Argument
In mathematics and logic an argument is a sequence of statements ending in a conclusion. All statements in an argument but the final one are called premises (or assumptions or hypothesis). The final statement is called the conclusion.
The logical form of an argument can be abstracted from the content of the argument. For instance, the argument
| If Socrates is a human being, then Socrates is mortal; |
| Socrates is a human being; |
| Therefore, Socrates is mortal. |
has the abstract form
| If p then q; | |
| p; | |
| ∴ | q. |
Note that an argument form is called valid if, and only if, whenever statements substituted that make all the premises true, then the conclusion is also true.
Formally,
|
An argument is an assertion that a given set of propositions p1, p2, ..., pn, called premises, yields (has a consequence) another proposition q, called conclusion. Such an argument is denoted by p1, p2, ..., pn ⊥ q |
An argument is an assertion that a given set of propositions p1, p2, ..., pn, called premises, yields (has a consequence) another proposition q, called conclusion. Such an argument is denoted by
p1, p2, ..., pn ⊥ q
To say that an argument is valid means that no matter what particular statements are substituted for the statement variables in its premises, if the resulting premises are all true, then the conclusion is also true. To say that an argument is valid means that its form is valid. The fact about a valid argument is that the truth of its conclusion follows necessarily (or by logical form alone) from the truth of its premises.
The notion of a valid argument is formalized as follows:
| An argument p1, p2, ..., pn ⊥ q is said to be valid if q is true whenever all the premises p1, p2, ..., pn are true. |
An argument that is not valid is called fallacy or invalid.
Now the propositions p1, p2, ..., pn are true simultaneously if, and only if, the proposition p1∧ p2∧ ...∧ pn is true. Thus, the argument p1, p2, ..., pn ⊥ q is valid if, and only if, q is true whenever p1∧ p2 ∧ ... ∧ pn is true or, equivalently, if the proposition (p1 ∧ p2 ∧ ... ∧ pn) → q is a tautology.
Formally,
| The argument p1, p2, ..., pn ⊥ q is valid if, and only if, the proposition is (p1 ∧ p2 ∧ ... ∧ pn) → q tautology. |
Algorithm to Test an Argument form for Validity:
| Step 1. | Identify the premises and conclusion of the given argument. | |
| Step 2. | Construct a truth table showing premises and conclusion. | |
| Step 3. | Find critical rows in which all the premises are true. | |
| Step 4. | In each critical row, determine whether the conclusion is also true. | |
| Step 5. | IF (the | conclusion is true) THEN |
| the argument form is valid | ||
| ELSE | ||
| the argument form is invalid | ||
As an example, show that the following argument form is valid.
| p ∨ (q ∨ r) | |
| ~r | |
| ∴ | p ∨ q |
Step 2. Construct the truth showing the truth values of all the premises and the conclusion:
| Premises | Conclusion | |||||
| p | q | r | p ∨ (q ∨ r) | ~r | p ∨ q | |
| T | T | T | T | F | T | |
| T | T | F | T | T | T | Critical Row |
| T | F | T | T | F | T | |
| T | F | F | T | T | T | Critical Row |
| F | T | T | T | F | T | |
| F | T | F | T | T | T | Critical Row |
| F | F | T | T | F | F | |
| F | F | F | F | T | F | |
In each row where the premises are both true the conclusion is also true, so the argument is valid.
Another example, shows that the following argument form is invalid.
| p → q ∨ ~r | |
| q → p ∧ r | |
| ∴ | p → r |
Step 2. Construct the truth showing the truth values of all the premises and the conclusion:
| Premises | Conclusion | ||||||||
| p | q | r | ~r | q ∨ ~r | p ∧ r | p → q ∨ ~r | q → p ∧ r | p → r | |
| T | T | T | F | T | T | T | T | T | C. Row |
| T | T | F | T | T | F | T | F | T | |
| T | F | T | F | F | T | F | T | T | |
| T | F | F | T | T | F | T | T | F | C. Row |
| F | T | T | F | F | F | T | F | T | |
| F | T | F | T | T | F | T | F | T | |
| F | F | T | F | F | F | T | T | T | C. Row |
| F | F | F | T | T | F | T | T | T | C. Row |
In the fourth row of the above truth table the premises are true and the conclusion is false; hence the argument form is invalid.
Valid Argument Forms.
1. Modus Ponens
The fact that this argument form is valid is called modus ponens. The term modus ponens is Latin meaning "method of affirming" since the conclusion is an affirmation.
