Logical Arguments
Robert Howard Kroepel
Copright © 2004
Lakeside Studios
20 South Shore Road
New Durham, New Hampshire USA 03855
Logical Arguments
Logical arguments are formal structures of thought [required
orders
or sequences of thinking] by which verifiable, falsifiable and verified
Premises
lead to logical [valid and true] Conclusions.
In general, logical arguments consist of verifiable/falsifiable
Premises which must be relevant to Conclusions; if the Premises are
relevant to the Conclusions, and, vice versa, the Conclusions are
relevant to the Premises, then the arguments are valid (regardless of
whether or not the Conclusions are true), and the Conclusions are (T)
true (proven by the Premises) if the Premises are verified or (F) false
(not proven by the Premises) if the Premises are not verified or
otherwise are falsified.
Logical arguments, in their basic forms, are one of three types: (1) P
= Q = X, (2) If P then Q, or (3) P1 + P2 + ... + Pn = Q.
1. P = Q = X Logical Arguments
P = Q = X logical arguments have a form of —
1. A Premise #1 which asserts P = Q,
2. A Premise #2 which asserts X = P,
3. A Conclusion which asserts X = Q.
P = Q = X logical arguments assert that if P = Q, and if X = P, then X
= Q.
P = Q = X logical arguments are most often arranged in the following
form:
1. Premise #1: P = Q.
2. Premise #2: X = P.
3. Conclusion: X = Q.
1. Premise #1:

P = Q.

2. Premise #2:

X = P.

3. Conclusion:

X = Q.

Here is the famous philosophical example in which the philosopher
Socrates is proven to be mortal:
1. Premise #1: (P) All men are (Q) mortal. [P = Q]
2. Premise #2: (X) Socrates is a (P) man. [X = P]
3. Conclusion: (X) Socrates is (Q) mortal.[X = Q]
1. Premise #1:

(P) All men are (Q) mortal.


[P = Q]

2. Premise #2:

(X) Socrates is a (P) man.


[X = P]

3. Conclusion:

(X) Socrates is (Q) mortal.


[X = Q]

2. If P, Then Q Logical Arguments
The If P, then Q logical arguments have one of two forms:
1. If P, then Q: P: therefore Q.
2. If P, then Q: notQ: therefore notP.
2.1. If P, Then Q: P: Therefore Q Logical Arguments
The If P, then Q: P: therefore Q logical arguments have a form, or
sequence, consisting of a Premise #1: If P, then Q, a Premise #2: P,
and a Conclusion: Q.
If P, then Q: P: therefore Q logical arguments are predictions.
Predictions consist of conditions and consequences.
In a prediction, the condition is If P and the consequence is then
Q.
If P, then Q: P: therefore Q logical arguments most often have the
following form:
1. Premise #1: If (condition) P, then (consequence) Q. [If the
condition, ... then the consequence ... .]
2. Premise #2: (Condition) P. [The condition occurs.]
3. Conclusion: (Consequence) Q. [The consequence follows.]
1. Premise #1:

If (condition) P, then (consequence) Q.


[If the condition, ..., then the consequence ...
.]

2. Premise #2:

(Condition) P.


[The condition occurs.]

3. Conclusion:

(Consequence) Q.


[The consequence follows.]

Here is a famous philosophical example of an If P, then Q: P: therefore
Q logical argument in which a specific window (the window) will break
if
a specific rock (the rock) hits it.
1. Premise #1: If (P) the rock hits that window, then (Q) the window
will break. [If P, ..., then Q ... .]
2. Premise #2: (P) The rock hits the window. [P]
3. Conclusion: (Q) The window breaks. [Q]
1. Premise #1:

If (P) the rock hits the window, then (Q) the
window will break.


[If P, ..., then Q ... .]

2. Premise #2:

(P) The rock hits the window.


[P]

3. Conclusion:

(Q) The window breaks.


[Q]

If P, then Q: P: therefore Q logical arguments are valid only for
specific conditions and specific consequences. The If P/rock, then
Q/window example is valid only if a specific rock is described in the
condition (P) and a specific
window is described in the consequence (Q).
The If P, then Q prediction could be restated as if (P) this
rock hits this window, then this window will break. Another
rock might not
break this window; another window might not break if this rock hits it.
1. Premise #1:

If (P) this rock hits that window, then (Q) that
window will break.


