parentheses and grouping: organizing truth table …selfpace.uconn.edu/class/phil102/u23.pdf ·...

Parentheses and Grouping:

Organizing Truth Table Columns

Philosophy and Logic

Unit 2, Section 2.3

Three connectives to play with

“It is not the case that P”

Flips the truth values.

~P

“Either P or Q”

False only if BOTH are False

P v Q

“Both P and Q”

True only if BOTH are True

P & Q

Even more fun: their

combinations

~(p & q) vs. ~p & q

On the left the entire conjunction (p & q) is

denied. You will figure out (p & q) first. Then

negate (~) that result.

On the right, only “p” is negated. You will figure

~p first. Then conjoin (&) that result with q.

Why we need parentheses

“We will pay all of your ground transportation

or half of your airfare and the full amount of

your conference registration.”

( G v A & R ) ?

“We will pay all of your ground transportation or

half of your airfare and the full amount of your

conference registration”

(G v A & R). Note the ambiguity:

( (G v A) & R)

Both we will pay either G or A and we will pay R

( G v (A & R) )

Either we will pay G or we will pay both A and R

Parentheses show grouping.

They say: do me first.

(p v q) & r vs. p v (q & r)

On the left, we first figure out the column for

(p v q). Then we conjoin (&) that result with “r”.

On the right, we first figure out the column for

(q & r). Then we disjoin (“v”) that result with

“p”.

(P v Q) & R

(P v Q) & R

(P v Q) R

(P v Q) & R

(P v Q) R

P Q

(P v Q) & R

(P v Q) R

P Q

RQP

(P v Q) & R

(P v Q) R

P Q

P v QRQP

(P v Q) & R

(P v Q) R

P Q

(P v Q) & R P v QRQP

(P v Q) & R

(P v Q) R

P Q

T...FT

FTT

TTT


(P v Q) & R

(P v Q) R

P Q

TT...FT

TFTT

TTTT


(P v Q) & R

(P v Q) R

P Q

T...TT...FT

FTFTT

TTTTT


(P & Q) v (~Q v P )

(P & Q) v (~Q v P )

(P & Q) (~Q v P)

(P & Q) v (~Q v P )

(P & Q) (~Q v P)

P Q

(P & Q) v (~Q v P )

(P & Q) (~Q v P)

P Q ~ Q P

(P & Q) v (~Q v P )

(P & Q) (~Q v P)

P Q ~ Q P

Q

(P & Q) v (~Q v P )

(P & Q) (~Q v P)

P Q ~ Q P

Q

QP

(P & Q) v (~Q v P )

(P & Q) (~Q v P)

P Q ~ Q P

Q

(P & Q)QP

(P & Q) v (~Q v P )

(P & Q) (~Q v P)

P Q ~ Q P

Q

~Q(P & Q)QP

(P & Q) v (~Q v P )

(P & Q) (~Q v P)

P Q ~ Q P

Q

(~Q v P)~Q(P & Q)QP

(P & Q) v (~Q v P )

(P & Q) (~Q v P)

P Q ~ Q P

Q

(P & Q) v (~Q v P)(~Q v P)~Q(P & Q)QP

(P & Q) v (~Q v P )

(P & Q) (~Q v P)

P Q ~ Q P

Q

FF

TF

FT

TT

(P & Q) v (~Q v P)(~Q v P)~Q(P & Q)QP

(P & Q) v (~Q v P )

(P & Q) (~Q v P)

P Q ~ Q P

Q

FFF

FTF

FFT

TTT

(P & Q) v (~Q v P)(~Q v P)~Q(P & Q)QP

(P & Q) v (~Q v P )

(P & Q) (~Q v P)

P Q ~ Q P

Q

TFFF

FFTF

TFFT

FTTT

(P & Q) v (~Q v P)(~Q v P)~Q(P & Q)QP

(P & Q) v (~Q v P )

(P & Q) (~Q v P)

P Q ~ Q P

Q

TTFFF

FFFTF

TTFFT

TFTTT

(P & Q) v (~Q v P)(~Q v P)~Q(P & Q)QP

(P & Q) v (~Q v P )

(P & Q) (~Q v P)

P Q ~ Q P

Q

TTTFFF

FFFFTF

TTTFFT

TTFTTT

(P & Q) v (~Q v P)(~Q v P)~Q(P & Q)QP

Nested parentheses. These look

complicated!

But the rule is simple: always do the

innermost stuff first.

p v (~q & r)

~q first.

Then conjoin (“&”) that with “r”

Finally, disjoin (“v”) that result with p.

More examples

p v ~(q & r)

(q & r) first.

Then its negation. (~)

Finally, disjoin (“v”) that result with p.

~(p v (q & r))

(q & r) first.

Then disjoin (“v”) that result with p.

Finally, negate (~) the result.

p v ~ (q & r)

p v ~ (q & r)

p ~ (q & r)

p v ~ (q & r)

p ~ (q & r)

(q & r)

p v ~ (q & r)

p ~ (q & r)

(q & r)

q r

p v ~ (q & r)

p ~ (q & r)

(q & r)

q r

RQP

p v ~ (q & r)

p ~ (q & r)

(q & r)

q r

(Q & R) RQP

p v ~ (q & r)

p ~ (q & r)

(q & r)

q r

~(Q & R)(Q & R) RQP

p v ~ (q & r)

p ~ (q & r)

(q & r)

q r

P v ~(Q & R)~(Q & R)(Q & R) RQP

p v ~ (q & r)

p ~ (q & r)

(q & r)

q r

FTF

P v ~(Q & R)~(Q & R)(Q & R) RQP

p v ~ (q & r)

p ~ (q & r)

(q & r)

q r

FFTF

P v ~(Q & R)~(Q & R)(Q & R) RQP

p v ~ (q & r)

p ~ (q & r)

(q & r)

q r

TFFTF

P v ~(Q & R)~(Q & R)(Q & R) RQP

p v ~ (q & r)

p ~ (q & r)

(q & r)

q r

TTFFTF

P v ~(Q & R)~(Q & R)(Q & R) RQP

More examples

• ~((p v q) & r)

– (p v q) first.

– Then conjoin (&) that result with “r”.

– Finally, negate (~) the result.

More examples

• ~((p v q) & r)

– (p v q) first.



(P v Q)RQP

More examples

• ~((p v q) & r)

– (p v q) first.



(P v Q) & R(P v Q)RQP

More examples

• ~((p v q) & r)

– (p v q) first.



~((P v Q) & R)(P v Q) & R(P v Q)RQP

More examples

• (~(p v q) & r)

– (p v q) first.

– Then negate (~) it.

– Finally, conjoin (&) that result with “r”.

More examples

• (~(p v q) & r)

– (p v q) first.



RQP

More examples

• (~(p v q) & r)

– (p v q) first.



(P v Q)RQP

More examples

• (~(p v q) & r)

– (p v q) first.



~(P v Q) (P v Q)RQP

More examples

• (~(p v q) & r)

– (p v q) first.



(~(P v Q) & R)~(P v Q) (P v Q)RQP

Why legalistic language sounds legalistic

For ( P & Q ) read:

Both P and Q

For ( P v Q ) read:

Either P or Q

( G v A & R) ?

( (G v A) & R): Both we will pay either G or

A and we will pay R

( G v (A & R) ): Either we will pay G or we

will pay both A and R

parentheses and grouping: organizing truth table …selfpace.uconn.edu/class/phil102/u23.pdf ·...

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