Discrete Structures

Lecture Notes 9/3/10

Last Update: 3 September 2010

Note: NEW or UPDATED material is highlighted


Note: A username and password may be required to access certain documents. Please contact Bill Rapaport.


Index to all lectures
…Back to previous lecture

  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:

    1. 2nd-order logic (I'll say more about this later)
    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)


    2. Talking about sentences requires us to talk about language
      & 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


  5. Syntax & Semantics of Our Language for Propositional Logic

    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 …)

Next lecture…


Text copyright © 2010 by William J. Rapaport (rapaport@buffalo.edu)
Cartoon links and screen-captures appear here for your enjoyment.
They are not meant to infringe on any copyrights held by the creators.
For more information on any cartoon, click on it, or contact me.

http://www.cse.buffalo.edu/~rapaport/191/F10/lecturenotes-20100903.html-20100903-3