[If P, ..., then Q ... .]

2. Premise #2:

(P) This rock hits that window.


[P]

3. Conclusion:

(Q) That window breaks.


[Q]

If P, then Q logical arguments can be generalized when the 'rocks' and
the 'windows' become generalized by their physical characteristics of
length, and weight (mass) :
1. Premise #1: If (P) these (types of) rocks hit those (types of)
windows, then (Q) those windows will break.
2. Premise #2: (P) These rocks hit those windows.
3. Conclusion: (Q) Those windows will break.
1. Premise #1:

If (P) these rocks hit those windows, then (Q)
those windows will break.


[If P, ..., then Q ... .]

2. Premise #2:

(P) These rocks hit those windows.


[P]

3. Conclusion:

(Q) Those windows break.


[Q]

If P, then Q: P: therefore Q logical arguments are descriptions of
causality—causeandeffect relationships, causal relationships. In
cause and effect relationships, causes cause effects. People, things
and events who or which are causes cause other people/things/events
who/which are effects.
The Fundamental Law of Physics/Logic
Charles Proteus Steinmetz.
Four Lectures on Relativity and Space
Dover Publications, Inc., 180 Varick Street, New York, NY 10014 1967
pp. 49–50.
The fundamental law of physics is the law of inertia. "A
body keeps the same state as long as there is no cause to change its
state." That is, it remains at rest or continues the same kind of
motion—that is, motion with the same velocity in the same
direction—until some cause changes it, and such cause we call a
'force.' " [Quotes in the original, but not attributed to anyone.]
This is really not merely a law of physics, but it is the fundamental
law of logic. It is the law of cause and effect: "Any effect must have
a cause, and without cause there can be no effect." This is axiomatic
and is the fundamental conception of all knowledge, because all
knowledge consists in finding the cause of some effect or the effect of
some cause, and therefore must presuppose that every effect has some
cause, and inversely. [Quotes in the original but
not attributed to anyone.]
The condition, P, describes the cause, and the consequence, Q,
describes the effect.
1 Premise #1: If (condition/cause) P, then (consequence/effect)
Q.
[If the condition/cause happens, ... then the consequence/effect
follows.]
2. Premise #2: (Condition/Cause) P.
[The condition/cause happens.]
3. Conclusion: (Consequence/Effect) Q.
[The consequence/effect follows.]
1 Premise #1:

If (condition/cause) P, then
(consequence/effect) Q.


[If the condition/cause happens, ... then the
consequence/effect follows.]

2. Premise #2:

(Condition/Cause) P


[The condition/cause happens.]

3. Conclusion:

(Consequence/Effect) Q.


[The consequence/effect follows.]

When you are using an If P, then Q: P: therefore Q logical argument you
are attempting to describe causality: If the condition/cause happens,
then
the consequence/effect happens. The condition causes the consequence.
That
is, the condition which is a cause causes the consequence which is an
effect.
In the examples, if the condition/causes of the rock hitting the window
(P) happens, then the consequence/effect will be the breaking of the
window
(Q).
Understanding conditions as causes and consequences as effects is the
key to understanding If P, then Q: P: therefore Q logical arguments.
2.2. The If P, Then Q: NotQ: Therefore NotP
Logical Arguments
If P, then Q: notQ: therefore notP logical arguments have a form
of a Premise #1: If P, then Q, a Premise #2: NotQ, and a
Conclusion: NotP.
If P, then Q: notQ: therefore notP logical arguments are predictions.
Predictions consist of conditions and consequences.
In a prediction, the condition is If P and the consequence is then Q.
If P, then Q: notQ: therefore notP logical arguments most often have
the following form:
1. Premise #1: If (condition) P, then (consequence) Q.
[If the condition, ... then the consequence.]
2. Premise #2: NotQ.
[The consequence did not occur.]
3. Conclusion: NotP.
[The condition did not happen.]
1. Premise #1:

If (condition) P, then (consequence) Q.


