Reflections: We need to know how to handle claims. Otherwise, we may come across assertions and be unsure of what exactly is being said and what in the world it means. Also, we need to know what we are asserting when we make claims and what others might justifiably infer on the basis of our statements.

Goals: My goals here are to help expand your range of skills and techniques so you can better function in the world. When you read, when you listen to others, when you sign contracts or vote--you are regularly confronted with claims and you regularly draw inferences. I want you to be able to handle these two activities with confidence that you are doing so in a way you can justify. What we will do in this chapter is go deeper into analysis and critical thinking skills by examining the different types of claims and learn techniques for handling those claims.

Handling Claims, Drawing Inferences

Propositions
A proposition, or claim, asserts something is or is not the case. These are all propositions: "The car rolled out into the street," "Chicago is in Illinois," "John Lennon was a Beatle." Propositions are not normally expressed as questions or exclamations, unless those are rhetorical forms of an assertion. In classical logic, moral claims (like " "Assault guns should be illegal") were not treated as propositions, because of the difficulty in assigning a truth-value. That does not mean such claims are just a matter of opinion. But you cannot say they are "True" or "False" with the degree of certainty attached to empirical claims.

Different Kinds of Propositions
There are three kinds of propositions: (1) Tautologies: Propositions that are always true or true by definition, (2) Contradictions: Propositions that are always false or false by definition, (3) Contingent Claims: Propositions that are not necessarily true or false, but are dependent on what is going on in the world to determine the truth-value. This would include claims for which the truth-value is unknown. A proposition is either simple or compound. A simple proposition is one that is at the atomic level--that is, it does not contains any of the logical connectives "and," "or", "not, "if...then," or "if and only if." A proposition that contains any of the five logical connectives is considered compound.

The Five Types of Compound Propositions are:

1. Conjunctions: Propositions of the form "A and B."
2. Disjunctions: Propositions of the form "Either A or B."
3. Negations: Propositions of the form "Not A."
4. Conditional claims: Propositions of the form "if A then B" or "B only if A."
5. Equivalence: Propositions of the form "A if and only if B."

Alternative Constructions: A proposition does not have to contain "and" to be a conjunction. It could have alternative words or phrases that function the same as an "and," so be on the lookout. When you spy an alternative, toss it out and replace it with "and" so we have a uniform way of setting up propositions. Alternatives to "and" include "however," "although," "also," "but," and "furthermore."

Disjunctions. These are propositions of the form "P or Q," where P and Q are each called disjuncts. A disjunction is a proposition that claims either one or the other, or both. Note: Disjunctions in logic are inclusive. The " Either/or or both" makes the claim an "inclusive or." This means a disjunction is true if either one or both of the two disjuncts are true. This contrasts with the everyday use of "either/or" which usually is treated as an exclusion not allowing both disjuncts to be true at the same time.

Negations. These are of the form "not P." A negated proposition has the opposite truth-value of the original statement. A negation of a proposition is true only if the proposition itself is false. Two special forms of negations are the "Neither/nor" and "Not both." Basically, with a "neither/nor" claim both options are eliminated (i.e., "Not this and not that"). With "not both" one of the two options is eliminated (i.e., "not this or not that"). We will learn how to handle these later in the chapter, along with other types of negated propositions.

Conditional Claims

These are of the form "If P then Q," where P is called the antecedent and Q is called the consequent. Other forms of conditional claims include "P only if Q," "Q is necessary for P," "A necessary condition of P is Q," "P unless Q," and "Without P, then Q." Propositions of the form "A only if B" assert that, "If B does not happen, then A won't happen either." In other words, "A only if B" can be rewritten "If not B then not A" or "If A then B."

Alternative Constructions: "P if Q," "Only if Q, then P," or "P only if Q," can all be rewritten in the "If . . . then" form. Be sure to make the necessary adjustments and add negatives when required. Below are some suggestions that should help clarify this:

Equivalence (Biconditional Propositions): Two propositions are equivalent if they assert the same thing. The resulting proposition is called a biconditional. "P is equivalent to Q" is the same as "If P then Q, and if Q then P." When that occurs, you can say "P if and only if Q." Any two equivalent propositions have the same truth-value; they are either both true or both false.

