[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 natural-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 that 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 sentence 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 with sentences, 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]
In all four examples, the end result is a substitution instance of the form p · q.
NOTE 1: in a substitution instance of a sentence form, every occurrence of the same variable in the form 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
“A É B” is a substitution instance of “p” because you can produce “A É B” by 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 all of these forms:
p
~p
~(p · q)
~(~p · q)
But it is not a substitution instance of either of the following forms:
p · q
~p · q
To check this, notice that you cannot replace “p” and “q” in either of those two expressions with well-formed formulae and thereby create “~(~A · B)”.
WALK-THROUGH: Substitution Instances (pp.59-61)
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 same steps:
(A É ~B) Ú (C º D)
- Every sentence is a substitution instance of: p [the atomic form of the sentence]
- 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.60) So the rest of the forms of which our sentence is a substitution instance will have the same main connective (in this example, “Ú”).
- Every sentence is a substitution instance of the form you get by replacing each constant in the sentence with a different 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.60) 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.
- 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]
[3.2.4.] Multiple Instances of the Same Variable or Constant.
To get a substitution instance of a form that contains multiple instances of the same variable, you must substitute every instance of that variable with the same sentence.
For example, substitution instances of “p Ú ~p” include:
A Ú ~A
B Ú ~B
(A · B) Ú ~(A · B)
but not
A Ú ~B
A Ú ~(B · C)
Question: Which sentences are instances of “p Ú (p · q)”?
1. A Ú (B · C)
2. A Ú (A · B)
3. D Ú (D · H)
4. D Ú (H · D)
Answer: Only sentences 2 and 3. Sentences 1 and 4 are not instances of “p Ú (p · q)”, because you cannot create either 1 or 4 by replacing each occurrence of p with the exact same sentence.
So in summary, when moving from form to substitution instance, every instance of a given variable has to be replaced with the same sentence.
But things are very different when moving from sentences to sentence forms…
· Some of the forms can be reached by replacing every occurrence of the same sentence with the exact same variable.
· But others can be reached by replacing different occurrences of the same sentence with different variables.
A simple example: “A Ú ~ A” is a substitution instance of…
p [atomic form]
p Ú q [basic form]
p Ú ~p [one-to-one form[1]]
p Ú ~q [expanded form]
Another example:
“A Ú (A · B)” is a substitution instance of all of the following:
p [atomic form]
p Ú q [basic form]
p Ú (p · q) [one-one logical form]
p Ú (q · r) [expanded form]
So, here’s another question: Which sentences are instances of “p Ú (q · r)”?
1. A Ú (B · C)
2. A Ú (A · B)
3. D Ú (D · H)
4. D Ú (H · D)
Answer: ALL OF THEM. This is because you can reach any of these four sentences by replacing every occurrence of a variable in that form with the same sentence – and that includes sentences 2 and 3, where different variables are replaced with the same sentence.
Another example (#6 from exercise 3-5, p.63):
“(A · B) º ~B” is a substitution instance of all of the following:
p [atomic form]
p º q [basic form]
p º ~q
(p · q) º ~q [one-one logical form]
(p · q) º r
(p · q) º ~r [expanded form]
[These points become relevant in exercise 3-5…]
EXERCISE 3-3 (p.62)
EXERCISE 3-4 (pp.62-63)
§ just write the letter of the form on the right; no need to write entire form
EXERCISE 3-5 (p.63)
§ obviously, you have to write out the forms in this exercise
[3.3.] “Tautologies, Contradictions, and Contingent Sentences.”
The truth tables we’ve worked with so far have not been complete truth tables, because they have displayed only one possible assignment of truth values to the sentence in question.
A complete truth table will display every possible interpretation. Here’s a simple example:
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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...
contradiction: a sentence that cannot possibly be true.
tautology: a sentence that cannot possibly be false.
contingent sentence: a sentence that can be either true or false.
We can tell which of these a given sentence is by examining its one-one logical form, the result of “[r]eplac[ing] each statement letter by a variable, making sure that the same letter is replaced by the same variable throughout.” (p.65)
For example, consider the sentence “A · ~A”.
To discover what sort of sentence this is, we will first identify its one-one logical form (replacing every instance of a given constant with the same variable)… [This is VERY important… you MUST use the sentence’s one-one logical form, not its expanded form!]
p · ~p
and then build a complete truth table for that form:
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There are only two possible assignments of truth values to this sentence form, and on both of them, the form is false (look directly underneath the main connective: two Fs, no Ts). So no matter whether “p” is true or false, “p · ~p” is false. So any sentence that has this form is a contradiction.
