Reasoning

Deductive reasoning

Deductive reasoning is a type of logical thinking in which a conclusion follows with certainty from premises or assumptions that are assumed to be true. This method of reasoning starts with a general statement, which is then used to derive a more specific conclusion.

Deductive reasoning can be analyzed by the repeated application of two strong syllogisms:

$$\begin{gather} \text{if A is true, then B is true}\\[2ex] \frac{\text{A is true}}{\text{therefore, B is true,}} \tag{1.1} \end{gather}$$

and its inverse:

$$\begin{gather} \text{if A is true, then B is true}\\[2ex] \frac{\text{B is false}}{\text{therefore, A is false.}} \tag{1.2} \end{gather}$$

For example,

$$\begin{align} A &\equiv \text{an animal is a bird}\\ B &\equiv \text{it has feathers} \end{align}$$

Premise 1: If an animal is a bird (A), then it has feathers (B).
Premise 2: A penguin is a bird (A).
Conclusion: Therefore, a penguin has feathers (B).

In this example, the conclusion is derived from the premises using logical steps that are based on a set of rules or principles. Deductive reasoning aims to draw logical conclusions that are necessarily true, provided that the premises are true.

We would like to reason in terms of the above syllogisms, but in almost all situations we do not have the right kind of information to allow this kind of reasoning.

Plausible reasoning

Inductive or plausible reasoning is a type of logical thinking that involves using specific observations or examples to draw a general conclusion. The conclusion follows not with certainty, but only with some likelihood.

In plausible reasoning, a conclusion is drawn based on a set of specific observations or data points, rather than starting with a general principle or premise. The goal is to identify patterns or trends in the data that can be used to make generalizations or predictions about future events or situations.

Consider a weaker syllogism of plausible reasoning:

$$\begin{gather} \text{if A is true, then B is true}\\[2ex] \frac{\text{B is true}}{\text{therefore, A becomes more plausible.}} \tag{1.3} \end{gather}$$

For example,

$$\begin{align} A &\equiv \text{it is raining}\\ B &\equiv \text{there are clouds in the sky} \end{align}$$

The evidence does not prove that A is true, but verification of its logical consequence gives us more confidence in A.

Another weak syllogism that uses the same major premise is

$$\begin{gather} \text{if A is true, then B is true}\\[2ex] \frac{\text{A is false}}{\text{therefore, B becomes less plausible.}} \tag{1.4} \end{gather}$$

For example,

$$\begin{align} A &\equiv \text{it is raining}\\ B &\equiv \text{the ground is wet} \end{align}$$

The evidence does not prove that B is false. However, one of the possible reasons for its being true has been eliminated, therefore we feel less confident about B.

Still weaker syllogism is given by

$$\begin{gather} \text{if A is true, then B becomes more plausible}\\[2ex] \frac{\text{B is true}}{\text{therefore, A becomes more plausible.}} \tag{1.5} \end{gather}$$

For example,

$$\begin{align} A &\equiv \text{there is smoke}\\ B &\equiv \text{there is a fire nearby} \end{align}$$

The presence of smoke often indicates the presence of fire, but it does not guarantee it. The conclusion that the presence of smoke becomes more plausible is reasonable because fire typically produces smoke. However, other factors such as controlled burning, smoke machines, or other sources of smoke could also be present.

Our brain not only decides whether something is more or less plausible, but it also evaluates the degree of plausibility. It analyses current information and old information to make a decision. In our reasoning we depend on prior information to evaluate the degree of plausibility in a new problem. Common sense is a complicated reasoning which depends hugely on prior information.

We can go through a long chain of reasoning of the first two types and the conclusions will have just as much certainty as the premises. With the weaker syllogisms (1.3)-(1.5), conclusions change in reliability as we move through several stages. However, in quantitative form conclusions can still approach the certainty of deductive reasoning.

Mathematicians try hard to invent arguments of the first kinds when publishing a new theorem, but the process of devising a new theorem involves weaker forms (follow up conjectures suggested by analogies).