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LOGOS


LOGIC
PROPOSITIONS

 

Not all sentences are declarative, however, but only those to which truth or falsity belongs. Thus truth or falsity does not belong to all sentences; for example, a prayer is a sentence but is neither true nor false. So let all sentences which are neither true nor false be dismissed here, for their consideration is more appropriate to rhetoric or to poetics. Our present inquiry is about declarative sentences [i.e., propositions (Gr. apophantikos logos)]

                                                                                                                      —Aristotle, On Interpretation, 17a 1-7

A. ETYMOLOGY
The English term proposition is historically derived from Latin, proponere and cognates: to put forward, to propose. In turn, proponere translates Aristotle’s Greek expression apophantikos logos: declarative sentence, statement, proposition.

B. GENERAL DEFINITION
Although precise formulation varies slightly with respect to theoretical details, there is general agreement among modern logicians for the following definition:

A proposition is ‘what is asserted’ by means of natural language (typically a declarative or indicative sentence) or ‘what is expressed by’ such a sentence. Any form of discourse that conveys or expresses a proposition is either true or false (truth-value).

Examples
1. The New England Patriots won the Super Bowl.

2. It is raining.
3. Water freezes at 32 degree F and boils at 212 degree F.
4. Either Eminem is a very talented musical artist or he is not.
5. Rembrandt and Picasso were masterful painters and philosophers.
6. Lewis Carroll (pseudonym Charles Dodgson) wrote Alice’s Adventures in Wonderland and was a logician.
7. If a deductive argument is sound, then it is valid.
8. Some Americans are atheists.
9. Corrupt politicians believe that “the end justifies the means.”
10. Nobody can predict all the future consequences of 9/11.

C. BASIC CHARACTERISTICS

1. All propositions make a factual claim. They assert that something is true or false, that a certain state of affairs actually exists, that p is the case (where p is a logical variable for any proposition whatsoever). As such, they claim to refer to (reference) or correspond to (correspondence) some specific state of affairs, fact or reality. Consequently, propositions are referential or descriptive.

2. Typically, propositions are conveyed by or expressed by means of declarative sentences, which assert (declare) that something is a fact. As such, the grammatical structure of the declarative sentence will constitute the minimal conditions of a proposition. This structure consists in a subject term and predicate term. The subject tells us what the subject is or does, whereas the predicate is the verb phrase that attributes a particular property or trait to the subject (predication).

Example: Hitler (subject) was a murderer (predicate).

3. All propositions are either true or false. This strictly disjunctive nature of the proposition is called its truth-value. They are strictly true (T, 1) or false (F, 0). Accordingly, the truth-value of the proposition reflects the logical law of excluded middle: any proposition is either true or false; and the logical law of contradiction: no proposition can be both true and false.

BASIC LAWS OF LOGIC

Noncontradiction/Contradiction

A is not non-A (~A)

Excluded Middle

X is either A or non-A (~A)

Identity
A =A


No statement can be both true and false in the same respect and at the same time.

Any statement is either true or false.

If any statement is true, then it is true.

 p ● ~p = false

p v ~p = true

p = p

 

Symbolic notation


4. Propositions are the linguistic formulation or result of a critical judgment. They advance or assert a definite position on the truth or falsity (true-value or status) of a specific state of affairs. Propositions are critical to the extent that they are based upon a discriminating evaluation of evidence for the truth or falsity of X. And they express a definite judgment about the truth-value of X based upon the evidence provided by the preceding critical inquiry.

5. Propositions are the ideal information, semantic content, or meaning conveyed by means of natural language. As such, they are not identical, equivalent, or reducible to any specific linguistic formulation, for the same proposition (semantic content) may be expressed by linguistic expressions. This difference may be based on the specific natural language employed (e.g., English, French, Spanish, etc.) or various grammatical and syntactical variations within a particular language. Accordingly, we refer to the identical information or same proposition expressed by various linguistic formulations as propositional content or ideal meaning. It is ideal because the proposition is not identical with any real sentence or discourse utilized to express it.

