Deductive arguments are evaluated in terms of their validity and soundness.
Definition: An argument is valid if it is impossible for its premises to be true while its conclusion is false. Or, the conclusion must be necessarily true, if the premises are true. So the premises provide a guarantee for the truth of the conclusion.
The deductive form of Modus ponens
The 1st form of deductive reasoning is modus (ponendo) ponens, also known as affirming the antecedent. We conclude Q from P by using modus ponens.
|P→Q||Is a conditional statement (implication) with antecedent (P) and consequent (Q)|
|P||Is the hypothesis|
|∴ Q||The conclusion is deduced from the statement and the hypothesis|
Affirming the consequent – a fallacy
If the conclusion (Q) is given instead of the hypothesis (P), there can be no valid conclusion; hence we have the fallacy of affirming the consequent.
Definition: An argument is sound if and only if it is valid and all its premises are true. Otherwise, the argument is unsound.
|All men are mortal.||all objects classified as ‘men’ have the attribute ‘mortal’|
|Aristotle is a man.||‘Aristotle’ is classified as a ‘man’ – a member of the set ‘men’.|
|Therefore, Aristotle is mortal||‘Aristotle’ must be ‘mortal’ because he inherits this attribute from his classification as a ‘man’.|
Valid, but unsound argument
A deductive argument can still be logically valid, if it has false premises, but then the argument is not sound. Trick arguments are based off of this.
|Everyone who eats carrots is a quarterback.||1st, major premise is false, but classifies correctly (everyone)|
|John eats carrots.||2nd, minor premise: ‘John’ is classified as a member of the set ‘carrot-eaters’|
|Therefore, John is a quarterback.||‘John’ inherits the attribute ‘is a quarterback’ from his membership; therefore through its form, the conclusion must be true.|
Generalizing – a fallacy
Generalizations often make unsound arguments, such as “everyone who eats carrots is a quarterback.” But, since everyone who eats carrots is not a quarterback, such arguments prove the flaw.
In this example, the first statement uses categorical reasoning (the theory of the categorical syllogism. This is the theory of two-premised arguments in which the premises and conclusion share three terms among them, with each proposition containing two of them.), saying that all carrot-eaters are definitely quarterbacks.
When a generalization about all instances of a kind is based on either too few examples which are not known to be typical or based on instances of a different kind, those are called ‘converse accident’, which is the opposite of accident: general.