Reference Manual
Chapter Two:
Deductive Concepts and Sentence Logic

.pdf Version for Printing

This reference provides some of the basic points made in Chapter Two. But it doesn't include everything of importance! Please spend the time working through all the tutorials! Often details for working homework problems -- the only good preparation for exams! -- is available in the tutorials. Even if you can do this weeks homework without doing them all, there may be material in later units that will be very hard without a clear understanding of all that going on in the tutorials.

Contents
: Section 1: Formal Fallacies; Section 2: The Expressions of SL: Syntax; Section 3: Truth Conditions for the Sentences of SL: Semantics; Section 4: Deductive Concepts; Section 5:Symbolization from English to SL

 

1. Formal Fallacies

Most of the formal fallacies we discuss are concerned with conditional statements. So, we begin there.

Conditional Statements

We'll call these "if...then..." and "only if" statements conditional statements or just conditionals.

Take these two examples:

1. If there is fire, then there is oxygen.

and

2. There is fire only if there is oxygen.

We need to see that these are two equivalent ways of expressing a conditional statement. Let's work on that. We'll start with some definitions of form.

Let's put it this in our placeholders:

 Form 1:     only if

and

Form 2:     If then

The box is the antecedent and the oval the consequent. Here's how 1 and 2 fit these forms.

In these forms, the antecedent comes either after "if" or before "only if". The consequent comes after "then" or "only if".

 

 

Now, of course, if we switch antecedent and consequent. around, we get something altogether new:

3. If there is oxygen, then there is fire.

While 1 and 2 are true in any circumstance, 3 normally is not true.

Because 1 and 2 are true while 3 is not, we can see that they do not mean the same thing. This is a hint at how we will be clear about "mean the same thing".

We'll call them logically equivalent.

A sentence P is logically equivalent to sentence Q if and only if it is not possible for one of P and Q to be true while the other is false.

The two conditionals with which we began this section are logically equivalent:

 

1. If there is fire, then there is oxygen.

and

2. There is fire only if there is oxygen.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fallacies

 

A fallacy is a poor argument, or type of argument, that nonetheless may appear convincing. 

A formal fallacy is a fallacy because of its form. In all instances that we'll see, these will be invalid deductive arguments that may appear valid if one is not careful.

 

Here's an argument that has a problematic form:

There is fire only if there's oxygen. There is oxygen here in the room, so there is fire.   

we can symbolize this as:

     Fire/oxygen argument:

F only if O
O
So, F.

This argument is clearly invalid. Right now as I'm writing, the premises are true but there is no fire. That is, the conclusion is plain false. An argument that can have true premises and a false conclusion is invalid.

 

Affirming the Consequent: the fallacy AC

Any argument based on the form of the Fire/Oxygen argument is fallacious. Here's the problematic form and an illustration of how the argument fits this form:

only if


                         

 

Again, anything base on this form is fallacious formal reasoning. The Sanchez argument is a case in point:

     Sanchez/Bank argument:

Sanchez stays only if she gets a raise.
She does get a raise.
So, she stays.

 

Denying the antecedent: the fallacy dA

Any argument based on this form:

Paul is from Quebec only if he's from Canada, but Paul is not from Quebec, so he's not from Canada.

is fallacious. Think about it: this conclusion is not inescapable. The conclusion could be false if Paul lives in Windsor, Ontario even while the premises are true.

Again, anything base on this form is fallacious formal reasoning.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Here again are the forms:

 

the fallacy of affirming the consequent or AC.

 

the fallacy denying the antecedent or DA

 

2. The Expressions of SL: Syntax

SL Sentences | Further Syntactical Concepts | Relaxing the Definition

Like English, our language SL contains "connectives", symbols which attach to sentences forming new sentences. For instance, '&', the ampersand, is a connective which cements two sentences together to form a new one. So, because 'A' and 'B' count as sentences of SL, 'A&B' does too.