The sentence in the above example, “A Ú ~B”, is contingent. This is shown by the truth table for its one-one logical form:
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Any sentence whose one-one logical form can be either true or false is a contingent sentence.
Finally, consider “A Ú ~A”. The one-one form of this sentence is “p Ú ~p”:
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There are only two possible assignments of truth values, and on both of them, the form is true (look directly underneath the main connective: two Ts, no Fs). So no matter whether “p” stands for a true sentence or a false sentence, “p Ú ~p” is true. So “A Ú ~A” is a tautology.
[3.3.1.] Steps to Follow in Constructing Complete Truth Tables.
In this section, I’m following the steps described in your textbook (pp.66-68) but using a different example:[2]
~[(H É ~B) Ú (~B É E)]
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 (H, B and E), so we need eight rows.
Step 2: Begin constructing the table, starting with the left-most columns (the “guide columns”) headed by the sentence variables, and the right column headed by the one-one logical form of the sentence.
Fill in the columns moving from the inside out:
§ in the innermost 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
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Step 3: Begin filling in the table underneath the sentence. Start by filling in truth-values directly underneath each atomic sentence.
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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 variables.
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Continue with other compound sentences in order of increasing scope:
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The final column to be completed will be the column underneath the main connective:
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Since this sentence is false on every possible assignment of truth values, it is a contradiction.
Exercise 3-6 (p.68)
§ 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]
[3.4.] “Logical Equivalences.”
In this section we will construct truth tables that specify the truth values of more than one compound sentence at a time. The task will be to determine whether two sentences are logically equivalent:
logically equivalent (df.) two sentences are logically equivalent if and only if it is impossible for them to have different truth values.
For example, "A · B" and "B · A" are logically equivalent. If one is true, so is the other; and if one is false, so is the other.
We can determine whether two sentences are logically equivalent using the tabular truth table method: two sentences are logically equivalent if and only if their one-one logical forms have the same truth values on every line of a truth table.
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EXAMPLES: p.69
Exercise 3-7 (p.70)
§ [do one or two even-numbered in class]
§ complete all of these for next time; check your even answers and we’ll look at the odds at the beginning of the next class [Watch out for #7, it’s especially tricky]
[3.5.] “Truth Table Test of Validity.”
We will now apply the tabular truth table method to determine whether arguments are valid or invalid.
Recall these definitions from the first week of class:
argument (df.): a set of statements some of which (the premises) are intended to serve as evidence or reasons for thinking that another statement (the conclusion) is true. In the words of the textbook: “A series of sentences, one of which (the conclusion) is claimed to be supported by the others (the premises).” (p.16)
validity (df.): A valid argument is one in which
1. the truth of the premises would guarantee the truth of the conclusion;
2. it is impossible for the premises to be all true and the conclusion to be false at the same time;
3. if the premises were true, then the conclusion would have to be true as well.
(These are three equivalent ways of defining validity. They all mean the same thing.)
Applied to arguments in sentential logic, this definition of validity means that an argument is valid if and only if there is no row of the truth table on which all its premises are true and its conclusion false.
So, we can tell whether an argument in sentential logic, such as
A, ~A Ú B /\B
is valid or invalid by constructing a truth table (for the one-one logical forms of each of the sentences in the argument) and checking for a row on which all of the premises are true and the conclusion is false:
§ if there is such a row, then it is possible for the premises to be all true and the conclusion false at the same time, and the argument is invalid
§ if there is not such a row, then it is not possible for the premises to be all true and the conclusion false at the same time, and the argument is valid.
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There is only one row in which both premises (p and ~p Ú q) are true. On that row (interpretation), the conclusion ("B") is also true. So this argument is valid.
Exercise 3-8 (pp.72-73)
§ complete this exercise for next time; check your answers to the even-numbered questions; we will review the odd-numbered problems at the beginning of class next time.
§ Make sure you indicate which rows show the argument to be valid or invalid
§ This is the most time-consuming set of problems you’ve been assigned up to this point
***This is the end of the material that will be covered on your first test.
Stopping point for Thursday January 26. For next time:
· exercises 3-3, 3-4, 3-5, 3-6, 3-7 and 3-8
· We will spend all of Tuesday’s class covering these exercises and reviewing for your first test, which is one week from today.