Logic Class Lesson 3: Propositional Arguments

Propositions are the “material of our reasoning”, according to Copi and Cohen. A proposition asserts that some condition is or isn’t the case. It is a statement, a declarative sentence, an assertion. Simple propositions point to a single fact: Q is P. This asserts that Q is a subset of P. For example, the statement, “purple is a color”, asserts that it is the case (it is true) that the set of “colors” contains “purple”.

This is merely an assertion of truth, and whether it is actually true remains to be verified independently; propositions (assertions) may be challenged.

In logic, arguments are not disputes nor are they debates. An argument is a statement or series of statements that attempts to establish that something is so.

A proposition is not an argument by itself. Arguments contain a group of propositions, where one proposition is asserted as a conclusion based on another proposition which is presumed valid, or several propositions presumed to be valid, called premises. Here is a sample of an argument which states that because [X] and [Y] are true, this is adequate support for the contention that the conclusion, [Z] is true:

Premise 1: It is the case that [X]. (This is a proposition or statement).
Premise 2: It is the case that [Y]. (This is a proposition or statement).

Conclusion: Because [X] and [Y], then [Z].

Arguments are classified as either inductive or deductive (note 1).

Inductive Arguments
Inductive arguments are based on premises that tend toward a conclusion because the premises observe a common characteristic that belongs to that conclusion. A famous example of induction is the Swan argument:

Because every swan I have observed is white, then it is probable that all swans are white.

Or,

Because instances [1] through [n] of Q are P, then it is probable that all instances of Q are P.

This is an inference drawn from individual members, 1 – n, of a set Q, that the character of the entire set is represented by the characteristics of this subset of individual members, P (where P includes members 1 – n), with some probability of accuracy. From the Swan argument it can be seen that an inductive argument never reaches a probability of 1.0, until 100% of the members of the set have been observed. White swans were always observed – until black swans were discovered in Australia.

Inductive arguments usually go from specific examples toward general cases (or laws). This has been used in natural science. For example, frogs all have certain features in common, salamanders have other features in common. Those features which are observed to be in common to frogs define the set [frogs], which logically speaking is a genus, with bullfrog being a species contained within the genus. Taking frogs and salamanders together, [amphibians] would be the genus and [frog] would be the species. These logical designations of genus and species differ from biological use of the terms, and are intended to define a superset / subset relationship.


Deductive Arguments
When an argument’s conclusion cannot be false because its premises are undeniably true and are necessary and sufficient to support the truth of the conclusion, the argument is deductive. Another way to say this is that a deductive argument produces a conclusion that is impossible to be false, based on the truth of the premises. An example of a deductive argument:

If [P] is true, then [Q] is true;
[P] is true;
Therefore, [Q] is true.

If [all the marbles in the jar are green], then [this marble in the jar is green];
All the marbles in the jar are green;
Therefore, this marble in the jar is green.

Note that the set is defined as “true” in the second premise. The argument proceeds from a general case which is declared valid to a subset of the general case which cannot be non-valid… or can it?

If the swan situation is declared deductively, it would look like this:

If [all swans are white] then [the next swan will be white];
All swans are white;
The next swan will be white.

It becomes obvious that the statement of the general case can be false, despite the valid format of the deductive argument. In this case the general case is false because the general case was based on an inductive conclusion. Induction does not produce absolute validity (until 100% of specific members are tested), so the second premise – all swans are white – is not a true general case. So deduction also has limitations: it is only as valid as its general case is valid. This is a serious consideration, because how are general cases determined? Very frequently through induction.

Perhaps there are cases where the general case of an inductive argument cannot be argued against, its validity being so obvious that it is unassailable. That would be the case for the swan argument, if and only if 100% of the swans that exist had been observed. At that point a valid deductive argument can be made that “any swan is either black or white”.

It is also possible to “define” categories to be valid.

A deductive statement with two premises, where the premises are defined categories used by biologists to classify animals:

If [frogs are amphibians] And [bullfrogs are frogs] then [bullfrogs are amphibians];
[frogs are amphibians] And [bullfrogs are frogs];
therefore, bullfrogs are amphibians.

The term amphibian is a class, not an animal per se; this class has been defined to include frogs, and frogs defined to include bullfrogs (but not toads). So the argument is hardly assailable because it is a serial definition. It is true by definition.

Here are some interesting points. The terms validity and invalidity, true and false, cannot apply to inductive conclusions, because the inductive process never produces certainty, nor can it produce truth. Those terms, validity and truth, can refer to the deductive conclusion though, because deduction should produce unquestionable truth, if the process being used is valid and the general case is true. In logic the term “truth” is restricted to mean “that which corresponds to a verifiable reality”, or “that which actually is the case”. And validity refers to whether the process itself is valid, not the conclusion.

Soundness
”When an argument is valid, AND all of its premises are true, it is called ‘sound’” (note 2). In science, the soundness of a proposed natural law is tested by deducing consequences of that law, and performing experiments designed to produce those consequences by creating an instance of the law in action. Deduction is about consequences, those eventualities that necessarily arise from the more general state or law. Even everyday consequences can be deduced, such as that heavy snowstorms impede traffic; as a consequence it will take more time to reach a destination.

In terms of consequence the simple deductive argument can be written as follows:

If [a law, P, is true], then [a necessary consequence, Q, of that law is true];
[ law, P, is true];
Therefore, [consequence Q is necessarily true].

This is simple deductive argumentation. There are more types of deductive arguments and these will be covered in the next lesson.

All questions and comments are welcome.

Note:
(1) There are other classes of argumentation too, such as abduction, which will be covered later.
(2) Copi and Cohen, Introduction to Logic, 12th Ed., pg 19.