CST 329 - Week 2

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"

^: The symbol for "And". So (Φ and Ψ) would be written as (Φ^Ψ). Two sentences linked by ^ are a conjuction, and each sentence is a conjunct. Alternative symbols for ^ are & and *.
 

Truth Table for (Φ^Ψ).

Identical to &/and comparison in Computer Science. 

 
Simplification
 
Adjunction
 
Logically Equivalent: We say two sentences Φ and Ψ are “equivalent” or “logically equivalent” when they must have the same truth value.
 
Tautologies: Sentences of the propositional logic that must be true. 
 
Tautology ((P^¬Q)→¬(P→Q))
 

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