Categorical Propositions: The Forms of Categorical Propositions are:

All P is Q.	This is called an "A" claim.
No P is Q.	This is called an "E" claim.
Some P is Q.	This is called an "I" claim.
Some P is not Q.	This is called an "O" claim.

Propositions containing a proper noun as the subject are considered A or E claims, relative to being positive or negative. Propositions of the form "x% of A is B" (where x is neither 100 nor zero) are treated as 'I" or "O" claims, according to the claim being in the positive or the negative.

Quantity and Quality: The quantity of a proposition answers the question, "How much?" The possible answer is "universal" or "particular" (i.e., all or some of it). The quality of a proposition answers, "Are you asserting something is or is not the case?" You are either affirming that it is the case, so the quality of the proposition is positive, or denying it, so the quality of the proposition is negative.

Symbolizing Propositions

Note: "P if and only if Q" could also be expressed "P is equivalent to Q," with the connective then referred to as equivalence.

Steps to Translating a Proposition

Step 1: Unpack the structure.We do this by examining the hierarchy of the connectives.
Step 2: Assign Variables To Component Propositions. Replace the antecedent and consequent with variables (A, B, C, etc.).
Step 3: Replace Component Propositions With The Assigned Variable.
Step 4: Put Symbols in Place of All the Logical Connectives.

Rules of Replacement for Ordinary Language

1. Only. "Only" functions as an exclusion narrowing down the territory of the predicated class. Any proposition of the form "Only P is Q" can be rewritten: If not P, then not Q. This is also equivalent to "All Q is P."

2. The Only. Another exclusion is "The only." Here it is the subject being restricted; not the predicate as above. Any proposition of the form "The only P is Q" can be rewritten: "If not Q then not P." This is also equivalent to "All P is Q." The Connection between "Only" and "The Only." "The only P is Q" is equivalent to "Only Q is P."

3. Unless. Propositions of the form "P unless Q" can be expressed as either a conditional claim or a disjunction. As a conditional claim it can be written, "If not Q then P." The restricted condition Q is the one thing that can stop P from happening. In other words, if you don't have Q, then P occurs. The second way to write "P unless Q" is in the form, "Either P or Q."

4. Sufficient. "P is sufficient for Q" asserts that Q will happen whenever P occurs. In other words, "P is sufficient for Q" is equivalent to "If P then Q."

5. Necessary. "P is necessary for Q" asserts that Q won't happen without P. That is, if you don't have P, you won't have Q. So if you have Q, you must also have P. P is necessary for Q. If not P then not Q. This is equivalent to: If Q then P.

6. The Evers: Whenever, Whoever, Whatever, Wherever, Never, and special constructions of However. Any proposition with the "--ever" construction should be treated as a universal claim. They can be rewritten as conditional claims. This is symbolized P ⇒ Q.

7. Negations: "P is never Q," It is not true that P, Not only P is Q, Not just P is Q, It is false that P. This is the same as putting the negative in front of the claim being negated.

Rules of Replacement for Ordinary Language

Formal Rules Of Replacement

The remaining Rules of Replacement are not simply focused on replacing one expression in ordinary English with an equivalent one, but focus on logical structure. These rules provide the means to translate from one logical form to another equivalent form.

8. DeMorgan's Laws: These are two special forms of negations. With the "Not both" construction, one of the choices is being denied--either the first option or the second one. With a "neither... nor..." construction, both options are being denied, the first choice and the second one. This can be expressed as follows:

DeMorgan's Law 1: Not both ~ (P & Q) = ~ P v ~ Q.
      → It is not true that both P and Q is the case.
      → Either P is not the case or Q is not the case.

DeMorgan's Law 2: Neither/nor ~ (P v Q) = (~ P & ~ Q)
      → It is not true that either P or Q is the case.
      → "P is not the case and Q is not the case.

9. Transposition. The rule of transposition allows us to flip the antecedent and consequent in a conditional claim--but doing so requires the terms to change to their opposites. (In short, "flip and switch.").

