an argument is inductive, its premises must contain all known evidence that is relevant to the conclusion

Fallacy of suppressed evidence: inductive arguments that fail contain all known evidence that is relevant to the conclusion

Fallacy can be intentional or unintentional

Tips for chapter 2:

Split up discussion of deductive and inductive arguments, and begin with more intuitive explanations of each

Deduction: often moves from general premises to specific conclusions

Example:

All men are mortal

Socrates is a man

Therefore, Socrates is mortal

Technical definition: an argument whose conclusion follows necessarily from its basic premises

Two criteria for evaluating deductive arguments: truth and validity

Soundness: a valid argument with all true premises

A deductive argument can fail for either being invalid or for having false premises

Validity: an argument which fits a valid argument form (such as modus ponens).

Testing premises for truth: in front of each premise, write a T, F or ? depending on whether it is true, false, or unknown; sound arguments cannot have premises that are either false or unknown

Example of valid argument with a false premise

(1) Either Uncle Bob is a tennis shoe, or Uncle Bob is a golfing shoe.

(2) It is not the case that Uncle Bob is a golfing shoe.

(3) Therefore, Uncle Bob is a tennis shoe.

Example of an invalid argument with all true premises

(1) If the President was the pilot of Air Force One, then he could fly in the presidential plane.

(2) The President can fly in the presidential plane.

(3) Therefore, the President is the pilot of Air Force One.

Ignore the discussion of relevance and logically necessary statements. These never occur in ordinary arguments, and the basic point can only be fully understood within the context of a full system of propositional logic as discussed later in the book

Induction: often moves from specific premises to general conclusions

Example: an argument whose conclusion is not necessary relative to the premises

(a) Rock 1 falls to the ground when I open my hand.

(b) Rock 2 falls to the ground when I open my hand.

Technical definition:

Inductive probability: how likely the conclusion is to be true given the truth of the premises

This is the inductive counterpart to deductive validity: inductive probability is a matter of degree, while with deduction an argument is either valid or it is not

Four criteria for evaluating inductive arguments

Truth: premises must be true (as with validity)

Inductive probability: argument must have high inductive probability

Relevance: the argument must be relevant to the conclusion

Total evidence: must contain all known evidence that is relevant to the conclusion

CHAPTER 3: PROPOSITIONAL LOGIC

3.1 Argument Forms

Terms

Formal logic: the study of argument forms, that is, abstract patterns of reasoning shared by many different arguments

Valid deductive argument: an argument whose conclusion cannot be false while the premises are all true

Propositional logic: the study of the logical concept of validity insofar as this is due to the truth-functional operators

Sentence letters: P, Q, R, place holders for declarative sentences

Three deductively valid argument forms

Disjunctive syllogism

Either P or Q

It is not the case that P

:. Q

Modus ponens

If P then Q

:. Q

Modus tollens

If P then Q

not Q

:. not P

3.2 Logical Operators

Negation (not)

Conjunction (and)

P & Q

Sentence letters are called “conjuncts”

Disjunction (or)

P v Q

Sentences letters are called “disjuncts”

Conditional (if-then)

P → Q

Biconditional (if and only if)

P ↔ Q

3.3 Formalization

Vocabulary of the language of propositional logic

Sentence letters

Logical operators

Brackets

Well-formed formulas (wff): a meaningful symbolic proposition

Wff Formation rules

1. Any sentence letter is a wff

2. If φ is a wff, then so is ~φ

3. If φ and ψ are wwfs, then so are (φ & ψ), (φ v ψ), (φ → ψ), (φ ↔ ψ)

Complex wwfs: built up from simple ones by repeated application of the formation rules

Atomic wwfs: sentence letters

Molecular (compound wwfs): all other wffs

Sub wff: a part of a wff that is a wff itself

3.4 Semantics of the Logical Operators

Terms

Truth value: the truth and falsity of a statement

Principle of bivalence: true and false are the only truth values and in every possible situation each statement has one and only one of them

Truth table: a summary of the truth value of a wff

Truth tables of logical operators

Negation

p ~ p

T F

F T

Conjunction

p q p & q

T T T

T F F

F T F

F F F

Disjunction

p q p v q

T T T

T F T

F T T

F F F

Conditional

p q p → q

T T T

T F F

F T T

F F T

Biconditional

p q p ↔ q

T T T

T F F

F T F

F F T

Material conditional:

The relation expressed by a conditional statement, which means the same thing as ~(p &~p)

Biconditional:

(p → q) & (q → p)

Truth tables for complex wwfs:

Find the truth values for the smallest subwffs and then use the truth tables for the logical operators to calculate values for increasingly larger subffs, until you obtain the values for the whole wff

3.5 Truth Tables for Wffs

Rules for truth tables of complex wffs

The column for any wff or subwff is always written under its main operator;

Circle the column under the main operator of the entire wff to show that the entries in it are the truth values for the whole formula

Three truth values of complex wffs

Tautologies: each line is true under the main operator

Truth-functionally inconsistent: each line is false under the main operator

Truth-functionally contingent: some lines are true, others false, under the main operator

Example: (p & q ) v ( p & r )

p q r p & q p & r ( p & q ) v ( p & r )

