§ 9. Equivalence

Truth-functional schemata are equivalent if they mutually imply each other. They will therefore agree case by case under truth-value analysis. You can say that no interpretation of literals will make the truth-values of one sentence unlike those of a sentence with which it is equivalent.

As implication is validity of the conditional, equivalence is validity of the biconditional.

Though logic is said to have the primary purpose of justifying inference, another perhaps equally important task of logic is to manipulate statements.

(1) P is equivalent to 1. ~(~p), 2. pp, 3. p ∨ p, 4. p ∨ pq, 5. p. p ∨ q, 6. pq ∨ p~q, 7. p ∨ q.p ∨ ~q.

***

i. Equivalence is mutual implication
ii. Any schema is equivalent to itself
iii. If one schema is equivalent to a second, and the second is equivalent to a third then the first is equivalent to the third.
iv. If one schema is equivalent to a second then the second is equivalent to the first...
v. Valid schemata are equivalent to one another and to no others, and similarly for inconsistent schemata.
vi S1 implies S2 if and only if S1 is equivalent to the conjunction of S1 and S2.
vii. S1 implies S2 if and only if S2 is equivalent to the alternation of S1 and S2.
viii. S implies each of S1 and S2 if and only if S implies the conjunction of S1 and S2.
ix. S1 and S2 each imply S if and only if the alternation of S1 and S2 implies S.

-p. 62

Putting a schemata for a literal is called substitution, but putting a schemata for another schemata is called interchange.

There are several laws of interchange. The first one is to:

Think of '...p...' as any schema containing 'p', and of '...q...' as formed from '...p...' by putting 'q' for one or more occurences of 'p'; then

'p↔q' implies '...p....↔....q...'

-p. 63-64, underline mine

The second law of interchange is:

If S1 annd S2 are equivalent, and S2' is formed from S1' by putting S2 for one or more occurences of S1, then S1' and S2 are equivalent.

-p. 64

so for example if 'p→q' and '~p(p~q)' are equivalent, then we can interchange 'p→q. ∨ r' and '~(p~q) ∨ r'. Once you decipher each matter its truth is evident, but the process of teaching yourself to think in certain patterns yields fruit when coming across complex expressions.

The third law of interchange is that interchange does not preserves validity, implication, equivalence, and inconsistency. Unlike substitution it also preserves consistency, non-validity, nonimplication, and nonequivalence.

Because of this, just like select obvious truth-functional reductions, Quine recommends we immediately supplant any schema of any of the seven forms found in (1) with p (or q or r, etc. as the case may be).

"Pure biconditionals are valid if and only no letter occurs in it an odd number of times" (p. 66)

§ 8. Words Into Symbols

This chapter is almost unnecessary for those to whom the translation of every day speech into truth-functional schemata is common-sensical. If this describes you and you are cramming for an exam, feel free to omit this reading.

Logical inferences get us from premises to conclusions in a very certain manner if done properly. But the premises and conclusions are not themselves grounded in logic. This is why it's important to understanding how to translate everyday language into logical language. This involves identifying and isolating each distinct statement in a piece of rhetoric and proceeding to substitute literals for them. It also requires the ability to identify the nature of the relationships between each of the statements in order to form schemata. Without performing these steps properly it is impossible to correctly test for logical implications.

Chapter one contains examples of how negation can be notated logically. And conjunction only requires reflection for its various every day forms to be understood (think of "and", "although", "but", etc.). Use of one of these forms over the other may provide extraneous interpretative tips about the speaker's disposition or beliefs, but cannot have any bearing on the truth or falsity of the claim itself.

Phrases like "if p then q", "p only if q", "q if p", "q provided that p", and "q in case p", all translate to the truth-functional "If then".

"P only if q" is different than the biconditional "p if and only if q".

"Unless" translates to "v", the logical symbol representing alternation.