Because '&' connects a pair of sentences, so we call it a binary connective. All connectives of SL except one are binary. The one exception is the tilde, '~'. It attaches to a single sentence. For example, '~A'. We call this a unary connective.

A synopsis of the first tutorial is presented in the following table. The connectives are listed in the column on the left.

 

  Connective Name Resulting Sentence Type Component Names Typical English Versions English Statement Symbolization in SL
& Ampersand Conjunction Conjuncts "and", "both ... and ... " Agnes and Bob will attend law school. A&B
> Horseshoe Conditional Antecedent, Consequent "if ... then ... " If Agnes attends, then Bob will. A>B
~ Tilde Negation Negate "it's not the case that", "not" Agnes will not attend law school. ~A
v Wedge Disjunction Disjuncts "or", "either... or... " Either Agnes or Bob will attend law school. AvB
= Triple Bar Biconditional Bicomponets1 "if and only if", "just in case" Agnes will attend law school just in case Bob will. A=B

 

SL Sentences

Our most important task is to say just what objects count as sentences of our language SL. We do this by showing how sentences are constructed. Let's start with SL without names, then add them later.

We will call the simplest sentences of our language the atomic sentences of SL. These building blocks include 'A','B','C','D'..., 'Z' (except we won't use the 'V' to avoid confusion with the wedge 'v').

The atomic sentences, as their name implies, are used as the basic building blocks for SL sentences. You use them to construct longer sentences, molecular sentences, like the following four sentences:

a. (A&B)
b. ~L 
c. (~L>C)
d. ((A&B)=(~L>C))

(Notice that we do not yet drop outside parentheses.) The rough idea behind the definition of an SL sentence can be put as follows:

One can construct a molecular sentence of SL either by taking any one sentence which is already constructed and adding a tilde on its left or by taking any pair of already constructed sentences, writing a binary connective between them, and surrounding the result with parentheses. (Atomic sentences count as "already constructed".)

We may now more precisely define just what counts as a sentence of SL. We give what is sometimes called an "inductive" or "recursive" definition. But all this means is that we define what counts as a sentence by showing how to construct one from basic parts.

Here's the definition. First,

i) All atomic sentences count as sentences of SL.

Second we say how we can build more complex sentences from any sentences which have already been built.

ii) If P is any sentence of SL, then so is ~P.
iii) If P and Q are any two sentences of SL, then '(P>Q)', '(P&Q)', '(PvQ)', '(P=Q)' are also sentences of SL.

When you construct a sentence by these rules, you may apply both of these clauses again and again. For example, think about following these rules to construct the sentence, (*):

(*)    ((A&B)>~C)

To construct this sentence following these rules, you would first apply the first clause i), noting that 'A' and 'B' are sentences of SL because they are atomic sentences. Then, think of setting P='A' and Q='B' so that clause iii) says that '(A&B)' is a sentence of SL. Also, 'C' is an atomic sentence, so is a sentence of SL by i). Thus by ii) '~C' is a sentence of SL. Finally, we can put the two sentences just highlighted together by yet another application of iii) to form '((A&B)>~C)' as desired.

 

Further Syntactical Concepts

The sentence (*) is an "if...then..." sentence, a conditional. In other words, (*) means that

if A and B are both true, then C is not.

Or to put the point in English, these sentences could express something like

If Agnes and Bob will both attend law school, then Carola will not.

Because this expresses a conditional, we will say that (*)'s main connective is the horseshoe. It's made up of antecedent '(A&B)' and consequent '~C' which we will call the two "immediate components". Its most basic sentential components we'll call its "atomic components" which in (*)'s case are 'A', 'B', and 'C'.

Now we need to carefully define these terms.

The sentential components of a sentence of SL are all components used in the building process in order to construct that sentence.

The atomic components of a sentence of SL are all atomic sentences used to construct that sentence.

The main connective of a sentence of SL is the last occurrence of a connective used to construct it.

The immediate component or components of a sentence of SL is (are) the sentential component(s) used in the final stage of its construction.

