### Logical Arguments

Robert Howard Kroepel
Lakeside Studios
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: not-Q: therefore not-P.

#### 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—cause-and-effect 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: Not-Q: Therefore Not-P Logical Arguments

If P, then Q: not-Q: therefore not-P logical arguments have a form of  a Premise #1: If P, then Q, a Premise #2: Not-Q, and a Conclusion: Not-P.

If P, then Q: not-Q: therefore not-P 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: not-Q: therefore not-P 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: Not-Q.
[The consequence did not occur.]
3. Conclusion: Not-P.
[The condition did not happen.]

 1. Premise #1: If (condition) P,  then (consequence) Q. [If the condition, ... then the consequence.] 2. Premise #2: Not-Q. [The consequence did not occur.] 3. Conclusion: Not-P. [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: (Not-Q) The window did not break.
[Not-Q]
3. Conclusion: (Not-P) The rock did not hit the window.
[Not-P]

 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: (Not-Q) The window did not break. [The consequence did not occur.] [Not-Q] 3. Conclusion: (Not-P) The rock did not hit the window. [The condition did not happen.] [Not-P.]

If P, then Q: not-Q: therefore not-P 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 not-Q 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 not-Q, then not-P 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: (Not-Q) That window did not break. [The consequence did not occur.] [Not-Q] 3. Conclusion: (Not-P) This rock did not hit that window. [The condition did not happen.] [Not-P.]

If P, then Q: not-Q: therefore not-P logical arguments are descriptions of causality—cause-and-effect 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) Not-Q.
[The consequence/effect did not occur.]
3. Conclusion: (Condition/Cause) Not-P.
[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: (Not-Q) The window did not break. [The consequence/effect did not occur.] [Not-Q] 3. Conclusion: (Not-P) The rock did not hit the window. [The condition/cause did not happen.] [Not-P.]

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 eyewitnesses--people who are not known to lie or deceive--and 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: Not-P. [(P) The rock did not hit the window.]
3. Conclusion: Not-Q. [(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. Not-Q; 3. Not-P, but, instead, the sequence is twisted and the form is invalidated—1. If P, then Q; 2. Not-P; 3. Not-Q. If (Not-P) this rock did not hit this window, then (Not-Q) this window might not have broken because it did not get hit by another rock.