Argument Form:
| If p then q, | |
| p | |
| ∴ | q |
Example of an argument of this form:
| If last digit of this number is a 0 then this number is divisible by 10, |
| The last digit of this number is a 0 |
| Therefore, this number is divisible by 10. |
To check the validity of this argument form, construct a truth table for the premises and conclusion.
| Premises | Conclusion | ||||
| p | q | p → q | p | q | |
| T | T | T | T | T | Critical Row |
| T | F | F | T | F | |
| F | T | T | F | T | |
| F | F | T | F | F | |
The first row (critical row) is the only one in which both premises are true, and the conclusion in that row is also true. Hence, the argument form (modus ponens) is valid.
2. Modus Tollens
The fact that this argument form is valid is called modus tollens. The term modus tollens is Latin meaning "method of denying" since the conclusion is a denial.
Argument form:
| If p then q, | |
| ~q | |
| ∴ | ~p |
Example of an argument of this form:
| If Zeus is human, then Zeus is mortal, |
| Zeus is not mortal, |
| Therefore, Zeus is not human. |
To check the validity of this argument form, construct a truth table for the premises and conclusion.
| Premises | Conclusion | ||||
| p | q | p → q | ~q | ~p | |
| T | T | T | F | F | |
| T | F | F | T | F | |
| F | T | T | F | F | |
| F | F | T | T | T | Critical Row |
The last row (fourth row) is the only one in which both premises are true, and the conclusion in that row is also true. Hence, this argument form (modus tollens) is valid.
Note that the validity of modus tollens can be shown to follow from modus ponens together with the fact a conditional is logically equivalent to its contrapositive.
3. Disjuctive Addition
These argument forms are used for making generalizations.
Argument Form:
| a. | p | b. | q | |||
| ∴ | p ∨ q | ∴ | p ∨ q |
For example, according to the first, if p is true, then, more generally, "p or q" is true for any other statement q.
| Premise | Conclusion | |||
| p | q | p | p ∨ q | |
| T | T | T | T | Critical Row |
| T | F | T | T | Critical Row |
| F | T | F | T | |
| F | F | F | F |
4. Conjunctive Simplification
These argument forms are used for particularizing.
Argument Form:
| a. | p ∧ q | b. | p ∧ q | |||
| ∴ | p | ∴ | q |
For example, the first one says that if both p and q are true, then, in particular, p is true.
| Premise | Conclusion | |||
| p | q | p ∧ q | p | |
| T | T | T | T | Critical Row |
| T | F | F | T | |
| F | T | F | F | |
| F | F | F | F |
5. Disjunctive Syllogism
These argument forms say that when we have only two possibilities and we can rule one out, the other must be the case.
| a. | p ∨ q | b. | p ∨ q | |||
| ~q | ~p | |||||
| ∴ | p | ∴ | q |
Truth Table:
| Premise | Conclusion | ||||
| p | q | p ∨ q | ~q | p | |
| T | T | T | F | T | |
| T | F | T | T | T | Critical Row |
| F | T | T | F | F | |
| F | F | F | T | F | |
For example, suppose we know that for a number x, x - 3 = 0 or x + 2 = 0. If we also know that x is not negative, then x≠ -2 and so x + 2 ≠ 0. By disjunctive syllogism we can conclude that x - 3 = 0.
6. Hypothetical Syllogism
This represent chains of if-then statements. From the fact that one statement implies a second and the second implies the third, we can conclude that the first statement implies the third.
Argument Form:
| p → q | |
| q → r | |
| ∴ | p → r |
Truth Table:
| Premises | Conclusion | |||||
| p | q | r | p → q | q → r | p → r | |
| T | T | T | T | T | T | Critical Row |
| T | T | F | T | F | F | |
| T | F | T | F | T | T | |
| T | F | F | F | T | F | |
| F | T | T | T | T | T | Critical Row |
| F | T | F | T | F | T | |
| F | F | T | T | T | T | Critical Row |
| F | F | F | T | T | T | Critical Row |
7. Dilemma: Proof by Division into Cases
It often happens that we know one statement or another is true. If we can show that in either case a certain conclusion follows, then this conclusion must also be true.
Argument Form:
| p ∨ q | |
| p → r | |
| q → r | |
| ∴ | r |
Truth Table:
| Premises | Conclusion | ||||||
| p | q | r | p ∨ q | p → r | q → r | r | |
| T | T | T | T | T | T | T | Critical Row |
| T | T | F | T | F | F | F | |
| T | F | T | T | T | T | T | Critical Row |
| T | F | F | T | F | T | F | |
| F | T | T | T | T | T | T | Critical Row |
| F | T | F | T | T | F | F | |
| F | F | T | F | T | T | T | |
| F | F | F | F | T | T | F | |
For example, consider the following argument:
| x > 0 or x < 0 |
| If x > 0, then x2 > 0 |
| If x < 0, then x2 > 0 |
| Therefore, x2 > 0 |
____________________________________________________________________________________
∴ therefore
Negation ~
Not equal ≠
And ∧
or ∨
implies →
equivalence ≡ ↔
XOR ⊕