William

Proofs → Derivations

We’re interested in mainly in derivations, not so much proofs.

Babies are illogical; nobody is despised who can manage a crocodile; illogical persons are despised; so babies cnanot manage a crocodile.

This seems unclear.

Derivate!
→

1. all babies are illogical people. (premise)

2. nobody is despised who can manage a crocodile. (premise)

3. illogical people are despised. (premise)

4. Babies are despied (from 1, 3).

5. Babies cannot manage a crocodile (from 2, 4)

Derivation:

• A sequence of annotated inference steps

• The validity of each inference step has to be absolutely clear

• All individual inference steps are necessarily truth-preserving

• … So the entire sequence is as well!

A derivation can be:

valid (each step is necessarily truth-preserving) & sound (necessarily valid + true)

In everyday language, a ”proof”, however is supposed to establish the truth of its conclusions.

Ie. – A proof has to both be valid as well as sound.

In the language of logic, a ”proof” does not have to be sound.

Each step in derivation has to be properly annotated.

We talk about what previous step it was based on, we use ”from” or similar.

We also add what kind of basic argument form we used to derive further.

Basic argument forms:

Hypothetical Syllogism (HS): If A then B, if B then C, So if A then C

Modus Tollens (MT): If A then B, Not B, so not A.

Modus Pollens (MP): If A then B, A, so B.

Always try to apply HS over MT – at least according to Lorenz. Idrk, feels like both are okay.

If A then B

If B then C

not C

If A then C (from 1, 2 by HS)

→ Not A (from 3, 4 by MT)

All arguments are valid If and only If the previous argument is valid and so on forever.

• Forms a net or continuum of validity.

These derivations are a kind of ideal and so arguments in everyday life (and in philosophy) are not like this, in their entirety.

It could however be a good exercise to ”massage” them into this idealised format.

Enthymemes:

if Roger is a logician, then he’s smart, Roger doesn’t even know that 1 + 1 = 2; therefore Roger is not a logician.

1. If roger is a logician, then he’s smart

2. Roger doesn’t know that 1 + 1 = 2

3. If Roger is smart, he knows that 1 + 1 = 2 (Hidden premise)

4. Roger is not a logician ( from, 1, 2, 3 *)

An enthymeme is an argument with unspoken premises (or conclusions)

• To assess its validity, these unspoken premises have to be reconstructed.

• Unspoken premises can be controversial.

In logic, we assume that everything is clear, and we work on the basis of the principle of charitability → if an argument can be rendered complete through adding a premise, and it makes intuitive sense to do, we do it.

Invalid derivations:

Eg. ”F”

1. All philosophers are logicians

2. all logicians are philosophers (from 1 ?!)

And. ”G”

1. All existentialist are philosophers

2. All logicians are philosophers

3. all existentialist are logicians

So

1. All existentialist are philosophers

2. All logicians are philosophers

3. all existentialist are logicians (from 2, by F)

4. all existentialists are lgicians (from 1, 3 by G)

Valid derivation:

And-Introduction

1. Lorenz is a logician

2. All logicians are silly

3. Lorenz is a logician and all logicians are silly (from 1, 2)

You can always but an ”and” in between two premises and the conclusion will be valid.

And-Elimination

1. Lorenz is a logician and all logicians are silly

2. all logicians are silly (from 1)

All-Elimination

1. Lorenz is a logician

2. All logicians are silly

3. Lorenz is silly (from 1, 2)

Circularity: Justifying a derivation from steps that come after the the previous premises.

Fallacy: not synonymous with invalid argument. Sometimes circular arguments are valid, like: P therefore P.

Indirect Argument (reductio ad absurdum: reduction to absurdity) shortened: RAA

• Establish the conclusion not-A

• by supporting that A

• And showing that thi leads to something absurd.

(Modus Ponens: If-Then elimination.)

1. A > B

2. ~B

3. A (supposition)

4. B (from 1, 3 by MP)

5. Contradiction

6. ~A (from 3-5, by RAA)