Discrete Structures

# Lecture Notes 9/3/10

 Last Update: 3 September 2010 Note: or material is highlighted

1. In this course, we'll look at two systems of logic:

1. propositional logic (sometimes called "0th-order" logic)
2. first-order predicate logic
Also just called:
1. 1st-order logic, or FOL
2. predicate logic

2. There are lots of other logics:

2. modal logics: the logic of necessity & possibility
3. epistemic logics: the logic of knowledge & belief
4. deontic logics: the logic of "ought"
5. many-valued logics (with more than just 2 truth-values)
6. non-monotonic logics: the logic of how to change your mind

3. The language of logic is like a programming language

1. Def: A proposition isdef a (declarative) sentence that is either true or else false.

• there are only 2 "truth values": true (T), false (F)
• a proposition has at least one truth value ("tval")
• a proposition has at most one truth value

• E.g.:

• 2+2=4; tval = T
• 2+2=5; tval = F
• Today is Monday; tval depends on when the sentence is said—I wrote this on a Friday, so, when I wrote it, tval = F
• George Washington was the first US president; tval = T

1. This definition rules out:

1. questions ("Is 2+2=4?") and commands ("Let x=2").

2. And it rules out sentence fragments ("Hey, you!"; "Today is"; these don't have tvals)
& "open" (or incomplete) sentences ("2+x=3") (this doesn't have a tval, but gets one once x is assigned a value)

3. And it rules out self-contradictory sentences that are "both" T & F or are neither T nor F (e.g., "This sentence is false")

2. Some logicians distinguish between:
• sentences (which are like an instance of a data structure)
• and propositions (which are like an abstract data type or a Java class):

Consider these English, French, and German sentences:
• "It's snowing"
• "Il neige"
• "Es schneit"
Here we have 3 sentences that express 1 proposition (the proposition that cold, wet, white stuff is precipitating)

& talking about language requires us to talk about grammar & meaning

1. Syntax is the study of relationships among symbols

• e.g., the grammar of a language

2. Semantics is the study of relationships between the symbols of a language and "the world"

1. i.e., the meanings of the symbols

2. The semantics of propositional logic:

• propositions only have 2 meanings: T, F
• Let p be a proposition.
Then tval(p) = T if & only if p describes/corresponds to/matches (part of) the world correctly;
Else tval(p) = F

4. Before presenting the syntax & semantics of our language for propositional logic, please

FORGET OR IGNORE WHATEVER YOU THINK YOU KNOW ABOUT PROPOSITIONAL LOGIC
& LEARN **OUR** LOGIC AND RULES

• This is just like my asking you to ignore your knowledge of the syntax of, say, Java when programming in, say, C++.

5. 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=4
tval(p)=T
iff
p describes the world correctly
molecular
propositions:

negation

¬p It's not the case that p.
Not p
Today isn't Monday.
2+2≠4
Use truth tables for the semantics of molecular propositions:

INPUTOUTPUT
p¬p
TF
FT

i.e., tval(¬p)=F iff tval(p)=T
& tval(¬p)=T iff tval(p)=F

conjunction (pq) 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/PO/P
pq(p ∧ q)
TTT
TFF
FTF
FFF

(more to come …)

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