Appeals to reasons through a logical argument in a technical presentation usually focus on informal logic through natural language arguments. These will usually be based on formal logic; however, the short period of time available to the speaker usually prevents detailed arguments and requires the speaker to interpolate certain (hopefully obvious) points, make assumptions, and use less-than-precise defintions.
Unfortunately, it is these interpolations, assumptions and imprecisions which lead to logical errors. These errors in logic have been observed since the time of the ancient Greeks and the most common errors have been categorized and listed fallacies.
To disucss logic, we must look at both formal and informal logic. Presented is an interpretation which the author has formed over time; however, it is by no means definitive.
Formal logic is associated with symbols
which may take on value of
either true/T or false/F and usually one unary and four binary
operations: not (
), and (
), or (
), implies (
), and equivalent to (
); though more
may be defined.
| T | F |
| F | T |
| T | T | T | T | T | T |
| T | F | F | T | F | F |
| F | F | F | F | T | T |
| F | T | F | T | T | F |
An expression is any expression containing symbols combined with logical operators. Certain expressions are always true regardless of the value of any of the symbols; such expressions are called tautologies. Examples of tautologies include De Morgan's laws:
Therefore, showing one side is equivalent to demonstrating the other.
Another tautology is:
In this case, if it can be shown that the left-hand side is true, it follows that the right-hand side must also be true.
Another common tautology is
and if it can be shown that either side is true or false, the other side must have the same value.
An example of a statement which may initially appear to be a tautology but which is not is
This is the fallacy of denying the antecedent.
An example of a use of this falacy would be to say "it is raining implies that there are clouds in the sky is true; consequently it is not raining imples there are no clouds in the sky". By assuming this logical statement to be a tautology, the speaker would be committing a formal fallacy: there is an error in the formal logic.
A similar fallacy is the expression
This is known as the fallacy of affirming the consequent.
Another example of a statement which may initially appear to be a tautology but which is not is
There are values of the symbols for which the entire statement is false. The example given on Wikipedia is "fish are not dogs and dogs cannot fly; therefore fish can fly".
A common fallacy is to use
Again, this logical expression is not true for all values of of the symbols.
This author will define informal logic as the application of statements which could be called real-world statements, the truth of which may be brought to question. For example, the statement "the sun is shining" is only true during certain hours of they day and when the cloud cover is less than a particular value. It also has many implied assumptions including "visible light from the sun is passing through the atmosphere where the majority of the photons reaching this point are coming directly from the sun". It may be argued that such a statement is still not precise enough and that further precision is needed; however, this would make communication impossible. Consequently, it is necessary to allow certain interpolations and imprecisions on the part of both the speaker and the audience.
This author will define a formal fallicy as a case where the individual statements are sufficiently well defined but where those statements are placed into a logical expression which is not a tautology.
For example, "if it is raining, there is a cloud in the sky; therefore if there are no clouds in the sky, it is not raining" is a valid argument. It is, however, incorrect to say, "if it is raining, there are clouds in the sky; therefore, if it is not raining, there are no clouds in the sky".
An error in informal logic is where a statement is not
necessarily true. Recall that
is a tautology; however, this can be abused:
we can all agree that a person with height 2 m is "tall", and yet
it could be argued that if k m is tall, then k m − 1 mm
is also "tall". This implies that 1.999 m is also "tall". Repeating
this weaker statement multiple times, it follows, therefore, that
everyone is tall.
A valid argument must have both a formal structure and have acceptably accurate informal components. The speaker must understand the appropriate level of precision for the audience: too precise and the the audience will become bored, too vague and the audience will become frustrated. The speaker could be correct, but too vague a statement will be noted by audience members.
Since the time of Aristotle, fallacies in informal logic have been studied and categorized according to the most common. The speaker in preparing his or her presentation is almost certainly aware of logical fallacies and it is a mistake to believe that the audience will not recognize these mistakes. In some cases, it may be overlooked as a minor oversight; however, in other cases, the use of an inappropriate fallacy at an opportune time will be seen as deliberate and manipulative by the audience.
A generalization which ignores relevant exceptions is said to be a fallacy of accident.