A central principle to the absolutely imperative art of translating words into symbols is to assure that you do not assign more than one interpretation to the same expression. "Violation of this principle was known traditionally as the fallacy of equivocation" (p. 56).

Though translating vocabulary into logical terms may come with little effort to some, the broader task at hand may be more involved. When sentences begin to agglutinate and the semantics of words and phrases are contingent on others, translation of day to day rhetoric into schemata requires additional levels of critical thinking.

[Note: I will henceforth use "***" to mark sections that I intend to return to and annotate at a later time.]

***

There are three phases to paraphrasing statements logically:

1. The translation of words into symbols.
2. Rephrasing clauses to avoid the fallacy of equivocation.
3. Organizing paraphrased clauses correctly in order to form a whole compound.

It's usually best to look for the big picture of a statement first, then work your way inward.

***

§ 7. Implication

One "truth-functional schema is said to imply another if there is no way of so interpreting the letters as to make the first schema true and the second false... whether a truth-functional schema S1 implies another, S2, can be decided always by taking S1 as antecedent and S2 as consequent of a conditional, and testing the conditional for validity... implication is validity of the conditional." (p. 46).

Once a logician encounters a false outcome of a conditional, it is clear that the conditional is invalid, and that its antecedent does not imply its consequent.

Quine lays out the following four rules of schematic implication:

(i) Any schema implies itself.
(ii) If one schema implies a second and the second a third then the first implies the third.
(iii) An inconsistent schema implies every schema and is implied by inconsistent ones only.
(iv) A valid schema is implied by every schema and implies valid ones only.

Implication always transmit validity.

I really like when Quine says

We must be able to think up schemata which imply or are implied by a given schema and promise well as links in a proposed chain of argument. Such products can be checked mechanically by truth-value analysis, but thinking them up is an unmechanical activity.

-p. 47

It brings to mind the worn-out quotation of Einstein, "Imagination is more important than knowledge". Here, even in the midst of pursuing the discipline of meticulous, exhaustive, systematic, and well-defined logical analyses, we see the glory of knack and of imagination and of art.

He continues

Facility in it depends on grasping the sence of a simple schemata clearly enough to be able, given a schema, to conjure up quite an array of fairly simple variants which imply or are implied by it. Given 'p v q', it should occur to us immediately that 'p' and 'q' and 'pq' and '~p→q' imply it and that 'p v q v r' and '~p→q' are implied by it... Such flashes need not be highly accurate, for we can check each hunch afterward by truth-value analysis. What is important is that they be prolific, and accurate enough to spare excessive lost motion.

No doubt repertoire is an aid to virtuosity in contriving implications, but understanding is the principal thing. When simple schema are sufficiently transparent to us, we can see through them by the light of pure reason to other schemata which must come out true if these do, or which can not come out true unless these do.

This is important to note, but I would like to comment that I enjoy Quine's writing style immensely and love the rhetorical beauty with which he teaches logic. He goes on to say "Readiness with implications is aided also, no doubt, by ease of checking" and offers a shortcut to what he calls the "full sweep" (truth-value analysis the long way"), which he calls the "fell swoop".

The fell swoop may be executed when a schema is "visibly verifiable by one and only one interpretation" of its literals (p. 48). "Now when S is such a schema, the question whether S implies a schema S' can be settled simply by supplanting [the literals] in S' by the values which make for truth of S, and resolving. If we come out with... a valid schema, then S implies S'; otherwise not" (ibid.).

The fell swoop may be reversed as well. When S' "is a schema thus falsifiable by one and only one interpretation, the question whether a schema S implies S' can be settled simply by supplanting [the literals] in S by the values which make for falsity of S', and resolving. If we come out with an inconsistent schema, then S implies S'; otherwise not" (ibid.)>

§ 6. Consistency and Validity

Consistency

A truth-functional schema is "consistent" when at least one of its possible outcomes is true, otherwise "inconsistent".

Validity

A truth-functional schema is "valid" if each of its possible outcomes is true, otherwise "invalid".

