Philosophy homework

REFERENCE NOTES FOR CHAPTER 7 SECTION 1 [PAGES 279-303]

Terms

· Atomic Statement: A statement that does not have any other statement as a component.

· EXAMPLE: “I like hats.”

· NOTE: Does not have any logical operators in the statement.

· NOTE: Will always be symbolized by a single variable, such as “P”.

· Compound Statement: A statement that has at least one atomic statement as a component.

· EXAMPLE: “I do not like hats.”

· EXAMPLE: “Either I like hats, or I like cats.”

· EXAMPLE: “If I like hats, then I like cats.”

· NOTE: Will always have at least one logical operator, and can have more.

· NOTE: May have only one variable, provided a negation is attached to it, such as “~P”.

· Main Logical Operator: The logical operator that governs the largest component or components of a compound statement.

· Minor Logical Operator: A logical operator that governs a smaller component or components.

· Well-formed formula (WFF): A grammatically correct symbolic expression.

· Statement variable: A lowercase letter (e.g. a variable) that serves as a placeholder for any statement.

The Logical Operators Take 2 [BOOK PAGES 281-290]

NEGATION: “NOT” = ∼

EXAMPLE: ∼P

DISJUNCTION: “OR” = ∨

EXAMPLE: P ∨ Q

CONJUNCTION: “AND” = ●

EXAMPLE: P ● Q

CONDITIONAL: “IF...THEN…” = →

EXAMPLE: P → Q

BICONDITIONAL: “...IF AND ONLY IF…” = ↔

EXAMPLE: P ↔ Q

The conclusion symbol (not a logical operator)

Conclusion = “∴” ( or “/ ” )

Disambiguating Statements and Operator Scope

Sometimes, you will have compound statements with multiple operators. When this happens, the English and symbolic expressions might be ambiguous, meaning that they could allow two or more interpretations. That’s bad for regular everyday language, and it’s really bad for math, for a variety of reasons. Consider, for example, what would happen if we didn’t have PEMDAS (the mathematical rules for order of operations). It would be impossible to find the ‘correct’ solution to basic arithmetic problems like “3+4 x 5”

So, we need a way to disambiguate potentially ambiguous statements in logic. Consider, for example, the following statement: “It’s not true that if you like cats, then you hate dogs”.

If we were to symbolize this right now, we might write it like this:

IF NOT P, THEN Q

or

∼P ➝ Q But this is technically mistaken. When P is “you like cats” and Q is “you hate dogs”, that expression says that if you don’t like cats, then you hate dogs. But that means something different!

The statement is saying that a person can like both. Here is the correct way to symbolize it:

∼(P ➝ Q)

The parenthesis make the negation operator (∼) attach to the entire conditional statement, rather than just the “P”. What this means is that the negation needs to be treated as the main operator. Our first mistake was to treat the conditional (➝) as the main operator. That changed the meaning.

Parentheses are used to dictate the scope of operators, and they can determine which operator is the main operator, and which are minors.

Now, here is an example of a scary and confusing expression:

(P ➝ M) v (T ● Q)

What is the main operator in this expression? It is whatever operator governs the largest component. In this case, it is the disjunction OR operator, the “v”. This disjunction governs two compound statements, each of which is a disjunct. (~P ➝ M) is one disjunct, and (T ● Q) is the other.

NOTE: If there is only one operator outside of all parentheses and brackets, that operator will be the main operator.

NOTE: The negation is only the main operator when: (a) it is the only operator outside of all parentheses and brackets, or (b) it is the only operator in the statement.

Here is a slightly less scary symbolic expression: (~P ➝ Q)

What is the main operator in this expression? It is the conditional, the “➝”. The conditional governs the relation between the antecedent (~P) and the consequent (Q). The negation only governs the truth of the atomic statement P.

1. It’s not true that if we have freewill then we are always responsible for our actions.

~(F -> R)

 

2. Either Cthulhu exists and the world will end, or the world won’t end.

  (C * E) v ~E

3. It’s not the case that reality is real and things are as they seem.

  ~(R * S)

4. I’m Pinocchio and I’m not Pinocchio if and only if I am not a puppet.

P * (~P <-> ~ U)

 

5. The largest celestial body is not a star, nor is it a planet.

~S v ~P

~(S * P)

 

6. If nothing exists except the universe (and all else is mere convention) then it is not the case that you and I are different.

(N * C) -> ~D

 

7. If Pigs can fly but pigs can’t fly then pigs can fly or FSM is real.

(P * ~P) -> (P v R)