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Logical Sentences and Logical Arguments

Date: 05/16/2008 at 11:30:40
From: Anne
Subject: What is the difference between infers and implies

What is the difference between A |- B and A -> B?  They seem to mean
the same thing to me; if you know that A is true then you know that B
is also true.

If you have A->B->C then you can say A->C (transitivity I think it's
called).

If you write A|-B->C ... this is where I get confused.  It seems to
say the same as A->B->C to me, but there must be a difference else why
the two different symbols?

I tried Googling this, but all you get is the difference in common
speech, not the difference in terms of mathematical logic.



Date: 05/16/2008 at 16:37:45
From: Doctor Achilles
Subject: Re: What is the difference between infers and implies

Hi Anne,

Thanks for writing to Dr. Math.  Great question!  This is a very
common source of confusion.

I will try to introduce all the terms I use, but if you want a
complete list of terms, you can refer to:

  A Crash Course in Symbolic Logic
    http://www.mathforum.com/dr.math/faq/symbolic_logic.html 

(A -> B) is a logical sentence.  It roughly translates to the English
sentence, "If A, then B", although a more precise translation may be:

  "A is not true at the same time that B is false"

Logical sentences can be true or false.  The sentence (A -> B) is true
unless the subsentence A is true and the subsentence B is false.

If, instead, I were to write:

  A |- B

The best English translation I have for that is:

  "I have a logical proof that B can be deduced from A"

Strictly speaking, in logic something of the form:

  A |- B

is not classified as a "sentence".  I would call it an "argument", but
other logicians may have other terms for it.

Unlike logical sentences, logical arguments of the form:

  A |- B

are not "true" or "false", but rather "valid" or "invalid".

Before I go on, I need to define a few terms as I use them.

In general, I use the capital letters: A, B, C, D to denote specific
logical sentences.  They denote specific propositions.

For example:

  A = "My dog is white"
  B = "The sky is cloudy"

Therefore, the sentence:

  (A -> B)

is true as long as either:

  the subsentence "My dog is white" is false

or

  the subsentence "The sky is cloudy" is true

But, if I were to dye my dog's fur white on a cloudless day, then the
sentence (A -> B) would become false.

Thus a logical sentence, such as:

  (A -> B)

as discussed above is true under some conditions, and false under
other conditions.

By contrast, the logical argument:

  A |- B

is *always* invalid.

The reason is that there is nothing about the arbitrary sentences:

  "my dog is white"
  "the sky is cloudy"

that makes it logically necessary that B always follows from A.

It doesn't matter that sometimes the sentence (A -> B) is true.  The
fact that there is no logical necessity for B to follow from A makes
the argument invalid.  No logical proof is possible.

However, we can make another more complicated sentence:

  (A ^ (A -> B))

This sentence, roughly translated to English is, "A, and if A then B", or:

  "My dog is white, and if my dog is white then the sky is cloudy"

The complicated sentence (A ^ (A -> B)) is also true under some
conditions and false under others.  It turns out that it is false if
A is false or if B is false, but it is true if both A and B are true.

The logical argument:

  (A ^ (A -> B)) |- B

is a valid argument.  Remember, roughly speaking, it says:

  "I have a logical proof that B can be deduced from (A ^ (A -> B))"

It is possible to generate such a proof, therefore:

  (A ^ (A -> B)) |- B

is a valid argument.

In general a valid argument is one where the structure of the first
sentence makes it necessary that the second sentence must be true if
the first sentence is assumed to be true.

Let me introduce two other logical terms.

  1) A "tautology" is a logical sentence that is always true.  The
     most commonly cited tautology is the sentence (A -> A).

  2) A "contradiction" is a logical sentence that is always false.
     The most common example of a contradiction is (A ^ ~A).

Another example of a tautology is the complicated sentence:

  ((A ^ (A -> B)) -> B)

This sentence turns out to always be true.

It is very common to confuse the tautological sentence:

  ((A ^ (A -> B)) -> B)

With the valid logical argument:

  (A ^ (A -> B) |- B


But there are at least three important distinctions.