Alternatively, using an exceptional case to state a general rule is said to be the converse fallacy of accident.
A line of reasoning which presents a conclusion which does not address the actual question at hand is an irrelevant conclusion and may also be termed a red herring. Specific cases include appeals to a person, to the majority, to fear, or to authority.
Examples of these include
Implication is a tool of formal logic: if A is true then B must follow (A → B). For example, if it is raining, then there are clouds in the sky. To affirm the consequent is to reverse the implication: if B is true then A must follow. Again, for example, one may present the previous implication and then state: since there are clouds in the sky, it must be raining.
The logical negation of the implication if A is true then B must follow is if B is false then A must also be false (¬(A → B) ≡ ¬B → ¬A.) For example, if there are no clouds in the sky, it must not be raining. To deny the antecedent is to use the argument if A is false, then B must be false. For example, because it is not raining, there are no clouds in the sky.
To beg the question is to show that a line of reasoning is valid by making .....
Using symbolic logic may be very rigorous, but it is very difficult to follow and the use of such logic would lead to a very tedious and boring presentation. Symbolic logic makes use of statements such as A or B, A implies B, A and B, and not A. These are represented with the operations A ∨ B, A → B, A ∧ B, and ¬A. The following are two very different examples of symbolic logic.
The following is an example of symbolic logic taken from the book Real-Time Systems: Scheduling, Analysis, and Verification by Albert Cheng.
Given the statements:
then is possible to prove that if the air conditioning is not on and the heater is not on then the temperature is comfortable ((¬A ∧ ¬G) → OK).
This can now be written as:
(H → A) ∧ (C → G) ∧ ((¬H &and ¬C) → OK)
≡ (¬H ∨ A) ∧ (¬C ∨ G) ∧ (¬(¬H ∧ ¬C) ∨ OK)
≡ (¬H ∨ A) ∧ (¬C ∨ G) ∧ (H ∨ C ∨ OK)
≡ (¬H ∨ A) ∧ (G ∨ H ∨ OK) (resolution)
≡ A ∨ G ∨ OK (resolution)
≡ (A ∨ G) ∨ OK
≡ ¬(A ∨ G) → OK
≡ (¬A ∧ ¬G) → OK
To present such a formal proof in a techncial presentation would either loose or bore most of the audience and, in all likelihood, would go no further to achieving the objective.
Another interesting example of symbolic logic comes from the book Symbolic Logic by Lewis Carroll (cf. Alice in Wonderland). Consider the following statements:
Reading these four statements, what can you deduce about the romances in this library? First, we may rewrite the statements as:
Using the relationship (A → B) ≡ (¬A ∨ B) and resolution (i.e., ((A ∨ B) ∧ (¬A ∨ C)) ≡ (B ∨ C)), we may deduce that:
(¬Recommend → ¬Healthy) ∧
(Bound → Well-Written) ∧
(Romances → Healthy) ∧
(¬Bound → ¬Recommend)
≡
(Recommend ∨ ¬Healthy) ∧
(¬Bound ∨ Well-Written) ∧
(¬Romances ∨ Healthy) ∧
(Bound ∨ ¬Recommend)
≡
(Recommend ∨ ¬Romances) ∧
(¬Bound ∨ Well-Written) ∧
(Bound ∨ ¬Recommend)
≡
(Recommend ∨ ¬Romances) ∧
(¬Recommend ∨ Well-Written)
≡
Well-Written ∨ ¬Romances
≡
¬Romances ∨ Well-Written
≡
Romances → Well-Written
Therefore, all the romances in this library are well-written. While this statement is obvious once the logical steps were performed, it is not not the line of reasoning which would normally be used to get to this conclusion.
Alternatively, one could rewrite and reorder the expressions as:
In this way, it is a little easier to see that all romances in this library are well-written.
To conclude, informal logic is necessary for technical presentations and for most human interactions, but by resorting to informal logic, people subject themselves to inadvertant errors due to oft-applied logical fallacies. It is the responsibility of the presenter to be aware of these fallacies and to avoid them. Using a logical fallacy to prove a point will render the presentation useless.