# LPS 29 Lecture Notes - Lecture 4: Propositional Calculus, Wnew-Fm, Logical Form

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12 Oct 2016

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LPS 29 - Lecture 3 – Propositional Logic

3.3 Formalization

Language of propositional logic: language consisting of symbolic notation

Formalization turns a word sentence into a sentence or argument form

Formalization of a simple sentence

o Today is cold. (P)

o Today is not cold. (~P)

Formalization of sentences with several logical operators

o Example 1: Today is not both very hot (P) and very cold (Q)

~ (P & Q)

Note: Brackets are needed in this case because ~ acts as a negative

sign, and only applied to the sentence letter it is attached to.

~P & Q means: Today is not very hot and today it is very cold

o Example 2: Today it is either hot (P) and sunny (Q) or cold (R) and windy (S).

(P & Q) V (R & S)

Note: Brackets must specify which sentence letters fit together to properly

reflect the meaning of the sentence

“P & Q V R & S” could mean different things such as

o (P & (Q V (R & S))), ((P & Q V R) &S)

o Both of these have a different meaning than (P & Q) V (R

& S)

Review: Vocabulary of language of propositional logic:

o Sentence Letters: Capital letters that represent a specific sentence/part of a

sentence. Can contain numerical subscripts.

o Logical Operators: ~, &, V, ,

o Brackets: (,)

Sentence letters are nonlogical symbols because they can represent different sentences

Logical Operators and Brackets are logical symbols because they always represent the

same thing

Well-formed Formula (wwf): Formulas that are meaningful

o Ex: (P V (Q & R)) is a wwf

o Ex: ((V P & R) S) is not a wwf

Three Rules that Qualify a WFF

o Any sentence letter is a WFF

o If P is a WFF, then its negation, ~P is a WFF

o If P and Q are WFFs, then so are sentences with logical operators that put them

together. (P & Q), (P V Q), (PQ), (P Q)

WWFs cont…

o Note: a form with many negation signs is a WFF!

~~~~~~~~~~~~~~P is a WFF

Even numbers of ~ means it is not a negation

Odd numbers of ~ means it is a negation

o Note: Always bracket wwfs with binary operators (V & )

WWFS that are not bracketed are called unofficial and not really WWFS

o Atomic WWFs: Sentence letters (ex. P, Q, R, S)

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o Molecular/Compound WFFS: all other wffs that aren’t atomic.

o Subwff: part of a wff that is itself a wff [ex. (~P & R) in this case, P is a subwff of

(~P & R)]

o Main operator: the binary operator that includes the scope of the entire formula

Ex: ~(P V (Q & R))

The scope of “&” is (Q & R)

The scope of “V” is (P V (Q & R))

The scope of “~” is ~(P V (Q & R))

In this case, “~” is the main operator because its scope is the entire

formula

Note: if the main operator is “&”, it is called a conjunction. If the main

operator is a “~”, it is called a negation…etc.

Practice Examples:

Identify WFFs/ not WFFS

a. (R & M V P)

b. (Q P)

c. ~(P 7 Q) R

d. ~~~~~~~~(P & Q)

e. (~(Q) & P)

f. ((P & Q) S) V (S (P & Q))

g. (((~Q V ~P) ~(~P R) > (~Q & ~S))

h. ~(~P V ~~~~~~Q)

i. ((P & Q) R S)

a. No, there are no brackets to specify the formula

b. Yes

c. No, missing brackets around the entire formula

d. Yes

e. No, sentence letters will never require brackets by themselves

f. No, missing bracket at the end

g. No, missing brackets at the end

h. Yes

i. No, there are no brackets around R and S to specify the formula

Put together complex formulas:

j. Either Sam came or Grace came

k. If either Sam or Rachel came, then Isabel did not come

l. Grace came if and only if Sam did not come

m. If Isabel came if and only if Cole came, then Sam and Grace came

n. If Sam came, then either Beck did not come or Jack did not come

o. Either Grace did not come, or if Same came, then Rachel came

p. Grace came, but Sam did not come if Jack came

q. If Grace did not come, then neither Olivia nor Rachel came.

r. Either Sam came if Grace came, or Rachel came if and only if John did not come.

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j. (S V G)

k. ((S V R) ~I)

l. (G ~S)

m. ((I C) (S & G))

n. (S (~B v ~J))

o. (~G V (S R))

p. (G (J ~S))

q. ((G S) V (~J R)) Note: order of formula does not matter in biconditionals

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