[If the condition, ... then the consequence.]

2. Premise #2:

NotQ.


[The consequence did not occur.]

3. Conclusion:

NotP.


[The condition did not happen.]

Here the philosophical example in which a specific window (the window)
did not break because a specific rock (the rock) did not hit it.
1. Premise #1: If (P) the rock hits the window, then (Q) the window
will break.
[If P, ... then Q]
2. Premise #2: (NotQ) The window did not break.
[NotQ]
3. Conclusion: (NotP) The rock did not hit the window.
[NotP]
1. Premise #1:

If (P) the rock hits the window, then (Q) the
window will break.


[If the condition, ... then the consequence.]
[If
P, ... then Q]

2. Premise #2:

(NotQ) The window did not break.


[The consequence did not occur.] [NotQ]

3. Conclusion:

(NotP) The rock did not hit the window.


[The condition did not happen.] [NotP.]

If P, then Q: notQ: therefore notP logical arguments are valid only
for specific conditions and specific consequences. The If P/rock, then
Q/window example is valid only if a specific rock is described in the
condition (P) and a specific window is described in the consequence
(Q). The If P, then Q prediction for notQ could be restated as if (P) this
rock does not
hit that window, then that window will not break.
Another rock
might break this window; another window might break if this rock hits
it.
But if notQ, then notP means that window did not break, which means
this
rock did not hit it (or that window would have broken).
1. Premise #1:

If (P) this rock hits that window, then (Q) that
window will break.


[If the condition, ... then the consequence.]
[If
P, ... then Q]

2. Premise #2:

(NotQ) That window did not break.


[The consequence did not occur.] [NotQ]

3. Conclusion:

(NotP) This rock did not hit that window.


[The condition did not happen.] [NotP.] 
If P, then Q: notQ: therefore notP logical arguments are descriptions
of causality—causeandeffect relationships.
The condition, P, describes the cause, and the consequence, Q,
describes the effect.
1. Premise #1: If (condition/cause) P, then (consequence/effect) Q.
[If the condition/cause happens, ... then the consequence/effect
follows.]
2. Premise #2: (Consequence/Effect) NotQ.
[The consequence/effect did not occur.]
3. Conclusion: (Condition/Cause) NotP.
[The condition/cause did not happen.]
1. Premise #1:

If (P) the rock hits the window, then (Q) the
window will break.


[If the condition/cause, ... then the
consequence/effect.]
[If P, ... then Q]

2. Premise #2:

(NotQ) The window did not break.


[The consequence/effect did not occur.] [NotQ]

3. Conclusion:

(NotP) The rock did not hit the window.


