PHIL 4160: Symbolic Logic
Dr. Robert Lane
Lecture Notes: Tuesday February 4, 2014

 

 

The material in these notes will not be covered on your first test; it is relevant to what youll eb required to do on your second test.

 

[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 Anne is an actress) and are either true or false.

 

On the other hand, sentence variables are place-holders for constants. We use them 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 notethese 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 q p q p 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  C, to get the sentence ~A  (B  C)

 

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

 

~[A (B  C)]  [~(B  D) 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  (q  p)

 

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

 

This is not: A  (B C)

 

(Its not, because the two ps in the sentence form have been replaced with different constants; the first has been replaced with A, and the second has been replaced with C.)

 

 

NOTE 2: Any compound sentence is a substitution instance of 2 or more sentence forms. E.g., A  B is a substitution instance of both of these forms:

  p q

  p

 

A  B is a substitution instance of p because you can produce A B by replacing the only variable in p with the well-formed sentence A  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.61-63)

Here youll 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 ~B) (C D)

 

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

 

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

 

~p

p q

p q

p q

p 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 ~B) (C D) is , that sentence is a substitution instance of p 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 sentences 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, ).

 

  1. 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 sentences expanded form:

(p ~q) (r s)

 

No sentence is a substitution instance of a form that contains more logical structure, or a different kind of logical structure, than the sentences expanded form. (p.62) The expanded form is a sort of limit to the complexity of the forms youre 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 sentences expanded form.

 

  1. 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 q

 

p stands for the conditional: A ~B

 

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

 

q stands for the biconditional: C D

 

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

 

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

(p q) (r 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 ~q) r

 

and of (p ~q) (r s)

 

which is the expanded form.

 

 

So our example: (A ~B) (C D)

is a substitution instance of each of the following:

 

p [its atomic form]

p q [its basic form]

(p q) r

p (q r)

(p ~q) r

(p q) (r s)

(p ~q) (r 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 more than one instance of the same variable, you must substitute EVERY instance of that variable with the same sentence.

 

For example, substitution instances of p ~p include:

 

A ~A

B ~B

(A B) ~(A B)

 

but not

 

A ~B

A ~(B  C)

 

 

Question: Which sentences are instances of p (p  q)?

 

1.      A (B C)

2.      A  (A  B)

3.      D (D  H)

4.      D (H  D)

 

Answer: Only sentences 2 and 3. Sentences 1 and 4 are not instances of p  (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  ~ A is a substitution instance of

 

p [atomic form]

p q [basic form]

p ~p [one-to-one logical form[1]]

p ~q [expanded form]

 

Another example:

 

A (A B) is a substitution instance of all of the following:

 

p [atomic form]

p q [basic form]

p (p q) [one-one logical form]

p (q r) [expanded form]

 

So, heres another question: Which sentences are instances of p (q  r)?

 

1.      A (B C)

2.      A  (A  B)

3.      D (D  H)

4.      D (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.65):

 

(A  B) ~B is a substitution instance of all of the following:

 

p [atomic form]

p q [basic form]

p ~q

(p q) ~q [one-one logical form]

(p q) r

(p q) ~r [expanded form]

 

 

[These points become relevant in exercise 3-5]

 

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

 

EXERCISE 3-5 (p.65)

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

 

 

 

Stopping point for Tuesday February 4.

 

For next time (Thursday February 6): your first exam! The material covered on this exam ends in the online notes for Thursday January 23. The material in todays notes will be covered on your second exam.

 

For Tuesday February 11:

         complete exercises 3-3, 3-4, and 3-5 we will cover at least SOME of the odds at the start of class;

         read ch.3 secs. 2-6.

 

 



[1] The one-one logical form of a sentence is 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.67) This notion will become very important soon, when we begin using truth tables to determine whether a given sentence is contingent, a contradiction, or a tautology.



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