Critical thinking final part 1
Deductive Arguments Handout
1. A deductive argument is an argument with the form if the premises are true the conclusion is necessarily true. The argument is valid when it has this form, invalid when it fails to have this form, and sound when it is both valid and has verified true premises.
2. To detect a deductive argument, look for indicator words that signify necessity, like ‘it must be the case’, ‘it necessarily follows that’, ‘it cannot be otherwise’, ‘we can be absolutely certain’, etc. Regardless of the presence of indicator words, consider whether it makes sense to think that the premises could be true and the conclusion false at the same time without a contradiction or paradox. If it’s not thinkable, then you have a good reason to believe that the argument is deductively valid. Always ask, “If the premises are true, does the conclusion follow? If so, how? Necessarily? Probabilistically?”
3. The following list contains some common and useful deductive argument types. Some of these are important because of their rigid formal structure alone, while others are more like strategies. ‘Syllogism’ is simple a traditional term for three statement argument, two premises and one conclusion. Below, all lower-case, italicized letters are variables that represent statements; upper-case S, P, M represent categories of noun-ish stuff.
a. Disjunctive Syllogism: An argument that uses an either-or statement, then eliminates one possibility, making the other follow as a conclusion. The first form below is a basic one, but the second strategy of arguments by elimination can be more diverse.
■ P1: Either p or q. ■ P2: Not p. *Both of these forms are always valid ■ C: q. ■ P1: It is the case that either p, q, r, s, or t. ■ P2: It is not the case that p, q, and s are the case. ■ P3: Not t. ■ C: Therefore, r must be the case.
b. Affirming the Antecedent: An argument that uses an if-then statement to set up a series of hypothetical conditions before introducing the first condition in reality, allowing the inference of the following hypothetical conditions in reality.
■ P1: If p then q. ■ P2: p. *AA (or “Modus Ponens”) is always valid ■ C: q.
c. Denying the Consequent: An argument that uses an if-then statement to set up a series of hypothetical conditions before asserting that the final consequent is not the case, allowing the inference that the first antecedent must not be the case.
■ P1: If p then q. ■ P2: Not q. *DC (or “Modus Tollens”) is always valid ■ C: Not p.
Prof. Eckel, U. Toledo, FA17
d. Affirming the Consequent and Denying the Antecedent: Both of these argument forms are invalid and result from mistaken hypothetical reasoning; they are worth bringing up since they are often compelling due to similarities with AA and DC.
■ P1: If p then q. ■ P2: q. *AC is always invalid ■ C: p. ■ P1: If p then q. ■ P2: Not p. *DC is always invalid ■ C: Not q.
e. Hypothetical Syllogism: An argument composed of a chain of hypothetical (if-then) statements as premises, where the antecedents and consequents of each are linked, allowing the first premise’s antecedent to be linked to the last premise’s consequent in the conclusion, itself a hypothetical statement. Premises aren’t limited to two.
■ P1: If p then q. ■ P2: If q then r. *HS are always valid ■ C: If p then r.
f. Categorical Syllogism: These arguments are very diverse, with no single form. These use categorical statements to associate certain noun-terms with others.
■ P1: All S are M. ■ P2: All M are P. ■ C: Some S are P. *CS forms are valid or invalid ■ P1: No M are P. ■ P2: Some M are not S. ■ C: All S are P.
g. Arguments by Mathematics: These arguments use the principles and definitions of mathematics in statement form to produce argumentative structures.
■ P1: There were 12 bears on the ridge at noon today but exactly 3 left five minutes later. ■ C: There were still 9 bears on the ridge at 12:05 pm that day. ■ P1: The radius of the first circular clearing in my woods is is unknown. ■ P2: The radius of the second circular clearing in my woods is ¾ of the first clearing's
radius. ■ C: The ratio of the first to second clearing’s area is 16 to 9.
h. Arguments by Definition: These arguments provide certain definitions of words, playing out the definition’s implications to make conclusions based on general concepts and particular instances.
■ P1: A planet is defined as a celestial body orbiting the sun with enough mass/gravity to be round, and which has not cleared the neighborhood of its orbit.
■ P2: Pluto has not cleared the neighborhood of its orbit. ■ C: Pluto is not a planet. ■ P1: Human life begins at conception. ■ P2: Deliberately terminating the development of a zygote is killing a living thing. ■ C: Abortion, even in the zygote stage, is murder.
Prof. Eckel, U. Toledo, FA17