Introductory LogicPHI 120

Presentation: "Intro to Formal Logic"

Please turn off all cell phones!

Please take out your book.

Homework1. Study Allen/Hand Logic Primer

– "Well-formed Formula," pp. 6-7– "Binary and Unary Connectives," p. 7– "Parentheses Dropping Conventions," p. 9– ("Denial,“ – logically opposite sentences, p. 7)

2. Handout on Class Web Page:– Truth Tables Handout

3. Watch At Home: – “Basic Concepts Review” presentation

Bring this handout to class

from now on!

New Unit

Formal (Symbolic) Logic

Today:Basic Grammar of Sentences

Sentential Logic

SYMBOLIC ELEMENTS OF THE LOGICPart I

Expressions any sequence of symbols in the logic

Sentences (WFFs)expressions that are well-formed

The Well-Formed Formula

An initial distinction

Sentences: two basic kinds i. atomic or simple

i. cannot be broken into simpler sentencesii. no connectives

ii. complexi. made up of simpler sentencesii. they always contain some connective

Symbolic Elements of the Logic

1. Atomic sentences

2. Connectives (or Logical Operators)

3. Parentheses ( … )

Symbolic Elements of the Logic

1. Atomic or Simple Sentences• Sentence variables

– Examples:» P e.g., “John dances on the table.”» Q e.g., “The table will be broken.”» R e.g., "James is the man next to the wall over

there.

Symbolic Elements of the Logic2. Connectives (or Logical Operators)

~ the tilde “it is not the case that …” or simply "not"

& the ampersand “ … and … ”

v the wedge “either … or … ”

-> the arrow “if … then … ”

<-> the double arrow “ … if and only if … ”

Examples:

~P

P & Q

P v Q

P -> Q

P <-> Q

Symbolic Elements of the Logic

3. Parentheses– Examples:

1. ( P & (Q -> R ))

2. P & (Q -> R)

3. P & Q -> R

4. P & (Q & R)

P and (if Q then R)

P and (if Q then R)

If P and Q then R

See page 9: “parentheses dropping conventions”

P and (Q and R)

Outermost parentheses unnecessary

Inner Parentheses When necessary?

Parenthesis Dropping

1. Drop parentheses surrounding sentence.

2. Drop embedded parentheses only if unambiguous.

KINDS OF VARIABLESExcursus

Kinds of Variables

• Sentence Variable: P, Q, R, S, T, ...– an element of the formal language

– stands for any simple (atomic) sentence in natural language

• Metavariable: Φ (Phi) or Ψ (Psi)– not an element of the formal language

– stands for the any WFF

– used to represent logical form

The 6 Sentences (WFFs)(pages 6-7)

1) Atomic Sentence (P, Q, R, S, …)2) Negation ~Φ3) Conjunction Φ & Ψ4) Disjunction Φ v Ψ5) Conditional Φ -> Ψ6) Biconditional Φ <-> Ψ7) and nothing else

Unary

Binary

READING SYMBOLIC LOGICPart III

(Order of Operations)

The Key to Recognizing SentencesBinding Strength

See page

9

Strongest~

& and/or v->

<->Weakest

P = We are studying symbolic logic.

Q = It is interesting.

P = We are studying symbolic logic.  ~P = We are not studying symbolic logic.  ~~P = It is false that we are not studying symbolic logic.

Recognizing Negations

• The ~ attaches to the symbol directly to the right of it.

Examples:~P~~P~(P & Q)~P & ~Q~(~P & ~Q)NB: the middle statement is not a negation

(Note the parentheses)

Strongest~

& and/or v->

<->Weakest

~Φ~Φ

P = You study hardQ = You will do well on the examsR = Your GPA will go up

Conjunctions and Disjunctions

• The & or v connects two WFFs.

Examples:P & QP v QP & (Q v R)(P & Q) v RP & (Q -> R)(P -> Q) v R

(Note the parentheses)

Strongest~

& and/or v->

<->Weakest

Φ & Ψand

Φ v Ψ

Φ & Ψand

Φ v Ψ

P = You study hardQ = You will do well on the exams

P = You study hardQ = You will do well on the examsR = Your GPA will go up

Conditional Statements

• The -> connects two WFFs.

Examples:P -> QP -> ~QP -> (Q -> R)(P -> Q) -> RP -> Q v RP & Q -> R

(Note the parentheses)

Strongest~

& and/or v->

<->Weakest

Φ -> ΨΦ -> Ψ

P = You study hardQ = You will do well on the examsR = Your GPA will go up

Biconditionals

• The <-> connects two WFFs.

Examples:• P <-> Q• P <-> ~Q• P <-> Q & R• P v Q <-> R• P -> Q <-> R• P <-> (Q <-> R)

(Note the parentheses)

Strongest~

& and/or v->

<->Weakest

Φ<->ΨΦ<->Ψ

Parentheses and Ambiguity

What kind of statement is this?

P v (Q & R)

P v Q & R

(unambiguous)

(ambiguous)

Strongest~

& and/or v->

<->Weakest

Summary1. Elements of Symbolic Logic

– (i) Variables, (ii) Connectives, (iii) Parentheses

2. Sentences (or WFFs)– Atomic– Complex

3. Key to Reading Symbolic Logic– Binding Strength of Connective

Homework1. Study Allen/Hand Logic Primer

– "Well-formed Formula," pp. 6-7– "Binary and Unary Connectives," p. 7– "Parentheses Dropping Conventions," p. 9– ("Denial,“ – logically opposite sentences, p. 7)

2. Handout on Class Web Page:– Truth Tables Handout

3. Watch At Home: – “Basic Concepts Review” presentation

Bring this handout to class

from now on!