T T T T T T

T T F T F T

T F T F T T

T F F F F F

F T T F F F

F T F F F F

F F T F F F

F F F F F F

3.6 Truth Tables for Argument Forms

Rules:

Display each premise and conclusion as a separate wff

The argument is valid when for each line that all premises are true the conclusion also is

Example: disjunctive syllogism

p q p v q, ~p ˫Q

T T T F T

T F T F F

F T T T T

F F F T F

3.7 Refutation Trees

General

Refutation trees provide a more efficient algorithm than truth tables for determining truth value

A refutation tree is an analysis in which a list of statements is broken down into sentence letters or their negations, which represent ways in which the members of the original list may be true. Since the ways in which a statement may be true depend on the logical operators it contains, formulas containing different logical operators are broken down differently.

All wffs containing logical operators fall into one of the following ten categories:

Negation — Negated negation

Conjunction — Negated conjunction

Disjunction — Negated disjunction

Conditional — Negated conditional

Biconditional — Negated biconditional

Steps

Construct a list consisting of its premises and the negation of its conclusion.

Break down the wffs on the list into sentence letters or their negations.

If we find any assignment of truth and falsity to sentence letters which makes all the wffs on the list true, then under that assignment the premises of the form are true while its conclusion is false. Thus we have refuted the argument form; it is invalid.

If the search turns up no assignment of truth and falsity to sentence letters which makes all the wffs on the list true, then our attempted refutation has failed; the form is valid.

Refutation Tree Rules

Negation (~): If an open path contains both a formula and its negation, place an “x” at the bottom of the path.

Negated Negation(~~): If an open path contains an unchecked wff of the form ~~p, check it and write p at the bottom of every open path that contains this newly checked wff.

Conjunction (&): If an open path contains an unchecked wff of the form p&q, check it and write p and q at the bottom of every open path that contains this newly checked wff.

Negated Conjunction (~&): If an open path contains an unchecked wff of the form ~(p&q), check it and split the bottom of each open path containing this newly checked wff into two branches, at the end of the first of which write ~p and at the end of the second of which write ~q.

Disjunction (v): If an open path contains an unchecked wff of the form pVq, check it and split the bottom of each open path containing this newly checked wff into two branches, at the end of the first of which write p and at the end of the second of which write q.

Negated Disjunction (~v): If an open path contains an unchecked wff of the form ~(p v q), check it and write both ~p and ~q at the bottom of every open path that contains this newly checked wff.

Conditional (→): If an open path contains an unchecked wff of the form p→q, check it and split the bottom of each open path containing this newly checked wff into two branches, at the end of the first of which write ~p and at the end of the second of which write q.

Negated Conditional ~(→): If an open path contains an unchecked wff of the form ~(p→q), check it and write both p and ~q at the bottom of every open path that contains this newly checked wff.

Biconditional (↔): If an open path contains an unchecked wff of the form p↔q, check it and split the bottom of each open path containing this newly checked wff into two branches, at the end of the first of which write both p and q, and at the end of the second of which write both ~p and ~q.

Negated Biconditional (~↔): If an open path contains an unchecked wff of the form ~(p↔q), check it and split the bottom of each open path containing this newly checked wff into two branches, at the end of the first of which write both p and ~q, and at the end of the second of which write both ~p and q.

Tips for Chapter 3

Section 3.6: truth tables

For testing validity, there is a shorter truth table method of just hunting for a row that is a counterexample rather than bothering with filling out the whole table

Section 3.7: refutation trees

Basic point: if the crappy (negated) version of the argument fails, then the original version is valid; if the crappy version succeeds, the original version is invalid

The crappy version fails when all paths end in a contradiction (e.g., both P and ~P)

The crappy version succeeds when at least one path ends in a consistency (e.g., both P and P)

Closed path: one which ends in a contradiction and is designated with an X

Open path: one that ends with a consistency (which in our class is designated with an !)

Begin working on non-branching paths

Conditionals: treat as disjunctions with the material implication rule of inference:

(p → q) ↔ (~p v q)

Biconditionals: treat as a disjunction where each branch is a conjunction, using the material equivalence rule of inference:

(p ↔ q) ↔ [(p & q) v (~p & ~q)]

CHAPTER 4: THE PROPOSITIONAL CALCULUS

4.1 The Notion of Inference

Terms:

Propositional calculus (sentential calculus): the system of inference rules designed to yield a proof of the valid argument forms of propositional logic

4.2 Nonhypothetical Inference Rules

Nolt’s definitions:

Negation elimination (~E): From a wff of the form ~~p, infer p

Conditional elimination (→E): From a conditional and it’s antecedent, infer its consequent

Conjunction introduction (&I): From any wwfs p and q, infer p&q

Conjunction elimination (&E): From a conjunction, infer either of its conjuncts

Disjunction introduction (vI): From a wff p, infer the disjunction of p with any wff

Disjunction elimination (vE): From wffs of the form pvq, p→r, q→r, infer r

Biconditional introduction (↔I): From wffs of the form p→q and q→p, infer p↔q

Biconditional elimination (↔E): From wffs of the form p↔q, infer either p→q or q→p

