# Preliminary [Pre-test]

Logical System
###### 6/20/12

You need to know some propositional logic to be able to understand the tree tutorials to come. In particular, you need to know about the symbols used in propositional logic, truth tables, satisfiability, consistency, and semantic invalidity (by counter example). You do not need to know propositional rules of inference and derivations.

Howson  will give you enough background.

Alternatively you could look at the first five propositional tutorials in Easy Deriver

Here are some quizzes to help you judge whether you have enough background.

#### Truth tables

Here the idea is to evaluate the truth value of a formula (and all its connectives), from truth values for its atomic components. So, for example, there might be the formula A&~(B&C) with A true, B true and C false and the task is to work out the values for the &s, ~s etc. To shorten things up a little bit, instead of writing 'A&~(B&C) with A true, B true and C false' we just write 'T&~(T&F)'.

#### Satisfiable

Here the idea is to try to produce a valuation for the atomic propositions that will make the whole formula true (or 'satisfy' it). This cannot always be done (for example A&~A is not satisfiable). And, if it can be done, sometimes there are different ways of doing it (for example, A∨B is satisfiable with A true and B false or with B true and A false or with A true and B true). Here you 'toggle' the atomic propositions (not the connectives) to set them true or false.

#### Consistent

(Semantic) consistency and satisfiability are very similar notions. Here, three propositions are considered (they are shown separated by commas) and the task is to produce an assignment of truth values which will show them all to be true at the same time. Thus they will have been shown to be simultaneously satisfiable or consistent. Of course, not all sets of formulas are simultaneously satisfiable, and when they are sometimes there are several assignments that will prove it.

#### Invalidity

Arguments can be shown to be invalid by showing that all the premises can be true and the conclusion false at the same time i.e the premises and the negation of the conclusion are simultaneously satisfiable. Here arguments are written 'premise1, premise2∴ conclusion' and the challenge is the get the premises to be true and the conclusion false. Of course, some arguments are valid so it cannot always be done.