Patently valid schemata may be reduced to ⊤, and patently invalid schemata to ⊥. "Patent" in this case is arbitrary, but classroom rules may be applied for consistency (get it?).

Quine names these two categories of patently inconsistent schemata: (a) conjunctions in which some part appears both plain and negated as component of the conjunction, and (b) biconditionals in which some part appears both plain and negated on either side of the biconditional.

Also note that p→~p is actually counted as consistent.

Quine names these two categories of patently valid schemata: (a) alternations in which some part appears both plain and negated as component of the alternation, and (b) conditionals or biconditionals whose two sides are alike.

Substitution of schemata for letters preserves validity!
Substitution of schemata for letters preserves inconsistency!
Substitution of schemata for letters DOES NOT preserve consistency!

-Jonathan Charles Wright and W. V. Quine

§5. Truth-Value Analysis

The "truth functions of given components are all the compounds constructed from them by means exclusively of conjunction and negation (and the dispensable further connectives 'v', '→', '↔')" (p. 33, hyperlinks mine). "P" is also counted as a truth function of "p".

Schemata

Truth functions containing literals are not statements, as each literal is semantically empty. Literals and truth functions are therefore called "schemata", and each is a singular "schema".

The "interpretation" of a schema is either 1. its truth value, or 2. the statement it represents.

"⊤" ("tee", or even "true") stands for "true", and "⊥" (there is no established pronunciation for the "Up Tack", but the monosyllabic "false" suffices) stands for "false" ("F" is needed for different purposes later).

Resolution of Schemata

"Resolution" is the process whereby schemata may be reduced logically. Obviously "~⊥" resolves simply to "⊤". More complex schemata can resolve according to at least nine rules, which Quine lays out on page 34 as

(i) Delete ⊤ as a component of conjunction...
(ii) Delete ⊥ as a component of alternation...
(iii) Reduce a conjunction with ⊥ as a component to ⊥.
(iv) Reduce an alternation with ⊤ as a component to ⊤.
(v) Drop ⊤ as antecedent of a conditional...
(vi) Reduce a conditional with ⊥ as antecedent, or ⊤ as consequent, to ⊤...
(vii) If a conditional has ⊥ as consequent, reduce the whole to the negation of the antecedent.
(viii) Drop ⊤ as component of a biconditional...
(ix) Drop ⊥ as component of a biconditional and negate the other side...
(emphases and quotation marks omitted by me)

Truth Value Analysis

Evaluating schemata for each possible interpretation of its literals is called truth-value analysis. "We make a grand dichotomy of cases by putting first '⊤' and then '⊥' for some chosen letter..." (p.37). "It is better to choose the letter which has the most repetitions, if repetitions there be, and to adhere to this plan also at each later stage... This strategy tends to hasten the disappearance of letters, and thus to minimize work." (p. 38).

[Note: At this point in my studying I have chosen to significantly downsize my annotation ambition, as well as to come back to link more of the text in each post to other posts for the sake of speeding up the process here and now.]

§3. The Conditional

In addition to negation, conjunction, and alternation, another connective role is played in rapid, precise dialogue: conditionality. Conditional statements come in the form

If p then q.

The "p" component is referred to as the "antecedent" and the "q" component the "consequent".

Material Conditionals

Conditionals of this form are called "material conditionals", and may be written like the following:

p ⊃ q

Conditionals, insofar as they are treated as proper truth functions, admit as wholes of truth when the antecedent and consequent are both true, and when the antecedent is false (regardless of the truth of the consequent). Conditionals, when treated as truth functions, admit as wholes of falsity only when the antecedent is true and the conditional is false.