4.3 Hypothetical Rules (rules using assumptions

Negation introduction (~I): Given a derivation of an absurdity from a hypothesis p, discharge the hypothesis and infer ~p

Assume p; get q&~q ˫ ~p

Negate the intended inclusion, draw an explicit contradiction from the negation, infer the intended conclusion

An explicit contradiction (or an “absurdity”) is p&~p

Also called Reductio ad absurdum (reduction to absurdity)

Conditional introduction (→I): Given a derivation of a wff q from hypothesis p, discharge the hypothesis and infer p → q

Assume p; get q ˫ p→ q

Guidelines for hypothetical rules

1. Each hypothesis introduced into a proof begins a new vertical line

2. No occurrence of a formula to the right of a vertical line may be cited in any rule applied after that line has ended

3. If two or more hypotheses are in effect simultaneously, then the order in which they are discharged must be the reverse of the order in which they are introduced

4. A proof is not complete until all hypotheses have been discharged

Proof Strategies

1. When conclusion is an atomic formula: If no other strategy is immediately apparent, hypothesize the negation of the conclusion for ~I. If this is successful, then the conclusion can be obtained after the ~I by ~E.

2. When conclusion is a negated formula: Hypothesize the conclusion without its negation sign for ~I. If a contradiction follows, the conclusion can be obtained by ~I.

3. When conclusion is a conjunction: Prove each of the conjuncts separately and then conjoin them with &I.

4. When conclusion is a disjunction: Sometimes (though not often) a disjunctive conclusion can be proved directly simply by proving one of its disjuncts and applying vI. Otherwise, hypothesize the negation of the conclusion and try ~I.

5. When conclusion is a conditional: Hypothesize its antecedent and derive its consequent by→I.

6. When conclusion is a biconditional: Use →I twice to prove the two conditionals needed to obtain the conclusion by ↔I.

Intuitive presentation of ten rules

Negation Introduction (~I – indirect proof IP)

Assume p

Get q & ~q

˫ ~p

Negation Elimination (~E – version of DN)

~~p → p

Conditional Introduction (→I – conditional proof CP)

Assume p

Get q

˫ p → q

Conditional Elimination (→E – modus ponens MP)

p → q

˫ q

Conjunction Introduction (&I – conjunction CONJ)

˫ p & q

Conjunction Elimination (&E – simplification SIMP)

p & q

˫ p

Disjunction Introduction (vI – addition ADD)

˫ p v q

Disjunction Elimination (vE – version of CD)

p v q

p → r

q → r

˫ r

Biconditional Introduction (↔I – version of ME)

p → q

q → p

˫ p ↔ q

Biconditional Elimination (↔E – version of ME)

p ↔ q

˫ p → q

˫ q → p

4.4 Derived Rules

Important derived rules

Modus Tollens (MT)

p → q

˫ ~P

Hypothetical Syllogism (HS)

p → q

q → r

˫ p → r

Disjunctive Syllogism (DS)

p v q

˫ q

Absorption (ABS)

p → q

˫ p → (p & q)

Constructive Dilemma (CD)

p v q

p → r

q → s

˫ r v s

Repeat (RE)

˫ p

Contradiction (CON)

˫ Any wff

4.5 Theorems

Theorems: wffs that are tautologies (whose instances are logically necessary)

They are provable without making any nonhypothetical assumptions

The symbol “˫” designates a theorem

Theorems can sometimes be stipulated in lines of proofs

Examples:

˫ ~(P & ~P)

˫ P → (P v Q)

˫ P → ((P → Q) → Q)

˫ P ↔ ~~P

˫ P v ~P

4.6 Equivalences (also called rules of inference)

Equivalence: a biconditional that is a theorem

Important equivalences

De Morgan’s Law (DM)

~(p & q) ↔ (~p v ~q)

~(p v q) ↔ (~p & ~q)

Commutation (COM)

(p v q) ↔ (q v p)

(p & q) ↔ (q & p)

Association (ASSOC)

[p v (q v r)] ↔ [(p v q) v r]

[p & (q & r)] ↔ [(p & q) & r]

Distribution (DIST)

[p & (q v r)] ↔ [(p & q) v (p & r)]

[p v (q & r)] ↔ [(p v q) & (p v r)]

Double Negation (DN)

p ↔ ~~p

Transposition (TRANS)

(p → q) ↔ (~qF~p)

Material implication (MI)

(p → q) ↔ (~p v q)

Material Equivalence (ME)

Version 1: (p ↔ q) ↔ [(p & q ) v (~p & ~q)]

Version 2: (p ↔ q) ↔ [(p → q ) & (q → p)]

Exportation (EXP)

[(p & q) → r) ↔ (p → (q → r)]

Tautology (TAUT)

p ↔ (p & p)

p ↔ (p v p)

Tips for Chapter 4

CHAPTER 5: THE LOGIC OF CATEGORICAL STATEMENTS

5.1 Categorical Statements

Four standard forms of categorical statements

d u

A: All S is P (All students are people)

d d

E: No S is P (No students are pelicans)

u u

I: Some S is P (Some students are Polish)

u d

O: Some S is not P (Some students are not polite)

Four requirements

1. A quantifier (all, no)

2. A subject term (S)

3. A copula (is, are)

4. A predicate term (P)

Definitions

Form: A, E, I or O (form of a statement)

Term: subject and predicate terms

Requirements: quantifier (all, no), subject term (S), copula (is, are), predicate term (P)

Distribution: when what's said about S or P applies to all S or P

Quality: affirmative (A, I) negative (E, O)

Quantity: universal (A, E) particular (I, O)

Existential import: S term is committed to existence in I and O forms.