For example, to build (*) one first builds '(A&B)' and '~C', then finally puts these together with the horseshoe. Thus our definitions lead to the conclusion that (*)'s main connective is indeed the horseshoe and its antecedent and consequent are the immediate components.

 

Relaxing the Definition of a Sentence

To save just a little time and to make our SL constructions easier to read we allow two alterations to the above definition:

  1. Outside parentheses may be dropped.
  2. Brackets, "[" and "]" may be used in place of the left and right parentheses respectively.

 

 

 

3. Truth Conditions for the Sentences of SL: Semantics

Truth conditions are usually given by way of tables. The following table summarizes the five separate tables given in tutorial 2. Like the tables we saw in the tutorial, read these from left to right along any one row. For example, in row one we are thinking about the possibility in which both P and Q are true: in that case only ~P is false.

  P Q P&Q PvQ P>Q P=Q ~P
row one: T T T T T T F
row two: T F F T F F F
row three: F T F T T F T
row four: F F F F T T T

Notice that the column under ~P looks a little different from what we say in tutorial 2. What you saw there was:

  P ~P
row one: T F
row two: F T

You should take a moment to see that there is no difference. Both say that '~P' is false exactly when 'P' is true.

You will need to memorize the table for all five connectives; you should do so immediately! To help, keep the following in mind. Conjunctions are true in only one row. Disjunctions and conditionals are false in only one row. The biconditional works a lot like the equal sign it resembles! Finally, the negation simply "reverses" truth values.

 

4. Deductive Concepts

The main point is tutorial 2.4 is that much of deductive logic has to do with what is possible and what is not. But "possible" means many things to people. So, let's be clear.

What is logically possible is what is allowed by language.

So, this notion of possibility is relative to a given language. At this point we should be thinking about logical possibility in English or in SL. So, it's logically possible that George W. Bush is a professor  (or that 'Pg' be true). Our language allows this possibility.

Still, in a different sense, it's also true to say that it's impossible that George W. Bush (now in 2006) be a professor. We know he's now U.S. president and not teaching. Just keep in mind that this sort of possibility is different from logical possibility. It's not language that excludes George W. Bush from that wonderful profession of higher learning! It's that he has another job.

So, keeping in mind that when we say "possible" or "impossible" we mean what language allows or disallows; notice how much of deductive logic can be defined in terms of these notions. Look first to arguments.

An argument is valid just in case it is not possible that its conclusion be false while its premises are all true.

This is just our old definition. Keep two things in mind:

  1. This definition of validity applies only to arguments. We'll not say that a single sentence is valid. Be careful as each of the definitions in this section applies only to a specific type of linguistic object or objects.
  2. Because possibility is logical possibility, this definition means that the language won't allow the possibility that a valid argument's conclusion be false while the premises are true.

 

Next, think about a single sentence. The next three definitions apply only to a sentence in isolation.

A sentence P is logically true if and only if it is not possible that P be false.

A sentence P is self contradictory  if and only if it is not possible that P be true.

A sentence P is contingent if and only if P is neither logically true nor self-contradictory.

 

But we can also compare one sentence with another. The next definition applies to a pair of sentences.

Two sentences P and Q are logically equivalent if and only if it is not possible for one of P and Q to be true while the other is false.

 

Finally, one can consider a set or collection of sentences.

A collection of statements is logically inconsistent if and only if it is not possible for all these statements to be true together.

The collection is logically consistent if and only if it is possible for all these statements to be true together.

We'll see more such definitions in chapter 4. But this gives you the idea.

5. Symbolization from English to SL

In the tutorials, you will find many examples of "symbolization", i.e., of translating a compound English sentence into a molecular sentence of SL with roughly the same meaning.

(**) For example, anything in English of the form "P only if Q" may be symbolized as 'P>Q'. Or equivalently, it may be symbolized as '~Q>~P'.