First, a sentence can have the connective "->" as part of a 
subsentence.  For example:

  (A -> B)

is a subsentence of the more complicated sentence:

  (A ^ (A -> B))

However, you can NEVER put the symbol "|-" within a larger logical
statement.  So if you were to write:

  (A ^ (A |- B))

That would be totally gobbledygook.  It makes as much sense to a
logician as if you were to write:

  &*k23*^tt++===BBB@

Second, while some sentences, such as:

  (A -> A)

  ((A ^ (A -> B)) -> B)

and

  (A ^ ~A)

always are true or always are false, many logical sentences, such as:

  B

  ~A

  (A v B)

  (A -> B)

and

  (A ^ (A -> B)

are SOMETIMES true and SOMETIMES false.

However, for logical arguments such as:

  A |- B

  A |- A

  (A -> B) |- B

and

  (A ^ (A -> B)) |- B

there is NO SOMETIMES ABOUT IT.  They are either ALWAYS valid or
ALWAYS invalid.


Third, you can NEVER have the symbol "|-" occur more than once in a
logical argument.  So if you were to write:

  A |- B |- A

it would be total gobbledygook.

However, the symbol "->" can occur many times in a logical sentence.  So:

  (A -> (B -> A))

is a perfectly acceptable logical sentence that happens to also be a
tautology.  Although (as a side note) the similar looking sentence:

  ((A -> B) -> A)

is an acceptable logical sentence, but it happens to be true under
some conditions and false under others.

Even contradictions, such as:

  (A ^ ~A)

are still acceptable logical sentences, they just happen to always be
false.  But they still are sensible and don't count as gobbledygook.


So why are the connective "->" and the symbol "|-" so often confused?

The confusion, I think, comes from the fact that the logical argument:

  something |- somethingElse

if and only if the logical sentence:

  (something -> somethingElse)

is a tautology.

So, for example:

  A |- A

Is a valid logical argument and:

  (A -> A)

is a tautology.

But even though these two expressions are related, they are
fundamentally different types of logical expressions.

The sentence:

  (A -> A)

happens to be a tautology, but it is first and foremost a *logical
sentence* and therefore it can be part of any number of logical
expressions.  In other words, the sentence:

  (A -> A)

Can be a subsentence of the larger tautology:

  (B -> (A -> A))

or it can be a subsentence of the larger contradiction:

  (~(A -> A) -> (B -> B))

or it can be a subsentence of the larger (sometimes true and sometimes
false) sentence:

  ((A -> A) -> B)

or it can even be part of logical arguments, such as:

  (A -> A) |- B

  ~(B -> B) |- (A -> A)

or

  B |- (A -> A)

Two of which are ALWAYS invalid and one of which is ALWAYS valid.

In contrast, the logical argument:

  A |- A

While it is always a VALID argument, like all other arguments (valid
or invalid) it MUST ALWAYS stand alone.  If we try to include it in
other logical expressions, such as:

  A |- A |- (B -> B)

  A |- A |- B

or

  (A |- A) -> A

we have written total gobbledygook.


One final point.  Sentences come in three different types:

  1) Sentences like (A -> B) are "sometimes true and sometimes false".

  2) Sentences like (A -> A) are "tautologies" (always true).

  3) Sentences like (A ^ ~A) are "contradictions" (always false).

However, logical arguments only come in TWO different types.

  1) Arguments such as A |- A are VALID.

  2) Arguments such as A |- B are INVALID.

Notice that the argument:

  ~(B -> B) |- (A -> A)

is also INVALID.

Even though the sentence:

  (~(B -> B) -> (A -> A))

is a contradiction, there is no distinction made between one invalid
argument:

  A |- B

and another:

  ~(B -> B) |- (A -> A)

They are both considered equally invalid.

I hope this helps.  If you have other questions or you'd like to talk
about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 05/17/2008 at 06:29:24
From: Anne
Subject: Thank you (What is the difference between infers and implies)

Thanks so much.  This and the link you gave me have really helped me
get to grips with all this, and make much more sense than the book I
was using!
Associated Topics:
College Logic
High School Logic

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