That a false antecedent implies every statement sounds tricky and perhaps wrong at first. What you need to realize is that we are handling statements only on a truth-functional level - only looking at their truth-values and not the details of any actual semantic contents. Thus it is not so that "Louis is a woman" proves the truth of "God does not exist". Rather, the truth-value of the whole conditional is handled as if it were true, since the antecedent is known to be false (and there may be controversy over the consequent). This might feel weird to you, but it is no less weird, according to Mr. W. V. Quine, than treating the truth-value of a conditional with a false antecedent as false. In everyday language we typically consider ourselves as simply not being committed to any claims when our antecedents turn out false (except in cases where we're intentionally making a contrafactual conditional). But in formal logic we just treat conditionals with false antecedents as true compounds.

These same ends could be accomplished by the means presented in §2 ("Truth Functions").

In elaboration, Quine offers the example

If anything is a vertebrate, it has a heart.

The reason that the above statement is not like the other truth functions is that its component parts are not independent statements, whose truth or falsity sufficiently determines the truth or falsity of the whole. Quine explains that "it has a heart" is not a statement unto itself.

Thus, the above example should be seen as affirming a series of material conditionals:

If a is a vertebrate, a has a heart;
If b is a vertebrate, b has a heart,
etc.

Or, as Quine puts it,

No matter what x may be, it is not the case that x both is a vertebrate and does not have a heart.

As a method of logic then, the precise meaning of

p ⊃ q

is

~(p~q)

However situations arise wherein it doesn't seem right to treat conditionals as traditional truth functions in this manner..

Contrafactual Conditionals

It is not uncommon to use "if-then" in the subjunctive mood, instead of the indicative mood. Such cases tacitly carry the admittance of the falsity of the antecedent. For example

If I had not walked to Java this morning, then I would not be drinking coffee as I write this example.

Such contrafactual conditionals are intended to transmit information despite the falsity of their antecedents. Thus, some contrafactual conditionals with false antecedents may be argued to be true, while others may be argued to be false. They therefore do not qualify as being truth functional. Rather, there are kindred causal relationships to explore during examinations of contrafactual conditionals. Quine briefly follows this rabbit trail in §3.

The arguably ineffable relationship between the antecedent and consequent of a contrafactual conditional is similar to that which may be argued to exist between the components of any meaningful alternation. This phenomenon is extremely fertile ground for exploration and controversy.

Biconditionals

Simply put, a material biconditional is of the form

p if and only if q

or

p ≡ q

and it is true only when the truth values of the components match (either both p and q are true, or both p and q are false). This truth function is identical to affirming

(p ⊃ q)(q ⊃ p)

Like "v" and "⊃", the particular symbol and use of "≡" is superfluous, yet helpful.

§2. Truth Functions

Truth Functions

The truth or falsity of the components of a truth function sufficiently determine the truth or falsity of the truth function as a whole. Stating a truth function is simply a convenient and precise way of summarizing the truth values of several statements at once. For example:

I am watching The Two Towers and I am drinking a milk stout

is a truth function, because whether the statements

I am watching The Two Towers

and

I am drinking a milk stout

are true determine whether the compound is true. However the statement

I am watching The Two Towers because I am drinking a milk stout

is NOT a truth function, because two people can affirm both statements, or affirm either statement and deny the other, or even deny both statements, and yet still debate over whether the compound is true.

A truth function can be described by creating a chart of the truth values of each of its components. Besides "not", "and", and "or", one can make up one's own truth function by explaining when the compound will turn out true, and when it will turn out false.

Negation and conjunction alone are sufficient for all truth-functional purposes. Actually, negation and alternation alone may also be sufficient for all truth-functional purposes.

Truth Function Exercise

Quine asks his reader to make a statement that will be true only when any, but only, two out of three components are true, using only conjunction and negation. I came up with:

~(pqr)~(~p~q~r)~(~p~qr)~(~pq~r)~(p~q~r)

So, it is not the case that all three statements are true, it is not the case that all three statements are false, it is not the case that only r, it is not the case that only q, and it is not the case that only p - therefore the only situations that would render the compound true would be (pq), (qr), or (pr).

