Deductive Logic/Arguments and Validity

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Arguments and Validity[edit]

In general an argument is a series of statements typically used to persuade someone of something or to present reasons for accepting a conclusion. The general form of an argument in a natural language is that of premises (typically in the form of propositions, statements, or sentences) in support of a claim: called the conclusion. In a typical deductive argument, the premises are meant to provide a guarantee of the truth of the conclusion, while in an inductive argument, they are thought to provide reasons supporting the conclusion's probable truth.

This course focuses on deductive arguments. A deductive argument asserts that the truth of the conclusion is a logical consequence of the premises—if the premises are true, then the conclusion must be true. Examples are:

(Premises)   If today is Saturday, then I will wash my car.
    Today is Saturday.

(Conclusion)   Therefore, I will wash my car.
(Premises)   If today is Saturday, then I will wash my car.
    I am not washing my car today.

(Conclusion)   Therefore, today is not Saturday.
(Premises)   All men are mortal.
    Socrates is a man.

(Conclusion)   Therefore, Socrates is mortal.
(Premises)   All cows eat grass.
    Elsie is a cow.

(Conclusion)   Therefore, Elsie eats grass.

An expression which becomes an argument when its variables are appropriately replaced is called an argument schema. Argument schemata constitute the forms of arguments; they reveal nothing about the content of arguments. Examples of argument schemata are:

(a)          (b)
    If p then q
p

  p

p or q   q

Here we use the straight line as an abbreviation of “therefore”, and equivalently “hence, “so”, “thus”, etc. Some authors use the "therefore sign" ∴ rather than the straight line. Schemata (a), then is short for

p. Therefore p or q

Similarly, schemata (b) is short for

If p then q, and p. Therefore q.

We can illustrate these with examples. An example of schema (a) is:

Apples are fruit; therefore either Apples or Oranges are fruits.

An example of schema (b) is:

If it is Saturday, then I will wash my car. Today is Saturday; therefore I will wash my car.

Validity[edit]

A valid argument schema is one for which every uniform interpretation that may be assigned to its variables is such that if its premises are all true, then its conclusion must be true. If there is at least one uniform interpretation such that the premises are all true yet the conclusion is false, then this is an invalid argument schema.

The validity of an argument depends, however, not on the actual truth or falsity of its premises and conclusion, but solely on whether or not the argument has a valid logical form. The validity of an argument is not a guarantee of the truth of its conclusion. Under a given interpretation, a valid argument may have false premises that render it inconclusive: the conclusion of a valid argument with one or more false premises may be either true or false.

Logic seeks to discover the valid argument schema, those forms that make arguments valid. An argument schema is valid if and only if the conclusion is true under all interpretations of that argument in which the premises are true. Since the validity of an argument depends solely on its form, an argument can be shown to be invalid by showing that its form is invalid. This can be done by giving a counterexample of the same form of argument with premises that are true under a given interpretation, but a conclusion that is false under that interpretation.

The conclusion of a valid argument is not necessarily true, it depends on whether the premises are true. If the conclusion, itself, just so happens to be a necessary truth, it is so without regard to the premises.

Some examples:

  • All Greeks are human and all humans are mortal; therefore, all Greeks are mortal. : Valid argument; if the premises are true the conclusion must be true.
  • Some Greeks are logicians and some logicians are tiresome; therefore, some Greeks are tiresome. Invalid argument: the tiresome logicians might all be Romans (for example).
  • Either we are all doomed or we are all saved; we are not all saved; therefore, we are all doomed. Valid argument; the premises entail the conclusion. (Remember that this does not mean the conclusion has to be true; it is only true if the premises are true, which they may not be!)
  • Some men are hawkers. Some hawkers are rich. Therefore, some men are rich. Invalid argument. This can be easier seen by giving a counter-example with the same argument form:
    • Some people are herbivores. Some herbivores are zebras. Therefore, some people are zebras. Invalid argument, as it is possible that the premises be true and the conclusion false.

In the above case (Some men are hawkers...), the counter-example follows the same logical form as the previous argument, (Premise 1: "Some X are Y." Premise 2: "Some Y are Z." Conclusion: "Some X are Z.") in order to demonstrate that whatever hawkers may be, they may or may not be rich, in consideration of the premises as such.

The forms of argument that render deductions valid are well-established, however some invalid arguments can also be persuasive depending on their construction

A sound argument is a valid argument whose conclusion follows from its premise(s), and the premise(s) of which is/are true.

Assignment[edit]

Optionally, watch this video on Fundamentals of Deductive Arguments.