PHIL 2306 - Deductive & Inductive Arguments

Deductive Arguments

Validity

An argument is valid if and only if there is no logically possible situation in which the premises of that argument are all true and the conclusion is false.
If an argument is valid, then if the premises are all true, the conclusion must be true. The conclusion of a valid argument is a logical consequence of the premises.
If you use only valid arguments in your reasoning, then if you start with true premises, you will never end up with a false conclusion.
A logically possible situation need not be a situation that is likely to occur (or even one that is physically possible).
We can determine whether an argument is valid without determining whether the premises/conclusion are true.
Validity concerns the logical connections between the premises and the conclusion.
Valid arguments have truth-preserving forms.
You can use truth-preserving rules to derive their conclusions.
Invalid arguments lack truth-preserving forms.
You can’t use truth-preserving rules to derive their conclusions.

Example 1

Premise: The Sun has risen every day in the past.
Conclusion: The Sun will rise again tomorrow.

Example 2

Premise: Zach Blaesi is a billionaire at 2:31 p.m. Central Time on Feb. 1 in Austin, Texas.
Premise: Zach Blaesi is a movie star 2:31 p.m. Central Time on Feb. 1 in Austin, Texas.
Conclusion: Zach Blaesi is a billionaire and Zach is a movie star at 2:31 p.m. Central Time on Feb. 1 in Austin, Texas.

Valid Forms

Modus Ponens

If pointless suffering exists, then a morally perfect, omnipotent, and omniscient god does not exist.
Pointless suffering exists.
So: a morally perfect, omnipotent, and omniscient god does not exist.
General Form:
- If P, then Q.
- P.
- So: Q.

Modus Tollens

If God does not exist, then objective moral values do not exist.
It is not the case that objectives moral values do not exist (they do exist).
So: it is not the case that God does not exist (God does exist).
General Form:
- If P, then Q.
- ¬Q.
- So: ¬P.

Example

If Wittgenstein is a complete idiot, then he should become an aeronaut.
It is not the case that Wittgenstein should become an aeronaut.
So: it is not the case that Wittgenstein is a complete idiot.
General Form:
- If P, then Q.
- ¬Q.
- So: ¬P.

Reductio ad Absurdum

P1. …
Q. (Assume for the sake of argument.) …
These logically entail a contradiction: Pn and ¬Pn.
So: ¬Q.
We can derive a contradiction from our premises and the assumption, so if the premises are true, the assumption must be false. (This is one way to “prove a negative.”)

Example

A sentence “s” is true if and only if s. (Premise)
There isn’t anything that is true. (Assume)
The sentence “there isn’t anything that is true” is true. (From 1, 2)
There is something that is true. (From 3)
There isn’t anything that is true, and there is something that is true. (From 2, 4) – contradiction!
So: It’s not the case that there isn’t anything that is true.

Invalid Forms

Affirming the Consequent

If someone is enrolled at UT Austin, then they’re a college student.
[ACC student] is a college student.
So: [ACC student] is enrolled at UT Austin.

Denying the Antecedent

If Zach is psychology professor, then Zach is a professor.
Zach is not a psychology professor.
So: Zach is not a professor.
General Form:
- If P, then Q.
- ¬P.
- So: ¬Q.

Invalidity

An argument is invalid if and only if it is not valid.
In other words, an argument is invalid so long as there is at least one logically possible situation in which its premises are true and its conclusion is false.
The situation (or invalidating counterexample) need not be realistic or actually occur so long as it is coherent and doesn’t entail a contradiction.
We can show that an argument is invalid by producing an invalidating counterexample.
But we can’t show that an argument is valid by failing to produce a counterexample—maybe we just lack the imagination to do so. Tools from formal logic are needed.

Soundness

Validity does not tell us whether the premises or conclusion are true.
An argument is sound if and only if (i) it is valid and (ii) it has all true premises.
If the argument is invalid, and/or at least one of the premises is false, then the argument is unsound.
Sound arguments, unlike valid arguments, must have true conclusions.
In this course, we will frequently ask whether a valid argument is sound. If you think that it is unsound, then you will need to explain which premise is false.

Misleading Language

Johnny gives Daniel an argument for the existence of God.
Daniel: “Your argument is false!”
Johnny: “No, my argument is true!”

Inductive Arguments

Scientific Reasoning

Observe
Hypothesize
Experiment
Assess Data

Inductive Argumentation

An argument is inductively strong if and only if:
- It is invalid.
- The conclusion is likely to be true given that the premises are true.
The true premises of a strong inductive argument do not logically guarantee the truth of the conclusion, but they make it rational for us to be more or less confident in the truth of the conclusion.
Inductive reasoning is indispensable to science, philosophy, ethics, and everyday life.
Inductive strength concerns the evidential connection between premises and conclusion, or the conditional probability of the conclusion given the premises.

Example 1

Premise: The Sun has risen every day in the past.
Conclusion: The Sun will rise again tomorrow.

Example 2

Premise: 93% of Chinese people have lactose intolerance.
Premise: Lee is Chinese.
Conclusion: Lee has lactose intolerance.

Defeasibility / Ampliativity

The strength of an inductive argument can increase or decrease as new premises are added to the argument.
This is an important difference between mathematics and the empirical sciences.
Adding new premises to a valid argument will not make it more or less valid; validity is all or nothing.
But adding new premises to an inductive argument may make it more or less strong.

Example

Premise: Zach fell from the UT Tower.
Conclusion: Zach is dead.
Adding "He landed on a conveniently placed trampoline" makes the argument weaker.
Adding "The trampoline has a huge hole in its center" makes the argument stronger.

Cogency

Strength does not tell us whether the premises or conclusion of an inductive argument are true.
An argument is cogent if and only if (i) it is inductively strong and (ii) it has all true premises.
If an inductive argument is weak, and/or at least one of the premises is false, then the argument is uncogent.
Cogent arguments, unlike sound arguments, may have false conclusions, but that is unlikely.

Argument by Analogy

x and y are relevantly similar (they have properties F1,…, Fn in common—properties that are relevant to having a property G).
x has property G.
So: y also has property G.

Inference to the Best Explanation

We have a set of evidence E.
H1,…, Hn are available hypotheses compatible with E.
Hi provides the best explanation of E (and Hi is satisfactory/good enough as an explanation).
So: H_i is true.

Example

E: Evidence of a messy kitchen.
H1: Your roommate made himself a late dinner and was too tired to clean up.
H2: Someone burgled the apartment and took the time to make food on the job.
H3: Your roommate dirtied the dishes without eating just to make you form the false belief that he ate a late dinner.

Theoretical Virtues

Power: The theory makes several precise predictions, and it doesn’t have a track record of being adjusted simply to avoid disconfirmation and without making any new predictions that could disconfirm the theory (this is sometimes called “ad hoc”).
Coherence: The theory is internally consistent and fits well with other well-confirmed theories.
Explanatoriness: The theory explains available evidence; explains correlations by unifying them under general principles or laws; or enables us to detect and explain coincidences we otherwise wouldn’t have noticed or been able to explain.
Simplicity: The theory makes fewer assumptions than its competitors; posits fewer entities than its competitors; or posits less complicated principles or laws than its competitors.