Translating from ordinary language

Asterisk around unit class

Add "thing" to adjectives (e.g. some apples are red things)

Times, places, cases (e.g., some times are times when I am happy)

Boolian Notation

A: SP = 0 (no members in the class of S and non-P)

E: SP = 0 (no members in the class of S and P)

I: SP ≠ 0 (at least one member in the class of S and P)

O: SP ≠ 0 (at least one member in the class of S and non-P)

5.2 Venn Diagrams

Diagraming categorical propositions

Shade areas where nothing is contained in the set.

With “All S is P”, everything in the S circle is also in the P circle , so you shade the portion of S that is outside of P.

With “No S is P”, nothing in S is also in P, so you shade the portion of S that overlaps with P.

Place X within areas where something is contained in the set.

Square of Opposition

5.3 Immediate Inferences

5.4 Categorical Syllogisms

Example of syllogism:

1. All men are mortal (All men are mortal things)

2. Socrates is a man (All *Socrates* are men)

3. Socrates is mortal (All *Socrates* are mortal things)

Definitions

Mood: AAA, EIO, etc., (mood of syllogism)

Figures:

1st Fig. 2nd Fig. 3rd Fig. 4th Fig.

M - P P - M M - P P - M

S - M S - M M - S M - S

S - P S - P S - P S - P

Fifteen Valid Syllogistic Forms:

Fig. 1: AAA-1, EAE-1, AII-1, EIO-1

Fig. 2: AEE-2, EAE-2, AOO-2, EIO-2

Fig. 3: AII-3, IAI-3, EIO-3, OAO-3

Fig. 4: AEE-4, IAI-4, EIO-4

Validity with Venn Diagram:

Three circles for S P and M.

When placement of X is ambiguous, put it on a line.

Diagram all premises, see if diagram indicates conclusion.

Six Rules of Validity

1. Three terms: must have exactly 3 terms used unambiguously.

2. One distributed middle term: middle term must be distributed in at least one premise.

3. Distributed term-distributed term: term is distributed in conclusion iff it is distributed in premise.

4. One affirmative premise: must have at least one affirmative premise.

5. Negative-negative: negative conclusion iff negative premise.

6. Particular-particular: cannot conclude a particular from two universals

CHAPTER 6: PREDICATE LOGIC

6.1 Quantifiers and Variables

Individual constants (a, b, c . . . r, s, t)

Singular statements

Ms (Socrates is moral)

Ln (New York City is large)

~Vh (Hitler was not virtuous)

Compound statements

Sj → Tj (if John is a student then John pays tuition)

Gb v Jb (Bob is either a geek or a jock)

Nj & Lb (John lives in New York and Bob lives in L.A.)

Two quantifiers

Variables: u-z

₳: universal quantifier (“₳x” means “for all x”)

Ǝ: existential quantifier (“Ǝx” means “for some x”)

Four quantification statement forms:

A: all S is P (e.g., all students are people)

₳x(Sx → Px)

For all x, if x is S then x is P

Warning: do not use a conjunction with A statements, since it will not mean the same thing. E.g. ₳x(Sx & Px) means “everything is a student and a person”

E: no S is P (e.g, no student is a pelican)

₳x(Sx → ~Px)

For all x, if x is S then it is not the case that x is P

I: some S is P (e.g., some students are Polish)

Ǝx(Sx & Px)

For some x, x is S and x is P

O: some S is not P (e.g., some students are not pilots)

Ǝx(Sx & ~Px)

For some x, x is S and it is not the case that x is P

Examples of translated statements

Frogs are green: ₳x(Fx → Gx)

There is at least one green frog: Ǝx(Fx & Gx)

Green frogs exist: Ǝx(Fx & Gx)

Some frogs are not green: Ǝx(Fx & ~Gx)

Everything is a frog: ₳x(Fx)

Something is a frog: Ǝx(Fx)

Not everything is a frog: ~₳x(Fx)

Nothing is a frog: ₳x~(Fx) -- or, as per replacement rule below, ~Ǝx(Fx)

Everything is a green frog: ₳x(Fx & Gx)

6.2 Predicates and Names (constants)

Three relational forms

One-place non-relational predicates

Mj (John is a musician)

Two-place relational predicates

Lbc (Bob loves Cathy)

Three-place relational predicates

Gcfb (Cathy gave Fido to Bob)

Examples of relational statements:

Cindy is a musician: Mc

Bill is a musician: Mb

Cindy and Bill are musicians: Mc & Mb

Either Cindy or Bill is a musicians: Mc v Mb

Bill admires Cindy: Abc

Bill admires himself: Abb

Donna gossiped to Cindy about Bill: Gdcb

Examples of quantified relational statements

Bill admire nothing: ₳x~(Abx)