[The condition/cause did not happen.] [NotP.] 
When you are using an If P, then Q: not P: therefore not Q logical
argument, you are attempting to describe causality: If the
condition/cause happens,
the consequence/effect happens. The condition causes the consequence.
Restated:
The condition which is a cause causes the consequence which is an
effect.
In the examples, if the condition/causes of the rock hitting the window
(P)
happens, then the consequence/effect will be the breaking of the window
(Q);
but where the condition/cause of the rock hitting the window (P) does
not
happen (not P), then the consequence/effect of the window breaking (Q)
does
not happen (not Q). Understanding conditions as causes and consequences
as
effects is the key to understanding If P, then Q: not P: therefore not
Q
logical arguments.
3. Pi + Pn = Q Logical Arguments
Pi + Pn = Q logical arguments consist of a string of Premises (P) which
lead to the Conclusion (Q).
Note:
P = Premise (Condition/Cause)
i = identification number of a Premise
n = the final number in a series of numbers, i.e., the identification
number of the last Premise
Q = Conclusion (Consequence/Result/Effect)
The Pi + Pn = Q logical arguments quite often consist of Premises which
are observable facts or the Conclusions of other logical arguments;
there is no limit to the number of Premises in Pi + Pn = Q logical
arguments, and it is possible for Pi + Pn = Q logical arguments to have
several Conclusions.
Premises and Conclusions in Logical
Arguments
All logical arguments must have the following:
1. Verifiable/Falsifiable/Verified Premises.
2. A Conclusion Logically Related to the Premises.
1. Verifiable/Falsifiable/Verified Premises.
The premises must be verifiable (provable), falsifiable (refutable) and
verified (proven) as true
before they can be used as premises in a logical argument. If the
premises
are not verified, then there is a logical fallacy of the begged
question
or unanswered question—a question that is begging to be asked and
answered:
Is this premise true?
Verification of premises in a logical argument is accomplished by proof.
What is proof?
Proof
Proof consists of one or more of the following:
1. Physical Evidence: People/Things/Events comprised of
matter/energy
who/which are observable with the five perceptual senses of
sight/hearing/touch/smell/taste
(A) directly, possibly with the use of machines including telescopes,
microscopes,
audio amplifiers, etc. which augment the perceptual senses or (B)
indirectly
by their observed effects on observable people/things/events.
2. Eyewitness Reports of physical evidence
(people/things/events comprised
of matter/energy) by credible eyewitnessespeople who are not known to
lie
or deceiveand corroborated by credible corroborators.
3. Logical Arguments in which the premises are
verifiable/falsifiable/verified
(by physical evidence/credible eyewitness reports of physical evidence)
and
are relevant to the conclusions which are (A) valid if relevant to the
premises
and (B) true if the premises are verified.
The begged or unanswered question actually has several parts:
A. Is this premise verifiable (or falsifiable)?
B. Has the premise been verified?
C. How has it been verified? Physical evidence? Eyewitness reports?
Logical
argument(s)?
Unverifiable/unfalsifiable/unverified premises are not acceptable in a
logical
argument because they will invalidate a conclusion.
2. A Conclusion Logically Related to the Premises.
The conclusion must be logically related to the subject and content of
the
premises, otherwise there is a logical fallacy of a shift of focus.
Example:
Premise #1: All (P) men are (Q) mortal. [Observation: Verified Fact]
Premise #2: (X) Socrates is a (P) man. [Observation: Verified Fact]
Conclusion: (X) Socrates is (Q) mortal. [Conclusion: Verified: Socrates
died—No
Shift of Focus]
Example:
Premise #1: All (P) men are (Q) mortal. [Observation: Verified Fact]
Premise #2: (X) Socrates is a (P) man. [Observation: Verified Fact]
Conclusion: (X) Socrates is (Y) smart. [Conclusion: Invalid: Shift of
Focus]
A logical argument which has a valid form and verified/true premises
has a valid conclusion and is called a “sound argument.”
A logical argument which has unverified/false premises has an invalid
conclusion
and is called an “unsound argument.”
If P, then Q logical arguments can be twisted if their sequence/form is
violated.
Here is an If P, then Q logical argument which is invalid because of a
twist in the form:
1. Premise #1: If P, then Q. [If (P) the rock hits the window, then (Q)
the window will break.]
2. Premise #2: Q. [(Q) The window breaks.]
3. Conclusion: P. [(P) The rock hits the window.]
The reason this If P, then Q logical argument is invalid is the
sequence of P’s and Q’s in the form. The sequence should be 1. If P,
then Q; 2. P; 3. Q, but, instead, the sequence is twisted and the form
is invalidated—1.
If P, then Q; 2. Q; 3. P. If (Q) this window breaks, it could have been
broken
by some thing/event (Y) other than being hit by this specific rock (P).
Here is an If P, then Q logical argument which has been invalidated by
a twist in the form:
1. Premise #1: If P, then Q. [If (P) the rock hits the window, ... then
(Q) the window will break.]
2. Premise #2: NotP. [(P) The rock did not hit the window.]
3. Conclusion: NotQ. [(Q) The window did not break.]
The reason this If P, then Q logical argument is invalid is the
sequence of P’s and Q’s in the form. The sequence should be 1. If P,
then Q; 2. NotQ; 3. NotP, but, instead, the sequence is twisted and
the form is invalidated—1. If P, then Q; 2. NotP; 3. NotQ. If (NotP)
this rock did not hit this window, then (NotQ) this window might not
have broken because it did not get hit by another rock.