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Logic I. CSE 140 etc. Motivation. The representation of something very central to cognition:. Particular facts about the world (e.g. Fido is a dog , John is tall ) General rules about how things work when we reason with such facts
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Logic I CSE 140 etc.
Motivation The representation of something very central to cognition: • Particular facts about the world (e.g. Fido is a dog, John is tall) • General rules about how things work when we reason with such facts • Deriving rules to help manipulate the representations, and come to (valid) conclusions about the world.
Outline • Introduce basic propositional logic, and the truth tables and deduction rules that will be important throughout • Move on to predicate logic, with explicit relations (e.g. Taller-than(john,bill) • With predicate logic, an introduction to quantification (all, some, etc.)
Two Notions We’ll talk about two ways of using our logic. One involves meaning, that is Truth or Falsity. The other involves the notion of proof, a mechanical procedure for manipulating symbols. Entailment: A set of sentences G entails a sentence f if fis true when all sentences in G are true. G╞f Proof: If G is a set of sentences, and f follows by rules of inference, we have G├ f That is, G provides a proof of f, where proof is simply based on the form of sentences (I.e. we have a theorem-proving program)
Expanding on This • Entailment is a semantic notion; Proof is a syntactic notion, relying on the form of sentences • The idea behind the logic is to relate meaning and form such that proofs imitate entailment • For each logical connective, we will have both truth tables and proof rules
Preliminary Definitions Truth: The primary notion we are interested in is the truth-value of propositions. We’ll have T for true and F for false. Atomic Propositions: Our basic irreducible (atomic) objects; an atomic proposition cannot be broken into more smaller propositions. -Propositions are broken down in terms of Boolean Connectives -The Boolean Connectives are and, or, not, and if…then Example:John is tall is an atomic proposition. John is tall and the book is red is not; it contains and.
Some Notation Atomic Propositions: f, c, … Connectives: and or not if…then
Truth-Tables: And Idea: A formula (f )is true if and only if both f is true and is true. f (f ) T T T T F F F T F F F F We can illustrate this in a truth-table for the connective So, for instance, for John is tall and the book is red to be true, both John is tall has to be true, and The book is red has to be true.
Proof Theory for and Idea: We will have rules that eliminate as well as a rule that introduces Elimination: … (P1 . . . . . . Pi . . . Pn) … Pi That is, if we have a line in a proof conjoined with a bunch of ands, we can conclude one of the P’s
Introduction Introduction: : P1 : Pi : Pn : (P1 . . . . . . Pi . . . Pn) I.e., conjoin individual Ps, each of which is derived earlier
Truth Tables: Or Idea: A formula (f )is true if and only if either f is true or is true, or both are true. f (f ) T T T T F T F T T F F F Here are some examples. Assume that John is tall is True. John is tall or snow is green: T Rabbits bark like dogs or snow is green: F
Rules of Proof for or • Introduction: Intuitively, if f occurs as a line in a proof, conclude (fc). : Pi : (P1 . . . . . . Pi . . . Pn) Simply adding another element to something that has already been established.
Elimination Elimination: If from (fc) you can prove S from f and you can prove it from c, conclude S (proof by subcases). That is, if you can prove S from each of the subcases, you can conclude S from the whole. (P1 . . . Pn) • P1 • : • S • • Pn • : • S • S
Truth Tables: Not Idea: Negation simply flips the truth value of a formula: f f T F F T We read negation when prefixed to a formula P as “It is not the case that P” Example: P = Snow is green. (F) So P = It is not the case that snow is green (T)
Rules of Proof for Negation That is, double negation cancels out. Elimination: : P : P If P leads to a contradiction, then we can conclude P Introduction: : P : (Q Q) P
Truth Tables: If…then The truth table for the conditional connective treats it as what is called material implication, which approximates if…then in English. f (f )T T T T F F F T T F F T Note: If the antecedent (if part) is false, it does not matter what comes next; we have T in either case. Note 2: Only if the if part is true and the then part is false do we have F
Rules of Proof Elimination: If we have (f )and f, then introduce a new line that says : P : (P Q) : Q
Introduction for If…then • Introduction: If P occurs as an assumption in a proof, and Q occurs as a line in a proof, introduce a new line saying (P Q) and withdraw the assumption that P. : P : Q (P Q)
Multiple Connectives (Tables) Using P, Q, R, now as our atomic propositions. Point: The truth of a complex expression is calculated in a way that relies on the basic truth tables. P Q R P ( Q R) T T T T T T F T T F T F T F F T F T T F F T F F F F T F F F F F We use the truth table for or to calculate the value of the parenthesized part, then the t.t. for and for the rest.