Nothing admires Bill: ₳x~(Axb)

There is something which both Bill and Cindy admire: Ǝx(Abx & Acx)

If Bill admires himself, then he admires something: Abb → Ǝx(Abx)

A musician admires Bill: Ǝx(Mx & Axb)

Every musician admires Bill ₳x(Mx → Axb)

Everything is a musician that admires Bill: ₳x(Mx & Axb)

Tips to remember

1. Different variables do not necessarily designate different objects

2. Choice of variables makes no difference to meaning

3. The same variables used with two different quantifiers does not necessarily designate the same object in each case

4. Many, if not most, English sentences which mix universal and existential quantifiers are ambiguous (e.g., something is liked by everything)

5. The order of consecutive quantifiers affects meaning only when universal and existential quantifiers are mixed

6. Nested quantifiers may combine with the truth-functional operators in many equivalent ways

6.3 Formation Rules

Logical symbols

Logical operators: ~ & v → ↔

Quantifiers: ₳ Ǝ

Variables: lower case u-z (things, “for all x”)

Name letters (constants): lower case a-t (“b” for “Bob”)

Predicate letters: upper case letters (“A” for “is an acrobat”)

Wffs

1. Any atomic formula is a wff

2. If P is a wff, so is ~P

3 if P and Q are wffs, so are (P & Q), (P v Q), (P & Q), (P → Q), (P ↔ Q)

4. If P is a wff containing the name letter a, then any formula of the form ₳bPb/a or ƎbPb/a is a wff, where Pb/a is the result of replacing one or more of the occurences of a in P by some variable b not already in P

6.4 Models

6.5 Refutation Trees

6.6 Identity

Identity predicate: the symbol =, which means identical to

CHAPTER 7: THE PREDICATE CALCULUS

7.1 Reasoning in Predicate Logic

Terms:

Predicate logic: the study of the logical concept of validity insofar as this is due to the truth-functional operators, the quantifiers, and the identity predicate

Predicate calculus: the system of inference rules designed to yield a proof of the valid argument forms of predicate logic.

7.2 Inference Rules for the Universal Quantifier

Universal Elimination (₳E – universal instantiation UI)

Two forms (works with both variables and constants)

₳x(Fx)

˫ Fy

e.g., For all things, things fluctuate; therefore things fluctuate.

₳x(Fx)

˫ Fa

e.g., For all things, things fluctuate; therefore atoms fluctuate.

Example of use in a proof:

All humans are mortal; Socrates is human; therefore Socrates is mortal.

1. ₳x(Hx → Mx)

2. Hs / ˫ Ms

3. Hs → Ms (1 ₳E)

4. Ms (3, 2 →E MP)

Universal Introduction (₳I – universal generalization UG)

One form (works only with variables)

˫ ₳x(Fx)

e.g., things fluctuate; therefore, for all things, things fluctuate.

Not permitted with constants:

˫ ₳x(Fx)

e.g., atoms fluctuate; therefore, for all things, things fluctuate.

Example of use in a proof:

All dogs are mammals; all mammals are warm-blooded; therefore all dogs are warm-blooded.

1. ₳x(Dx → Mx)

2. ₳x(Mx → Wx) / ˫ ₳x(Dx → Wx)

3. Dx → ~Mx (1 ₳E)

4. Mx → Wx (2 ₳E)

5. (Dx → Wx) (3, 4 HS)

6. ˫ ₳x(Dx → Wx) (5. ₳I)

7.3 Inference Rules for the Existential Quantifier

Existential Introduction (ƎI – existential generalization EG)

Two forms (works with both variables and constants):

˫ Ǝx(Fx)

e.g., atoms fluctuate; therefore, there exists some thing that fluctuates

˫ Ǝx(Fx)

e.g., things fluctuate; therefore, there exists some thing that fluctuates

Example of use in a proof:

All frogs are green things; Alex is a frog; therefore, there is at least one green thing

1. ₳x(Fx → Gx)

2. Fa / ˫ Ǝx(Gx)

3. Fa → Ga (1 ₳E)

4. Ga (3, 2 MP)

5. ˫ Ǝx(Gx) (4 ƎI)

Existential Elimination (ƎE – existential instantiation EI)

One form (works only with constants)

Ǝx(Fx)

˫ Fa

e.g.: There exists some thing such that this thing is a frog; therefore Alex (which we will call him) is a frog.

Restriction: the existential name “a” must be a new name that has not occurred in any previous line

Not permitted with variables:

Ǝx(Fx)

˫ Fy

There exists some thing such that this thing is a frog; therefore things are frogs.

Example of use in a proof:

All guitarists are musicians; some guitarists are homeless; therefore some homeless are musicians. (“Alex” is a stipulated person)

1. ₳x(Gx → Mx)

2. Ǝx(Gx & Hx) / ˫ Ǝx (Hx & Mx)

3. Ga & Ha (2 ƎE)

4. Ga → Ma (1 ₳E)

5. Ga (3 SIMP)

6. Ma (4, 5 MP)

7. Ha & Ga (3 COM)

8. Ha (7 SIMP)

9. Ha & Ma ( CONJ)

10. ˫ Ǝx (Hx & Mx) (ƎI)

7.4 Theorems and Quantifier Equivalence Rules

Quantifier Equivalence rules (Quantifier Exchange QE)

₳x(Fx) ↔ ~Ǝx~(Fx)

~₳x(Fx) ↔ Ǝx~(Fx)

₳x~(Fx) ↔ ~Ǝx(Fx)

~₳x~(Fx) ↔ Ǝx(Fx)

7.5 Inference Rules for the Identity Predicate

CHAPTER 8: FALLACIES

8.1 Classification of Fallacies

Fallacies (in the broadest sense) are mistakes that occur in arguments and affect their cogency.

