Deductive Arguments: Validity and Soundness

Validity in Deductive Arguments

A deductive argument can have false premises and still be valid.
Validity depends on the structure of the argument, not the truth of the premises.
To determine validity, abstract away from the truth content of the premises and suppose they are true.
- If the conclusion necessarily follows from the assumed true premises, the argument is valid.
- This process extracts the logical structure of the argument.

Argument: "Only those who have HIV can have AIDS. Harold doesn't have HIV, so Harold cannot have AIDS."
This argument is valid.
If the premises are true, the conclusion necessarily follows.
There is no possible way for the conclusion to be false if the premises are true.

Argument: "Alice graduated from Yale. If she graduated from Yale, she has a superior intellect. Therefore, Alice has a superior intellect."
This argument is valid, despite the potential falsity of the second premise.
To evaluate, imagine a world where the premises are true.
In that world, if Alice graduated from Yale and anyone who graduates from Yale has a superior intellect, then Alice necessarily has a superior intellect.
It's crucial to abstract away from the truth content of the premises and just assume the premises are true to perceive whether or not conclusion follows the premise or premises.

Argument: "If you study hard, you'll get an A on the exam. You got an A on the exam. Therefore, you must have studied hard."
This argument is invalid.
A counterexample can be constructed where the premises are true, but the conclusion is false (e.g., someone is smart and lazy and gets an A without studying, or someone cheated).

The difference between the Alice and Yale example and the Studying Hard example lies in necessary and sufficient conditions.
In the Alice example, graduating from Yale is presented as a sufficient condition for having a superior intellect.
In the Studying Hard example, studying hard is a sufficient condition for getting an A, but getting an A is not a sufficient condition for having studied hard.
Analogy: "If you live in Beijing, then you live in China" vs. Living in China does not imply living in Beijing.

Assume all premises are true and check if the conclusion follows by necessity.
Construct a counterexample where the premises are true, but the conclusion is false.
Helpful way to determine whether an argument is valid.

Argument: "If pigs fly, then Lady Gaga was Darth Vader in a past life. Pigs most certainly fly. Therefore, Lady Gaga was Darth Vader in a past life."
This argument is valid.
If the premises are immovable truths, the conclusion must also be true.
Validity is a matter of structure, not the truth of the premises.
Build a mental wall between the concepts of truth and validity.

Argument 5: "Birds are robots. Every robot animal descended from dinosaurs. Therefore, birds descended from dinosaurs."
- Valid.
Argument 6: "Birds are mammals. Every mammal descended from octopi. Therefore, birds descended from octopi."
- Valid.
Argument 7: "Birds are descended from dinosaurs. Anything descended from dinosaurs is a dinosaur. Therefore, birds are avian dinosaurs."
- Valid.

Argument: "If it's raining, then the streets are wet. It's raining. Therefore, the streets are wet."
- Valid.
- Raining is the sufficient condition for wet streets

Argument: "Schmee is a dolphin. If Schmee is a dolphin, Schmee is a mammal. Therefore, Schmee is a mammal."
- Valid.
- The conclusion necessarily has to be true.

Argument: "If the light is on, then the switch has been flipped. The switch has been flipped. Therefore, the light is on."
- Invalid.
- Flipping the switch being flipped does not guarantee that the light is on.
- There might be scenario in which electricity is out, and the switch doesn't turn on the light.
- This is an example of affirming the consequent, a logical fallacy.

Argument: "If the bridge was destroyed, then the supplies never got through. The bridge was not destroyed. Therefore, the supplies got through."
- Invalid.
- The bridge being destroyed is the sufficient condition, the bridge not being destroyed doesn't guarantee that supplies got through.

Argument:
- "If the universe had a beginning, then it was caused to begin."
- "We know that the universe did have a beginning, so it was caused to come into existence."
- "If the universe was caused to come into existence, that cause must have been God."
- "Therefore, God caused the universe to come into existence."
Valid.
This is a complex argument with multiple inferences.
The argument is not sound because the premise "if the universe was caused to come into existence, that cause must have been God" is not proven true and can be viewed as false.
- Justifying premises often requires significant effort.

Argument: "I like philosophy. My philosophy teacher likes me. Therefore, I will pass my philosophy course with flying colors."
- Invalid.

The material requires effort, and the texts presented can be difficult.
It is okay and natural to be confused.
If one persists, with hard work, one will advance and be rewarded and gain understanding and clarity, and this insight and clarity can be used as the basis for further insight and clarity.