§1. Negation, Conjunction, and Alternation

Negation

A “statement” can be affirmed or denied.

Denying a statement is the same as affirming another statement.

For example, if you denied the previous statement you would be affirming the following:

Denying a statement is NOT the same as affirming another statement.

Sometimes it is more complicated to figure out what the “negation” of a statement is, but with a little effort the negation of a statement can be figured out and worded clearly.

In logic, the symbols “-” and “~” are signs of negation. So

~(I am in Cafe Doma)

could be rewritten as

It is not the case that I am in Cafe Doma.

The parenthesis work just like they do in math: to indicate that something should be dealt with as a whole.

Sometimes in logic “it is not the case” is shortened to just “not”. And statements are often represented with letters, especially starting with “p”.

So “~p” should be read in your mind as “it is not the case that p”, or just “not p”.

Note that sometimes a sign of negation is written above a letter rather than in front of it.

Conjunction

The “conjunction” of multiple statements forms a single statement. This is represented by writing each statement next to the others. For example,

(I am in Cafe Doma)(Lindsey is next to me)(It is 40 degrees Fahrenheit)

could be written as

I am in Cafe Doma, Lindsey is next to me, and it is 40 degrees Fahrenheit.

If each statement in the above conjunction were represented by a letter, it would be written as

pqr

After explaining all this Quine says,

The meanings of negation and conjunction are summed up in these laws. The negation of a true statement is false; the negation of a false statement is true; a conjunction of statements all of which are true is true; and a conjunction of statements not all of which are true is false.

- p. 2

Note that, just like mathematical addition and multiplication, the “associative” property holds true of conjunctions. This means that internal grouping is irrelevant. For example

2 + (2 + 2) = 6,
(2 + 2) + 2 = 6,
2 + 2 + 2 = 6, and
(2 + 2 + 2) = 6

all express the same statements.

Similarly, note that, just like addition and multiplication, conjunctions are “commutative”. This means that internal order is irrelevant. For example

1 + 3, and
3 + 1

express the same statements.

But, unlike multiplication and addition, logical conjunctions of statements are “idempotent”. All this means is that conjoining a statement to itself is unnecessary. For example

(I am in Cafe Doma)(I am in Cafe Doma)

or, in everyday language

I am in Cafe Doma, and I am in Cafe Doma

is simply redundant. But adding a number to itself is meaningful: 2 + 2 = 4.

Revolutionary, no doubt.

So “negation” is like “not”, and “conjunction” is like “and”. Next, Quine looks at “or”, and calls it “alternation”.

Alternation

In everyday language “or” can be ambiguous. In logic, there is a distinction be an exclusive "or", and a nonexclusive "or".

An exclusive "or" is like saying

Either p or q, but not both.

Whereas a nonexclusive "or" is like saying

Either p or q or both.

The exclusive "or" was represented in Latin with the word "aut", and the nonexclusive "or" by the word "vel".

"Alternation" references only the nonexclusive "or". It is sometimes simply written as "v"; reminiscent of it's Latin term.

Quine proposes to the reader that the nonexclusive use of the term be understood when he writes the ambiguous English "or".

So an alternation is true if at least one of its member statements is true.

In order to achieve an exclusive "or", one can write it out

p or q but not both

or

p~q v ~pq

This setup is also often written

p≡~q

Alternation is associative, cummutative, and indempotent like conjunction.

Combining negation, conjunction, and alternation provides for the ability to quickly write complex statements, but you have to be careful to think clearly in order to sort out the truth values of statements that combine conjunction and alternation.

On page 7 Quine lays out something helpful:

1. ~(pq): It is not the case that both p and q.
2. ~p~q: It is not the case that p and it is not the case that q.
3. ~(p v q): It is not the case that either p or q.
4. ~p v ~q: It is not the case that p or it is not the case that q.

(I added the numbers). He notes that 1 and 4 are the same statement, and 2 and 3 are the same statement.