Propositional Logic 6.1 Symbols and Translation. Symbols and Translation In Propositional Logic the...

Propositional Logic

6.1 Symbols and Translation

Symbols and Translation

In Propositional Logic the basic elements are statements and operators (also called connectives).

Letters are assigned to simple statements only.

A simple statement is a statement that has no other statement as a component, for example…

Symbols and Translation

Simple statement: The car is red.

Compound statement: The car is red and the track is fast.

In propositional logic, a simple statement can be represented by any convenient upper case letter. ‘C’ or ‘R’ would be natural for the simple statement above.

Symbols and Translation

When reading a sentence, we look for simple statements and operators or connectives. There are only 5 operators:

Operator Name Logical Function Used to Translate

~ Tilde Negation Not, it is not the case that

• Dot Conjunction And, also, moreover

V Wedge Disjunction Or, unless

→ Horseshoe (arrow)

Implication If … then …, only if

Ξ Triple Bar Equivalence If and only if

Symbols and Translation

With the operators, we can translate our compound statement:

The car is red and the track is fast.C • T

How should we translate the following?It is not the case that the car is red.

~C

Symbols and Translation

How should we translate these statements?Either Jo sits or Ed hums.

J v E

If you want a pop, then you should go up to the fridge.

W G

Symbols and Translation

How to translate (continued)…The Packers will lose if and only if Favre is a

spaz.L Ξ S

Symbols and Translation

The tilde, ~, is always placed in front of the proposition it negates.

Jo does not hum ~JThe other connectives always appear between

propositions.Ed sits and Jo hums E • J

How about …Ed sits and Jo doesn’t humE • ~J

Symbols and Translation

How would we translate this?Ed doesn’t sit and Jo doesn’t hum.

~E • ~J

How about this?It is not the case that Ed sits and Jo hums.

~(E • J)

Symbols and Translation

Our last example introduces the notion of a Main Operator.

Main Operator =df that operator in a compound statement that governs the largest component(s) in the statement.

So, in ~ (E v J), the main operator is…tilde,

because it governs E, J, and the wedge, while the wedge just governs E and J.

Symbols and Translation

How about …S • ~(G v L)

The main operator is…Dot.

Tilde only governs ‘G v L’

Symbols and Translation

What should we call this statement?S • ~(G v L)

Since the main operator is the dot, we call this statement …

A Conjunction

Symbols and Translation

Here is a table of names…~A is called Negation

A • B is called Conjunction (each simple statement is a conjunct)

A v B is called Disjunction (each simple statement is a disjunct)

A B is called Material Implication (each simple statement is called … nothing, they each have different names: A is the antecedent, B is the consequent)

A Ξ B is called Material Equivalence (the book provides no name for the simple statements associated with this connective; you can call them equivalents if you like)

Symbols and Translation

What should we call this statement?[K v (S F)] [(R • H) J]

It is a Conditional

Why? Because the main operator is the horseshoe (arrow).

Symbols and Translation

What should we call this statement?(N • P) v ~R

It is a Disjunction

Why? Because the main operator is the wedge.

Symbols and Translation

What should we call this statement?[(H • P) v L] ~R

It is a Mistake!

Why? Because any formula has to be asserting just one thing, complex as it may be…

Symbols and Translation

[(H • P) v L] ~R

This “statement” would be saying something like…

Harry sits and Peter sings, or Larry plays harmonica Randy does not run

This “statement” was fine up to ‘harmonica’Nothing tells us what relation ‘Randy does not run’

has to the rest of the statements.

Symbols and Translation

[(H • P) v L] ~R

This “statement” is, in the books terminology, not a Well-Formed Formula. (Wiff, Woof)

Wiffs, or Woofs, are required before we can use any tools to test arguments, just like Standard Form was required before we could use Venn Diagrams.

Symbols and Translation

Make sure you read section 6.1 completely.

The section is long, but it is important for grasping the easier stuff to come.