An Introduction to Propositional Logic

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An Introduction to Propositional Logic. Translations: Ordinary Language to Propositional Form. What is a Proposition?. Propositions are the meanings of statements. I have no money Ich habe kein geld.

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### An Introduction to Propositional Logic

Translations: Ordinary Language to Propositional Form

What is a Proposition?
• Propositions are the meanings of statements.
• I have no money
• Ich habe kein geld.
• Meanings are the thoughts, concepts, ideas we are trying to convey through speech and writing.
Simple Propositions
• Fast foods tend to be unhealthy.
• Parakeets are colorful birds.

(p. 290)

• Simple propositions are grammatically independent expressions of information.
Compound Propositions
• If fast foods tend to be unhealthy, then you shouldn’t eat them.
• Parakeets are colorful birds, and colorful birds are good to have at home.
• People are free, if and only if they can choose their actions and there are no forces compelling those actions.
The Focus of Propositional Logic
• Propositional truth is determined by consulting typical sources of information.
• Propositional logic is determined by examining how various propositions are related.
Types of relations between propositions
• 1 proposition is offered in support of another (simple argument)
• 1 proposition expressed the condition under which a 2nd proposition is true (conditional statement)
• 1 sentence offers two proposed alternatives, and a 2nd proposition negates one of these alternatives (disjunctive syllogism)
Propositional Symbols
• Symbolizing propositions allows us to focus on the relations between propositions (logic) rather than the content of those propositions (truth).
Module Objectives - 1
• Learn how to symbolize complex propositions
• Simple propositions, which express grammatically independent units of information, are easy to symbolize: “It is cold” = “C”
• Complex propositions are sentences which contain 2 or more simple propositions. These can be more difficult to symbolize.
Module Objectives - 2
• Learn how to determine the truth-value of complex propositions

Remember, an argument can only establish the truth of it’s conclusion if all its premises are true (and its reasoning is valid)

• “Fees are rising at UCLA” is either true or false
• “Either fees are rising or services are being cut back” could be true or false – depending on the actual situation regarding fees and services at UCLA.
Module Objectives - 3
• Create and interpret truth tables for both propositions and arguments (series of propositions).

Truth tables allow us to see all possible conditions under which a statement could be true and could be false.

Module Objectives – 4 & 5
• Learn to recognize common argument forms, and know when an argument form is valid or invalid
• Prove that an argument is valid or invalid when it doesn’t fit a common argument form.
Basics of Propositional Logic

All arguments are reducible to symbols, which represent either

elements of an argument

orways these elements are put together.

• All arguments contain statements, by definition. Each statement is represented by a “propositional variable” – p, q, r, s
• All arguments also contain connections, or ways in which individual propositions are related. Each of these connections are represented by one of five “operators”:

Putting propositional variables together with operators creates a “statement form,” or a symbolic blueprint identifying typical structures of English expressions.

Propositional Operators

~ (“tilde,” negation)

Not, it is false that, conjunctions like “don’t”

• (“dot,” conjunction)

And, also, but, in addition, moreover

v (“wedge,” either-or)

Or, unless

> (“implication” or “conditional,” if,then).

Is a sufficient (or necessary) condition of, if-then, implies, given that, only if

Ξ (“biconditional,” if and only if)

If and only if, is equivalent to, is a sufficient and necessary condition of

Note on the “Tilde”

1. All operators except the tilde must relate at least two propositions.

2. The tilde negates either a proposition directly, or an operator relating to propositions (by standing directly before a parentheses/bracket/etc.) .

Examples of “Tilde” Functions

~ p = not p; p is not true, etc

~ p ● ~ q = p is false and q is false; p and q are both false

~ ( p ● q ) = not both p and q (maybe one is true and one false)

Rules for Operator Types - 1

If there is more than one operator (excluding tildes), then some portion of the statement must be included in parentheses/brackets/etc.

Rules for Operator Types - 2

The tilde negates either a proposition directly, or an operator relating to propositions (by standing directly before a parentheses/bracket/etc.).

Tips for Translation
• Use “clue words”:
• “If, then”; “on the condition that”: >
• Both; and; also; etc: ●
• Either, or; or maybe both:v
• If one, then the other; if and only if; always occur together:
• Negation; it is not true that, not: ~