Examples

A. Different languages
It is raining. (English)

Está lloviendo. (Spanish)
Il pleut.  (French)
Es regent.  (German)

B.
Same language
The New England Patriots won the Super Bowl. (Active Voice)
The Super Bowl was won by the New England Patriots. (Passive Voice)
The Patriots won the Super Bowl. (Different words)
New England did not lose the Super Bowl. (Stated negatively)
The Pats are the Super Bowl winners of the 2001 NFL season. (Temporal index)

6. Propositions are the basic building blocks or constituent components of arguments. They function in three (3) distinguishable capacities: as premise, as conclusion, and as both premise and conclusion (subconclusion).

Example
: (1) Cats make good pets because (2) they are affectionate. So, (3) if you want a good pet, you should get a cat.

        1) Cats make good pets. (Subconclusion: both premise and conclusion)
        2) They (cats) are affectionate. (Premise)
        3) If you want a good pet, you should get a cat. (Conclusion)

7. There are two (2) basic types of proposition: simple and compound.


a) Simple: contains only one subject and one predicate
Example: Nice guys finish last. —Attributed to Leo Durocher

Symbolic notation: p

b) Compound: contains more than one subject, predicate, or both. As such, a compound proposition consists of two
or more component propositions. Three (3) basic types of compound proposition are distinguishable:

b.1) Conjunctive: formed by adding at least two (2) component propositions (called conjuncts), usually by putting the word “and” between them.

Examples

1. Michael Jordan is very wealthy. (simple proposition)
2. Michael Jordan is very wealthy and athletic. (conjunctive proposition)
3. Michael Jordan and Bill Gates are very wealthy.
4. Michael Jordan and Bill Gates are very wealthy and Americans.

Symbolic notation
: p ● q (the dot “●” = conjunction)

b.2) Disjunctive (or alternative): often formed by an “either-or” phrase or more simply by inserting the word “or” between the two component propositions (called disjuncts). There are two distinct types of disjunctive propositions: 1) Weak, Inclusive; and 2) Strong, Exclusive.

1) Weak, Inclusive: The inclusive “or” has the sense of “either, possibly both.” Accordingly, an inclusive disjunction is true when one or the other or both disjuncts are true. As such, it is false only if both disjuncts are false. This precise meaning of the weak disjunction is occasionally made explicit in legal documents by the phrase “and/or.”

Example
: Your driver’s license can be rescinded for repeated DWI convictions or for several incidents of driving without a registration.

Symbolic notation
: p v q (the wedge “v” = disjunction)

2) Strong, Exclusive: The exclusive “or” has the sense of “at least one and at most one.” Accordingly, an exclusive disjunction is true when one or the other, but not both, disjuncts are true. As such, it is false if both disjuncts are true or both are false. This precise meaning of the strong disjunction is occasionally made explicit in legal documents by the phrase “but not both.”

Example

Either Mike Tyson is emotionally stable or he is not emotionally stable.
Mike Tyson is either emotionally stable or he is not.

Symbolic notation
: p v (~p) (the tilde or curl “~” = negation)

b.3) Hypothetical, Conditional: Most often formed by an “if-then” statement, the hypothetical or conditional proposition combines two (2) component propositions by placing the word “if” before the first component (called the antecedent) and the word “then” before the second component (called the consequent). Accordingly, its basic meaning is the claimed relationship between the antecedent and the consequent, and precisely in that order. More specifically, the hypothetical proposition claims a relationship of implication: the truth of the antecedent implies the truth of the consequent. As such, the relationship of implication is unidirectional. It is not claimed that the consequent implies the antecedent.

