[3.2.] “Logical Form.”

 

[3.2.1.] A Reminder: Constants vs. Variables.

 

Remember the difference between sentence constants (A, B, C...) and sentence variables (p, q, r...).

 

Sentence constants stand for actual ordinary language sentences (e.g., “A” might stand for “Amy is an actress”) and are either true or false.

 

On the other hand, sentence variables are place-holders for constants which we use when we want to make a point about sentences in general rather than about any particular sentence or sentences (like when we construct truth-table definitions of connectives).

 

 

[3.2.2.] Sentence Forms & Substitution Instances.

 

sentence forms (df.): a sentence form is an expression that contains sentence variables and that can be turned into a sentence if all of its variables are replaced by well-formed sentences.

 

The following expressions are sentence forms (but note—these are not all possible sentences forms; there are an infinite number of possible sentence forms, and these are merely the simplest):

 

p          ~p        p · q                p Ú q                p É q               p º q

 

A sentence form is neither true nor false. But it can be turned into a sentence (which is either true or false) by replacing its variables, as the following examples show:

 

In  p · q  , replace p with A and q with B, to get the sentence      A · B

 

In  p · q  , replace p with B and q with A, to get the sentence      B · A

 

In p · q, replace p with ~A and q with B É C, to get the sentence                       ~A · (B É C)

 

            In p · q, replace p with ~[A º (B · C)] and q with [~(B É D) Ú A] · C, to get the sentence

 

                                    ~[A º (B · C)] · {[~(B É D) Ú A] · C}

 

In all four examples, the end result is a substitution instance of the form p · q.

 

NOTE 1: in a substitution instance, every occurrence of the same variable is replaced with the same sentence.  For example, consider the sentence form:

 

p É (q É p)

 

This is a substitution instance of that form:          A É (B É A)

 

This is not:                                                        A É (B É C)

 

 

NOTE 2: Any compound sentence is a substitution instance of more than one sentence form. E.g., “A É B” is a substitution instance of both of these forms:

§         p É q

§         p

 

So “A É B” is a substitution instance of “p” because you can produce “A É B” by doing nothing other than replacing the only variable in “p” with the well-formed sentence “A É B”.

 

 

[3.2.3.] Tricky Examples of Substitution Instances.

 

“~A” is a substitution instance of the form “~p” and of the form “p”

 

“~(~A · B)” is a substitution instance of:

§         p

§         ~p

§         ~(p · q)

§         ~(~p · q)

 

But it is not a substitution instance of either of the following:

§         p · q

§         ~p · q

 

To check this, notice that you cannot replace “p” and “q” in either of those two expressions with well-formed sentences and thereby create “~(~A · B)”.

 

 

WALK-THROUGH: Substitution Instances (pp.61-63)

Here you’ll find steps for identifying all the sentence forms of which a given sentence is a substitution instance.  I will use a different sentence to illustrate these steps:

 

(A É ~B) Ú (C º D)

 

1.      Every sentence is a substitution instance of:      p  [the atomic form of the sentence]

 

2.      Every compound sentence is a substitution instance of one of the five possible forms which have only a single connective:

§         ~p

§         p · q

§         p Ú q

§         p É q

§         p º q

Which of these forms is the right one is determined by the main connective of the sentence in question. Since the main connective of “(A É ~B) Ú (C º D)” is “Ú”, that sentence is a substitution instance of “p Ú q”. This is the basic form of the sentence.

 

“Other than the atomic form, every correct form should have the same main connective as the sentence’s basic form.” (p.62) So the rest of the forms of which our sentence is a substitution instance will have the same main connective (in this example, “Ú”).

 

3.      Every sentence is a substitution instance of the form you get by replacing each constant in the sentence with a variable. This is the sentence’s expanded form:

 

(p É ~q) Ú (r º s)

 

“No sentence is a substitution instance of a form that contains more logical structure, or a different kind of logical structure, than the sentence’s expanded form.” (p.62) The expanded form is a sort of limit to the complexity of the forms you’re looking for. A sentence cannot be a substitution instance of a form that is more complicated (or has a different sort of structure) than the sentence’s expanded form.

 

4.      Now to produce the rest of the forms, in a systematic way:

 

§         For each of the variables in the basic form, think about what sort of sentence (negation, conditional, etc.) that variable represents.