Six groups of fallacies, corresponding to the following six divisions of this chapter

8.2 Fallacies of Relevance

Fallacies of relevance occur when the premises of an argument have no bearing upon its conclusion.

In addition, such fallacies often involve a distractive element which diverts attention away from this very problem.

Often called non sequiturs (from the Latin phrase meaning "it does not follow")

1. Ad hominem (against the person) arguments try to discredit a claim or proposal by attacking its proponents instead of providing a reasoned examination of the proposal itself.

a. Ad hominem abusive: attack a person's age, character, family, gender, ethnicity, social or economic status, personality, appearance, dress, behavior, or professional, political, or religious affiliations. E.g., Jones advocates fluoridation of the city water supply; jones is a convicted thief; therefore we should not fluoridate the city water supply.

b. Guilt by association: the attempt to repudiate a claim by attacking not the claim's proponent, but the company he or she keeps, or by questioning the reputations of those with whom he or she agrees. E.g. Jones advocates fluoridation of the city water supply; Jones spends much of his free time hanging around with known criminals, drug addicts, and deviants; therefore, we should not fluoridate the city water supply.

c. You too (tu quoque): the attempt to refute a claim by attacking its proponent on the grounds that he or she is a hypocrite, upholds a double standard of conduct, or is selective and therefore inconsistent in enforcing a principle. E.g., Jones believes we should abstain from liquor; Jones is a habitual drunkard; therefore, we should not abstain from liquor.

d. Vested interest arguments attempt to refute a claim by arguing that its proponents are motivated by the desire to gain something (or avoid losing something). E.g., Jones supports the fluoridation bill pending in Congress; he does so because he owns a major fluoridation firm, which will reap huge dividends if the bill passes; therefore, we should not support this bill.

e. Circumstantial ad hominem: is the attempt to refute a claim by arguing that its proponents endorse two or more conflicting propositions. E.g., Jones says he abhors all forms of superstition; Jones also says that breaking a mirror brings bad luck; therefore, there probably is something to superstition after all.

2. Straw man arguments, by contrast, attempt to refute a claim by confusing it with a less plausible claim (the straw man) and then attacking that less plausible claim instead of addressing the original issue. E.g., There can be no truth if everything is relative; therefore, Einstein's theory of relativity cannot be true.

3. Appeal to force (ad baculum): the attempt to establish a conclusion by threat or intimidation. E.g., “If you don't vote for me, I'll tell everybody you are a liar.”

4. Appeal to authority (ad verecundiam): we accept (or reject) a claim merely because of the prestige, status, or respect we accord its proponents (or opponents). E.g., My teacher says that I should be proud to be an American; therefore, I should be proud to be an American

Testimonial version: celebrities appear in advertisements endorsing products

5. Appeal to the people (ad populum): we infer a conclusion merely on the grounds that most people accept it. E.g., Everybody believes that premarital sex is wrong; therefore, premarital sex is wrong.

Bandwagon effect: we are asked to join forces with others

6. Appeal to pity (ad misericordiam): we are asked to excuse or forgive an action on the grounds of extenuating circumstances. E.g., Officer, you shouldn’t give me a ticked since I was buying candy for my crying baby.

7. Appeal to ignorance (ad ignorantiam): X has not been proved, therefore x is false. E.g., no one has ever proved that God does not exist, therefore, God exists.

8. Missing the point (irrelevant conclusion; ignoratio elenchi): the premises of an argument warrant a different conclusion from the one that the arguer draws.

9. Red herring: an extraneous or tangential matter used purely to divert attention away from the issue posed by the argument. E.g., some members of the police force may be corrupt, but there are corrupt politicians, corrupt plumbers, etc.

8.3 Circular Reasoning

1. Circular reasoning (begging the question and petition principia): assuming what we are trying to prove (assumes its own conclusion). E.g., God exists since the Bible says he does, and the Bible was written by God

2. Question-begging epithets: phrases that prejudice discussion and assume the very point at issue. E.g., bleeding heart liberal, Neanderthal conservative

3. Complex question: presupposing an answer to a logically prior question. E.g., have you stopped beating your spouse

8.4 Semantic Fallacies

Semantic fallacies result when the language employed to construct arguments has multiple meanings or is excessively vague in a way that interferes with assessment of the argument.

1. Equivocation (ambiguity): the meaning of an expression shifts during the course of an argument. E.g., it is silly to fight over mere words; discrimination is just a word, it is silly to fight over discrimination

2. Amphiboly. E.g., arguments whose meanings are indeterminate because of loose or awkward sentence construction; e.g., “save soap and waste paper”, “Charles the First walked and talked, thirty minutes after he had his head chopped off.”