(P → Q) = (~ Q → ~ P)
If P then Q. → If not Q then not P.

10. Material Implication. Material Implication allows you to go from a conditional claim (IfÉthen) to a disjunction (Either/or), with one proviso. When we make the switch the first term is negated.

(P → Q) = (~ P v Q)
If P then Q. → Either not P or Q.

11. Exportation. This rule allows you to restructure a conditional claim with a conjunction in the antecedent. The form of exportation is this: [(A & B) → C] = [A → (B → C)]

If A and B, then C. → If A then, if B then C.

12. Equivalence. This is also known as a biconditional" or "if and only if claim. This rule allows us to set out the two component parts of a biconditional claim in two different ways.

"P if and only Q" can be written in two equivalent forms:
→ If P then Q, and, if Q then P.
→ If P then Q, and, if not Q then not P.
Translation of form 1:(P → Q) & (Q → P)
Translation of form 2:(P → Q) & (~Q → ~ P)

Square of Opposition: Drawing Inferences

Contrary: Two propositions are contraries if they cannot both be true, but could both be false. If one is true, then the other one is necessarily false.

Subcontrary: Two propositions are subcontraries if they cannot both be false but could both be true. This is true of the two particular claims. If one is false then the other must be true.

Contradictory: Two propositions are contradictories if they cannot both be true and they cannot both be false. All the categorical propositions have contradictories. "All P is Q" is opposite in truth-value to "Some P is not Q." "No P is Q" has an opposite truth-value to "Some P is Q."

Subaltern: When a universal claim is true and the subject class is not empty of members, we can conclude that the corresponding particular claim is also true. This is, called the subaltern. The process of going from the universal claim to its corresponding particular claim is called subalternation.

The Square of Opposition

The Obverse, Converse, Contrapositive There are three other key moves you can make in terms of drawing inferences. These are the obverse, the converse, and the contrapositive. For these we need to know one more thing--the complement of a class.

Complement: The complement of a class A is the class of those things not in A. So, for instance, the complement of the set of voters is the set of nonvoters. The complement of the set of noncitizens is the set of citizens. So, given any set A, the complement is the set non-A.

Obverse: The Obverse of a proposition involves two steps: First, change the quality (from positive to negative or vice versa); then change the predicate to its complement. The result is the Obverse. It has the same truth-value as the original claim

Converse: The Converse of a proposition is obtained by switching the subject and the predicate, when possible. We can take a converse on an E or I claim. However, the converse of an A claim is known as converse by limitation, for we must step down to an I claim. We can't take the converse of an O claim.

Contrapositive: To take the contrapositive of a proposition, follow these two steps: First, replace the subject with the complement of the predicate. Second, replace the predicate with the complement of the subject. The contrapositive cannot be taken on the I claim. It can only be taken on an A, E and O. The E claim is contrapositive by limitation: step down to an O claim. Don't be surprised with a strange-looking result. Once you verify the original sentence is A, O, or E, then just flip the subject and predicate, changing each one to the complement when you do the switch and, in the case of the E claim, move it down to an O claim.

Converse, Obverse, and Contrapositive

Converse: A, E, and I (We can't take the converse on the O claim)
Switch subject and predicate. Note: Converse of an A goes to an I

Forms of the Converse:

No P is Q.	Converse is	No Q is P.
Some P is Q.	Converse is	Some Q is P.
All P is Q.	Converse is	Some Q is P.
Some P is not Q.	No converse	N/A

Obverse: A, E, I, O (all claims)

Forms of the Obverse:

All P is Q.	Obverse is	No P is non-Q.
No P is Q.	Obverse is	All P is non-Q.
Some P is Q.	Obverse is	Some P is not non-Q.
Some P is not Q.	Obverse is	Some P is non-Q.

Contrapositive: A, E and O (We can't take the contrapositive on I claim)

Forms of the Contrapositive:

All P is Q.	Contrapositive is	All non-Q is non-P.
No P is Q.	Contrapositive is	Some non-Q is not non-P.
Some P is not Q.	Contrapositive is	Some non-Q is not non-P.
Some P is Q.	No contrapositive	N/A