What does this mean exactly? First, the hybrid form, "P only if Q", is shorthand for an English sentence "___ only if ___" where the blanks can be filled in by sentences symbolized by some SL sentences P and Q respectively. For example,

(***) There is fire only if oxygen is present

or "F only if O". This is logically equivalent to either "If there is fire then oxygen is present" or "If there is no oxygen, then there is no fire". So, you may symbolize (***) as either

F>O

or

~O>~F

This is all (***) comes to. But make sure you understand it before looking at the following review of the many examples discussed in the tutorial.

 

Equivalent English Forms (Each table element -- i.e., box -- below gives English forms instances of each of which can be symbolized by a sentence of any SL form on its right. Please note that there are many more English forms than can be covered below.) Equivalent SL Forms (Each table element below gives SL sentence-forms to guide in translating English sentences of forms found on the left. Please note that this is an incomplete list of possible symbolizations.) Example Applications (Each of the table elements below shows a way to apply the table elements on their left.)

If P, then Q.
If P, Q.
Provided P, Q.
Were P to hold, Q would be true.
Should P be true, Q.
P only if Q.
P is a sufficient condition for Q.
P implies Q.

P>Q
~Q>~P
If there is fire, then there is Oxygen" or "There is fire only if there is oxygen" may both be symbolized as 'F>O' or equivalently as '~O>~F'.
P if Q.
P provided Q.
P is a necessary condition for Q.
Q>P
~P>~Q
"Water is a necessary condition for life" or "there's water if there's life" may both be symbolized as 'L>W' or equivalently as ~W>~L'.
P if and only if Q.
P just in case Q.
P is necessary and sufficient for Q.
P=Q
(P>Q)&(Q>P)
"An argument is sound if and only if it is both valid and has true premises" may be symbolized as either 'S=(V&T)' or '[S>(V&T)] &[(V&T)>S]

Both P and Q.
P and Q.
P but Q.
Q and P.
Q but P.
P however Q.
P although Q.
P moreover Q.

P&Q
Q&P
"Sandra is both brave and careful", "Sandra is brave, moreover she is careful' or "Sandra is brave but careful" can all be symbolized as 'B&C' or 'C&B'.
Either P or Q.
Either Q or P.
P or Q.
Q or P.
At least one of P, Q.

PvQ
QvP
"Either the other team will score and tie up the game, or we win!" can be symbolized as '(S&T)vW'.
P unless Q.
Q unless P.
Unless P, Q.
Unless Q, P.
PvQ
~Q>P
~P>Q
"We win unless the other team scores" can be symbolized as 'WvS'.
Neither P nor Q.
Not-P and not-Q.
~(PvQ)
~P&~Q
"They neither scored nor tied the game" may be symbolized as either '~(SvT)' or '~S&~T'.
It's not the case that both P and Q.
Not both P and Q.
Either not-P or not-Q.
~(P&Q)
~Pv~Q
"Sandra is not both brave and careful" may be symbolized as either '~(B&C)' or '~Bv~C'.

Negation

There are lots of ways to indicate negation. Words like "not", "it's not the case that" are obvious examples. The prefixes "un" or "not" are sometimes indicative of negation. See the tutorial for more... .

Punctuation

We use parentheses in SL to group sentential components and so show the sentence's structure. English has many means to do this. Consider the following sentence of English:

The Russian president will be reelected if and only if either the opposition bows out or the gods intervene.

Is this symbolized as the following?

R=(BvI)

Yes, this is correct. But how do we know the parentheses group 'BvI'? Why not

(R=B)vI

If the second translation were intended, then the English sentence in question would have begun with an "Either". This would have indicated that the main connective was the "or". Instead, the word "either" comes just before "the opposition bows out" and groups that with "the gods intervene" to form the disjunction. (Thus the word "either" works much like a left hand parenthesis.)

Other ways to indicate a main connective at the beginning of a sentence are to use "Both", "Either", or "If". Commas are also frequently used to set off two immediate components of the main connective. Examples? How about: "If one of the British and French win, then both the Americans and the Russians loose." This would be

(BvF)>(A&R)

All of these basic rules must be used together when you get to more complicated sentences. But practice first on the simpler cases.