3. Vagueness: an argument whose meaning is indistinct, as opposed to having a multiplicity of meanings.

Double think: every sentence cancels out its predecessor and contradicts its successor

4. Accent: emphases that generate multiple and often misleading interpretations. E.g., I’m in favor of a missile defense system that effectively defends America.

8.5 Inductive Fallacies

Inductive fallacies occur when the probability of an argument's conclusion, given its premises —i.e., its inductive probability—is low, or at least less than the arguer supposes.

1. Hasty generalizations: fallaciously inferring a conclusion about an entire class of things from inadequate knowledge of some of its members. E.g., last Monday I wrecked by car; the Monday before that my furnace broke; therefore, bad things always happen to me on Mondays

2. Faulty analogy: comparing things in an analogy that are too dissimilar. E.g., The American colonies justly fought for their independence in 1776; today the American Football Alliance is fighting for its independence; therefore, the Alliance’s cause is also just

3. Gambler’s fallacy: the gambler falsely assumes that the history of outcomes will affect future outcomes. E.g., heads has come up heads five times in a row now, so there is a greater than 50/50 chance that tails will come up on the next toss

4. False Cause (post hoc ergo propter hoc – “after this, therefore because of this”): confusing a cause with an effect; inferring a causal connection based on mere correlation. E.g., the number of stroke victims in hospitals is directly proportional to the number of tar bubbles on the road; therefore, tar bubbles cause strokes

5. Suppressed evidence (cherry picking the evidence): intentionally failing to use information suspected of being relevant and significant. E.g., this is the highest rated printer, therefore it is the best choice for us (suppressing the fact that it costs three times more than competing brands)

8.6 Formal Fallacies

Formal fallacies occur when we misapply a valid rule of inference or else follow a rule which is demonstrably invalid.

1. Denying the antecedent (faulty modus tollens): denying the antecedent of an if-then statement, and then inferring that the consequent must also be denied. E.g., if it rains, the sidewalk will be wet; it’s not raining, therefore, the sidewalk is not wet

2. Affirming the consequent (faulty modus tollens): affirming the consequent in an if-then statement, and then inferring that the antecedent is true. E.g., if it rains, the sidewalk will be wet; the sidewalk is wet; therefore it must be raining.

8.7 Fallacies of False Premises

1. False dichotomy (excluded middle): making a false assumption that only one of a number of alternatives holds. E.g., either you’re for us or you’re against us; you’re not for us; therefore you must be against us.

2. Slippery slope: the conclusion of an argument rests upon an alleged chain reaction, suggesting that a single step will result in an undesirable outcome. E.g., Caffeine use leads to cocaine use, which leads to crack use; therefore you shouldn’t take caffeine.

CHAPTER 9: INDUCTION

9.1 Statement Strength

Strong statement: is true only under specific circumstances; the world must be just so in order for it to be true. E.g., every vertebrate has a heart. Bob’s house is the third on the left on main street

Weak statement: is true only under a wide variety of possible circumstances; it says nothing specific and demands little of the world for its truth. E.g., something is happening somewhere. Some people are sort of weird

Rules of relative strength

Rule 1: If statement A deductively implies statement B, but B does not deductively imply A, then A is stronger than B. E.g., all cows are horned; some cows are horned

Rule 2: If statement A is logically equivalent to statement B, then A and B are equal in strength. E.g., John loves Mary; Mary is loved by John

9.2 Statistical Syllogism

Two types of inductive arguments

Statistical inductive arguments: an inductive argument which does not presuppose the uniformity of nature.

E.g., 98% of college freshmen can read beyond the 6th grade level; David is a college freshman; therefore, Dave can read beyond the 6th grade level

Humean inductive arguments: an inductive argument which presupposes the uniformity of nature

E.g., Each of the 100 college freshman surveyed know how to spell logic; if we ask another college freshman, he or she will also know how to spell logic

Two interpretations of inductive probability

Logical: the percentage figure divided by 100

Subjective: inductive probability is a measure of a particular rational person’s degree of belief in the conclusion, given its premises

Statistical Syllogism (general to specific)

Allows us to arrive at a conclusion concerning a member of a set from statistics concerning a set of individuals

n percent of F are G; x is F; therefore, x is G

9.3 Statistical Generalization (specific to general)

Statistical generalization (specific to general): allows us to arrive at a conclusion concerning an entire population from a premise concerning a random sample of that population

Formula

n percent of s (number) randomly selected F are G;

therefore, about n percent of all F are G

Fallacy of small sample: the conclusion is too strong to be supported by sample number in the premises

9.4 Inductive Generalization and Simple Induction

Inductive generalization (specific to general): allows us to arrive at a conclusion concerning an entire population when it is not possible to obtain a random sample (e.g., they may involve future events)

Formula

n percent of s (number) thus-far-observed F are G;

therefore, about n percent of all F are G

Fallacy of biased sample: attempts to apply statistical generalization with a nonrandom sampling technique (a type of hasty generalization)

The success of statistical generalization depends on randomness of sampling

Simple induction

Inductive generalization where the population in the conclusion is reduced to one individual (the argument is strengthened by lessening the conclusion)