Examples

1.  If my car is out of fuel, then it will not run.
2.  If Tonya Harding is an excellent ethical example for everyone to follow, I’m the greatest philosopher who ever lived.
3.  You cannot continue to live if you stop drinking and eating.
4.  Unless I win the lottery, I’ll keep my day job.
5.  If p is true, then ~p is false.
6.  You will win the game if you score more points than the other team.
7.  If you miss class consistently your chances of success are diminished significantly.
8.  If Hitler was a warm and caring person, then I’m an alien from outer space.

9.  Life is worth living unless you never critically examine it.
10. If Patrick Kennedy is a bachelor, then he is unmarried.

Types of implication
:
1. Logical
2. Causal
3. Definitional
4. Decisional
5. Material

Symbolic notation: pq (the horseshoe ” = implication)

8. Propositions and Language: Various grammatical units may express a proposition. As such, one needs to judge whether a unit of discourse meets the minimal requirements (a factual claim containing a subject and predicate) of a proposition.

a) Declarative Sentence
1) An inductive argument is probabilistic.
2) Nonviolence is the answer to the crucial political and moral questions of our time.
                                                                                        —Martin Luther King, Jr., Nobel Prize address

b) Relative clause: Usually, a relative clause begins with a relative pronoun (who, whom, which, that) and, like all clauses, contains a subject and a predicate. Specifically, it relates to or modifies the subject or predicate of the main clause.

Examples
:

1. The Japanese, who eat lots of fish, have fewer heart attacks. (nonrestrictive modification of the subject)
2. The Japanese who eat lots of fish have fewer heart attacks. (restrictive modification of the subject)
3. Shaquille O’Neill is tall, like most basketball players. (modification of the predicate)

c) Noun clause: A phrase is a noun clause when the entire clause functions like a noun. Unlike a relative clause, it doesn’t modify anything (neither the subject nor the predicate) in the main clause. Rather, it is itself the subject or predicate of the main clause.

Examples
:

1. It is irrelevant whether Antoine is gay. (subject of proposition)
The fact that Antoine is gay is irrelevant.

2. The Pope knows that God exists. (predicate of proposition)
The Pope believes that God exists.

d) Propositions not in declarative form: Units of discourse other than declarative forms occasionally express propositions.

1) Rhetorical question (interrogative sentence): A question may suggest, imply, or assume a proposition when its author believes the answer is obvious or undeniable. As such, it can be an effective rhetorical technique since it calls upon the reader/listener to answer the question and thereby state the implied or indirect proposition.

Examples:

a) …what penalty can frighten a person who is not afraid of death itself?
                                        —Arthur Schopenhauer, “On Suicide” (1851)

b) …what is misery but the desire and possession of evil?
                                        —Plato, Meno, 78a

c) …he that loves not his brother whom he hath seen, how can he love God whom he hath not seen?
                                        — 1 John 4:20

2) Imperative sentence or phrase (command, prescription, advice, etc.): Occasionally an imperative of some kind is employed as a proposition.

Examples:

a) Neither a borrower nor a lenderer be
For loan oft loses both itself and friend,
And borrowing dulls the edge of husbandry.

—Shakespeare, Hamlet, act 1, scene 3

b) Since you want to excel in all your endeavors, you should work hard at everything you do.

c) Thou shall not kill. Therefore, if a serial murderer surprises you in your home you should under no circumstances kill him.

d) Hope empowers one in the face of adversity. Therefore, as the Rev. Jesse Jackson says, “keep hope alive.”

9. Propositional attitudes: These are the states of mind, attitudes, or dispositions that we exhibit toward particular propositions; they express the relation between a person and proposition. These attitudes typically precede the proposition and may be symbolized in the formula:  I (specific attitude) that p (p = any proposition

Examples

I believe that Mayor Cianci is guilty of political corruption.
I doubt that Mayor Cianci is guilty of political corruption.
I know that Mayor Cianci is guilty of political corruption.
I hope that Mayor Cianci is guilty of political corruption.
I suspect that Mayor Cianci is guilty of political corruption.
I am certain that Mayor Cianci is guilty of political corruption.


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