 

In our example, the basic form is:           p Ú q

 

“p” stands for the conditional:     A É ~B

 

So our sentence is also a substitution instance of              (p É q) Ú r

 

“q” stands for the biconditional:  C º D

 

So our sentence is also a substitution instance of              p Ú (q ºr)

 

Combining these gives another form, one that makes it explicit what sorts of sentences both “p” and “q” stand for:

            (p É q) Ú (r º s)

 

§         Now, for each of the variables in this form, think about what sort of sentence that variable represents.

 

“p” stands for the atomic sentence “A” -- so there is no more structure to be revealed by replacing “p” with anything.

 

“q” stands for the negation:        ~B

 

So our sentence is also a substitution instance of              (p É ~q) Ú r

 

and of                                                                           (p É ~q) Ú (r  º s)

 

which is the expanded form.

 

 

So our example:             (A É ~B) Ú (C º D)

                       

is a substitution instance of each of the following:

 

§         p                                                                            [its atomic form]

§         p Ú q                                                                      [its basic form]

§         (p É q) Ú r

§         p Ú (q º r)

§          (p É ~q) Ú r

§          (p É q) Ú (r º s)

§         (p É ~q) Ú (r º s)                                                   [its expanded form]

 

 

A detail not mentioned in the book and that becomes relevant in exercise 3-5 nos. 6, 7, 11 and 12:

 

When the same constant appears twice in a given sentence, e.g.,

 

A Ú (A · B)

 

...this increases the number of forms of which the sentence is a substitution instance. This disjunction is a substitution instance of all of the following:

§         p

§         p Ú q

§         p Ú (p · q)

§         p Ú (q · r)

 

We can form the sentence “A Ú (A · B)” by taking the third form in the list and replacing both occurrences of “p” with “A” and “q” with “B”. And we can form that sentence by taking the forth form in the list and replacing “p” and “q” with “A” and “r” with “B”. So “A Ú (A · B)” is a substitution instance of each of those forms.

 

 

EXERCISE 3-3 (p.64)

 

EXERCISE 3-4 (pp.64-65)

§         just write the letter of the form on the right; no need to write entire form

§         notice the closing bracket missing at the end of #11; this is corrected on the errata sheet.

 

EXERCISE 3-5 (p.65)

§         obviously, you have to write out the forms in this exercise

 

 

[3.3.] “Tautologies, Contradictions, and Contingent Sentences.”

 

Each row of a truth-table (except the very top row, which contains the sentence to be analyzed) represents an interpretation of that sentence:

 

interpretation: “a particular assignment of truth-values to each of the atomic constituents of a sentence or group of sentences.”[1]

 

Look back at any of the truth tables we have examined so far and you’ll see that the bottom row in that table specifies a truth value for each of the sentence’s constituent atomic sentences.

 

The truth tables we’ve worked with so far have not been complete truth tables, because they have displayed only one possible interpretation.

 

A complete truth table will display every possible interpretation. Here’s a simple example:

	A B A V ~B T T T T FT T F T T TF F T F F FT F F F T TF

 

 

 

 

 

 

 

 

Unlike the short truth tables from sec. 3.1, this complete truth table begins with columns specifying every possible combination of truth values for every atomic sentence in the compound sentence under analysis. Since there are two atomic sentences (A and B) and each of those is one of only two possible truth values (T and F), there are four possible interpretations of this sentence, i.e., four possible ways in which the truth values of these atomic sentences can vary. These are displayed in the four bottom-most rows of the chart.

 

We will be using complete truth tables to determine whether a given compound sentence is a...

 

contingent sentence: a sentence the truth-value of which “is not determined by logic alone but is contingent upon the state of the world. A sentence is contingent when on some interpretations it is true and on others it is false.”[2] The sentence in the above example, “A Ú ~B”, is contingent -- there are both Ts and Fs in the column immediately under its main connective.

 

tautology: a sentence that is guaranteed to be true because of its logical form.

	A A V ~A T T T FT F F T TF

 

 

e.g.   “A Ú ~A”

 

 

There are only two possible interpretations of this sentence, and on both of them, the sentence is true (look directly underneath the main connective: two Ts, no Fs). So no matter whether “A” is true or false, “A Ú ~A” is true.

 

 

contradiction: a sentence that is guaranteed to be false because of its logical form.