The argument gets stronger when the percentage is over 50, but weaker when under 50

Formula

n percent of the s thus-far-observed F are G

Therefore, if one more F is observed, it will be G

9.5 Induction by Analogy

Formula

F₁x & F₂x & . . . & F_nx [object x has several properties]

F₁y & F₂y & . . . & F_ny [object y has these same properties]

Gy [object y has an additional property]

Therefore, Gx [object x probably has that additional property too]

Relevant disanalogy: contrary evidence to analogical arguments

9.6 Mill's Methods

Two-step procedure for determining the cause of an observed effect

1. Formulate a list of suspected causes (including the actual cause)

2. Rule out by observation as many suspected causes as possible down to one

Four kinds of causes

Necessary cause (causally necessary condition): a condition needed to produce a certain effect

The effect will never occur without the cause (although the cause can occur without the effect)

e.g., fuel a necessary cause of fire (fire will never occur without fuel, although fuel can occur without fire)

Conditional statement: If you have a given effect (fire), then it will always be produced by a given cause (fuel)

Sufficient cause (causally sufficient condition): a condition which always produces a certain effect

The cause will never occur without the effect (although the effect can occur without the cause)

e.g., decapitation is a sufficient cause of death in higher animals (although death in higher animals can occur without decapitation)

Conditional statement: If you have a given cause (decapitation of higher animals), then it will always result in a given effect (death)

Necessary and sufficient causes: a condition that is needed to produce and will always produce a certain effect

The effect will never occur without the cause, nor the cause without the effect

e.g., the presence of a massive body is a necessary and sufficient cause for the presence of a gravitational field

Conditional statement: a given cause (massive body) will occur if and only if a given effect (gravitational field) accompanies

Causal dependence of one variable quantity on another: a variable quantity B is casually dependent on a second variable quantity A if a change in A always produces a corresponding change in B

e.g., the brightness of an object B varies inversely with the square of the distance A from that object

Conditional statement: if you change a given variable quantity (distance) then this will result in a change to another variable quantity (brightness)

Method of Agreement (necessary causes of E)

A deductive procedure for ruling out suspected causally necessary conditions, with the goal of narrowing the list down to one

e.g., several students get sick eating at the cafeteria; examine what they ate in common

Mill’s wording: “If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree, is the cause (or effect) of the given phenomenon.”

Method of Difference (sufficient causes of E)

A procedure for narrowing down a list of suspected sufficient causes for an effect E by rejecting any item on the list that occurs without E, with the goal of narrowing the list down to one

e.g., several students get sick eating at the cafeteria; examine what the non-sick students all ate

Mill’s wording: “If an instance in which the phenomenon under investigation occurs, and an instance in which it does not occur, have every circumstance in common save one, that one occurring only in the former; the circumstance in which alone the two instances differ, is the effect, or the cause, or an indispensable part of the cause, of the phenomenon.”

Method of Agreement and Difference (necessary and sufficient causes of E)

A procedure for eliminating items from a list of suspected necessary and sufficient causes of an effect E by simultaneously applying the methods of agreement and difference

Mill’s wording: “If two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instances in which it does not occur have nothing in common save the absence of that circumstance: the circumstance in which alone the two sets of instances differ, is the effect, or cause, or a necessary part of the cause, of the phenomenon.”

Method of Concomitant variation (quantities on which the magnitude of E is causally dependent)

A procedure for narrowing down a list of variable magnitudes suspected of being the cause of a specific change in the magnitude of an effect E, where a variable is rejected if it remains constant throughout the change in E

e.g., give differing portions of contaminated food to different people to see how sick they get

Mill’s wording: “Deduct from any phenomenon such part as is known by previous inductions to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents.”

9.7 Scientific Theories

Terms:

Scientific theory: an account of some natural phenomenon which in conjunction with further known facts or conjectures (auxiliary hypotheses) enables us to deduce consequences which can be tested by observation

Model: a physical or mathematical structure claimed to be analogous in some respect to the phenomenon for which the theory provides an account

Confirmation through successful prediction

A theory’s predictions are deduced from the theory (plus auxiliaries); if the prediction proves false, then either the theory or one of the auxiliaries must be false

If we are confident of the auxiliaries, then the fault rests with the theory

Confidence in a theory should never be absolute: even if all predictions so far are successful, there may be some untested prediction that is false

A theory becomes more probable with more successful predictions

Principle of scientific probability: If E is some initial body of evidence (including auxiliary hypotheses) and C is the additional verification of some of the theory’s predictions, the probability of the theory given E & C is higher than the probability of the theory given E alone

CHAPTER 10: THE PROBABILITY CALCULUS

10.1 The Probability Operator

10.2 Axioms and Theorems of the Probability Calculus

10.3 Conditional Probability

10.4 Application of the Probability Calculus

CHAPTER 11: FURTHER DEVELOPMENTS IN FORMAL LOGIC

11.1 Expressive Limitations of Predicate Logic

11.2 Higher-Order Logics

11.3 Predicate Logic with Function Symbols

11.4 Formal Arithmetic

11.5 Formal Definitions

11.6 Definite Descriptions

11.7 Modal Logic