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- First, here's the complete chart, including the old stuff from last
time & the new material from today:
**Syntax & Semantics of Our Language for Propositional Logic**-
Given propositions

*p*&*q*, we can generate (or construct) more complex propositions.NAME SYNTAX ENGLISH E.G. SEMANTICS atomic

proposition*p*,*q*, etc.(any "simple" grammatical

declarative sentence)Today is Monday.

2+2=4tval( *p*)=T

iff*p*describes the world correctlymolecular

propositions:negation

¬ *p*It's not the case that *p*.

Not*p*Today isn't Monday.

2+2≠4Use truth tables

for the semantics of

molecular propositions:INPUT OUTPUT p ¬p T F F T i.e., tval(¬

*p*)=F iff tval(*p*)=T

& tval(¬*p*)=T iff tval(*p*)=Fconjunction ( *p*∧*q*)*p*and*q**p*but*q**p*although*q*

etc.Today is Monday and 2+2=4.

Today is Monday but there's no school anyway.I/P O/P p q (p ∧ q) T T T T F F F T F F F F inclusive

disjunction(Latin "vel")

( *p*∨*q*)(Either) *p*or*q*(or both)*p*and/or*q*Today is Monday or 2+2=4. You'll pass if you study or if you already know the material.

I/P O/P p q (p ∨ q) T T T T F T F T T F F F exclusive

disjunction(Latin "aut")

( *p*⊕*q*)(Either) *p*or (else)*q*(but not both)Today is a weekday or a weekend.

You'll pass 191 or else you'll fail 191.I/P O/P p q (p ⊕ q) T T F T F T F T T F F F Note: tval(

*p*⊕*q*)=T iff tval(*p*)≠tval(*q*)

i.e., T iff*different*tvals.material

conditional( *p*→*q*)*p*is the "antecedent"

(= "goes before")*q*is the "consequent"

(= "follows together")If *p*, then*q*

*q*if*p*

*p*only if*q*

*p*is a sufficient condition for*q*

*q*is a necessary condition for*p*If today is Tuesday, this must be Belgium. - (title of famous movie)

If the sum of the digits of a number is divisible by 3,

then so is the number.I/P O/P p q (p → q) T T T (easy) T F F (obvious) F T T (!) F F T (!) biconditional ( *p*↔*q*)*p*if & only if*q**p*iff*q**p*is necessary & sufficient for*q**q*is necessary & sufficient for*p*A plane figure is a triangle iff it is a 3-sided polygon. I/P O/P p q (p ↔ q) T T T T F F F T F F F T Note: tval(

*p*↔*q*)=T iff tval(*p*)=tval(*q*)

i.e., T iff*same*tvals.

i.e., (*p*⊕*q*) is the negation of (*p*↔*q*)!

(later, we'll have a precise way of saying that) -
How to compute a truth table for a molecular proposition:
- Consider this molecular proposition:
((

*p*∨*q*) ∧ ¬(*p*∧*q*)) - It will have 4 possible truth values corresponding to
the 4 possible combinations of truth values of its atomic
propositions
*p*,*q* - But we don't want to show just the 2 "input" columns
of possible truth values for the atomic propositions

and the "output" column of the truth values for the molecular proposition. - Instead, we want to show how the output can be computed from the input (after all, we're computer scientists!).
- To do that, we need to show the "intermediate" columns where all the work is done.
- There will be one intermediate column for each "constituent" part of the molecular proposition.
- Our example molecular proposition is basically
a conjunction of the form (
*A*∧*B*), where:*A*is basically a disjunction of the form (*p*∨*q*), where:*p*and*q*are atomic propositions

- and
*B*is basically a negation of the form ¬*C*, where:*C*is basically a conjunction of the form (*p*∧*q*), where:*p*and*q*are atomic propositions

(in fact, the same ones as before)

- So, here is the full truth table, with our 2 input
columns of atomic propositions, our 3 intermediate columns of
"constituent" parts of the molecular proposition, and our output
column for the molecular proposition:
input intermediate output 1 2 3=(1∨2) 4=(1∧2) 5=¬4 6=(3∧5) *p**q*( *p*∨*q*)( *p*∧*q*)¬( *p*∧*q*)(( *p*∨*q*) ∧ ¬(*p*∧*q*))T T T T F F T F T F T T F T T F T T F F F F T F - Note that our molecular proposition has the same truth table
as exclusive or: (
*p*⊕*q*)- We'll come back to this later.

- Consider this molecular proposition:

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http://www.cse.buffalo.edu/~rapaport/191/F10/lecturenotes-20100908.html-20100908-2