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CST 329 - Week 5

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This was our fifth week in CST 329 - Reasoning with Logic. Chapter 11: Names and predicates First Order Logic:  A combinaton of Propositional Logic and Aristotle's.    Name:  A name refers to one specific thing. Proper names include: Abraham Lincoln and Venus . A thing can have multiple names, but one name can't refer to multiple things. General names such as cat are not proper names. Names are represented with lowercase letters starting from a.   Predicate:  A sentence fragment that describes one or more names. ...is 6 feet tall and ...is taller than... are predicates. Predicates are represented by upper case letters starting from F. Arity:  The arity of a predicate is the minimum number of things that can have the property or relation. The arity of ...is tall would be one, and the arity of ...is taller than... would be two. A predicate of arity n must be true or false of each n objects from its domain of discourse. In other words, it cannot be a...
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CST 329 - Week 4

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This was our fourth week in CST 329 - Reasoning with Logic Chapter 8: Reductio ad Absurdum Reductio ad Absurdum:  Often called an "indirect proof" or "indirect derivation". Similar to a subproof, but our assumption is the opposite of the conclusion we want to reach, and we create a contradiction within the indirect proof itself.   Basic structure of an Indirect Proof   An Indirect Proof We assume that ¬¬P, we arrive at both R and ¬R, we conclude that the assumption must be false, so ¬P.    A second example of an Indirect Proof Contradictory Sentence:  A sentence that must be false. Chapter 9: "... if and only if...", Using Theorems ↔ :  The symbol for a biconditional, (Φ↔Ψ). A biconditional  is true if both or neither of its contituents is true, and false otherwise.   Truth Table for a Biconditional     Equivalence   Bicondition     Many proofs for biconditionals look like the image above, with multiple subproofs ...
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CST 329 - Week 3

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This was our third week in CST 329 - Reasoning with Logic. Chapter 6 - Conditional Derivations Conditional Derivation or Subproof:  Subproofs allow us to prove conclusions when we would otherwise not be able to do so with a inference rule. When making a subproof we assume that the initial premise is true. Once the subproof is done we cannot use it other than to show the associated conclusion.   Proof with a subproof for  (Φ→Ψ)     Invalid proof using a subproof. In line 7 it references line 3, leading to a invalid argument.      Proof with a subproof using conjunctions   Proof with multiple subproofs    Theorem: A sentence that can be proved without premises.   Proof for a theorem/tautology   Chapter 7 - "Or" v:  The symbol for "Or". So  (Φ or Ψ) would be written as  (Φ v Ψ). Two sentences linked by an "Or" is called a disjunction, and each sentence is a disjunct. Disjunctions are an "inclusive or".  ...
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CST 329 - Week 2

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This was our second week in CST 329 - Reasoning with Logic. Chapter 4 - Proofs Inference Rules:  Formulas for creating valid arguments. After writing an inference rule, we can add another to continue the argument. Inference rules are made to always be valid arguments, so adding an inference rule correctly should never lead to an invalid argument.    Modus Ponens   Direct Proof:  Assuming that each line in an argument is numbered, we justify an argument by stating which inference rule was used with which lines to conclude it. Inference rules can be used for any previous lines in the argument, including lines inferred by an inference rule.   Direct Proof for Modus Ponens   Fitch Bar: As seen in the image above, an argument is written to the right of a vertical bar, with a horizontal bar separating the premises and conclusion. Named after logician Frederic Fitch.   Modus Tollens     Double Negation   Repeat Chapter 5 - "And" ^:  T...
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