	A A · ~A T T F FT F F F TF

 

 

e.g.   “A · ~A”

 

 

There are only two possible interpretations of this sentence, and on both of them, the sentence is false (look directly underneath the main connective: two Fs, no Ts). So no matter whether “A” is true or false, “A · ~A” is false.

 

 

[3.3.1.] Steps to Follow in Constructing Complete Truth Tables.

 

In this section, I’m following the steps described in your textbook (pp.68-70). Here’s a different example:[3]

 

~[(A É ~B) Ú (~B É C)]

 

Step 1: Determine the number of rows needed (in addition to the top row, where the sentence is displayed).

 

The formula for determining the number of rows is:

 

For a sentence with n constants, use 2ⁿ rows:

§         for a sentence with 1 constant, use 2 rows

§         for a sentence with 2 constants, use 4 rows

§         for a sentence with 3 constants, use 8 rows

§         for a sentence with 4 constants, use 16 rows

...and so on-- every time you add a constant, the number of rows doubles.

 

Our example contains three constants, so we need eight rows.

 

Step 2: Begin constructing the table, starting with the left-most columns (the “guide columns”) headed by the individual constants. Fill in the columns moving from the inside out:

§         in the first column, use alternating Ts and Fs

§         in the next column, use alternating pairs of Ts and Fs

§         in the final column, use alternating quartets of Ts and Fs

	A B C ~[(A É ~B) V (~B É C)] T T T   T T F   T F T   T F F   F T T   F T F   F F T   F F F

 

 

 

 

 

 

 

 

 

 

 

NOTE: You should set up your guide columns with the sentence constants in the same order in which they appear in the sentence. This will sometimes mean that they are not in alphabetical order, but that is OK. (This is an arbitrary choice, between order-they-appear and alphabetical-order. The important thing is that the entire class stick with one method of doing the guide columns.)

 

 

Step 3: Begin filling in the table underneath the sentence. Begin by filling in truth-values directly underneath each atomic sentence.

 

	A B C ~[(A É ~B) V (~B É C)] T T T    T    T      T   T T T F    T    T      T   F T F T    T    F      F   T T F F    T    F      F   F F T T    F    T      T   T F T F    F    T      T   F F F T    F    F      F   T F F F    F    F      F   F

 

 

 

 

 

 

 

 

 

 

 

 

Step 4: Begin adding the truth-values for the compound sentences that are components of the entire sentence. Start with the compound sentences with the smallest scope: the negations that operate on single sentence constants.

	A B C ~[(A É ~B) V (~B É C)] T T T    T   FT     FT   T T T F    T   FT     FT   F T F T    T   TF     TF   T T F F    T   TF     TF   F F T T    F   FT     FT   T F T F    F   FT     FT   F F F T    F   TF     TF   T F F F    F   TF     TF   F

 

 

 

 

 

 

 

 

 

 

 

 

Continue with other compound sentences in order of increasing scope:

	A B C ~[(A É ~B) V (~B É C)] T T T    T F FT  T  FT T T T T F    T F FT  T  FT T F T F T    T T TF  T  TF T T T F F    T T TF  T  TF F F F T T    F T FT  T  FT T T F T F    F T FT  T  FT T F F F T    F T TF  T  TF T T F F F    F T TF  T  TF F F

 

 

 

 

 

 

 

 

 

 

 

 

The final column to be completed will be the column underneath the main connective:

 

 

 

 

	A B C ~[(A É ~B) V (~B É C)] T T T F  T F FT  T  FT T T T T F F  T F FT  T  FT T F T F T F  T T TF  T  TF T T T F F F  T T TF  T  TF F F F T T F  F T FT  T  FT T T F T F F  F T FT  T  FT T F F F T F  F T TF  T  TF T T F F F F  F T TF  T  TF F F

 

 

 

 

 

 

 

 

 

 

 

 

 

Since this sentence is false on every possible interpretation, it is a contradiction.

 

Exercise 3-6 (p.70)

§         do all of these and check the answers to the even problems; we’ll do the odd ones in class next time [watch out for number 15 – it only has three constants, C, D and A]

 

 

Stopping point for Monday January 28. For next time, exercises 3-3, 3-4, 3-5, and 3-6, and read Ch.3 secs. 5-7 (pp.72-78).