Lecture Notes 9/3/10

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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

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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

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3. The language of logic is like a programming language
   (read the quote from Bertrand Russell)

   1. Def: A proposition is_def 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)

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   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

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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++.

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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: INPUT OUTPUT p ¬p T F F T i.e., tval(¬p)=F iff tval(p)=T & tval(¬p)=T iff tval(p)=F
   conjunction	(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

   (more to come …)