Logic — 2026-10-09
Forms of Inference
Argument form: all F are G, n is F, so n is G
Lorenz is a logician all logicians are smart, so Lorenz is smart.
Concrete argument instantiates the argument form.
The order of premises is not important, however the distinction between premises and conclusion does matter!
Dealing with argument forms
- all F are G, n is F, so n is G
- all X are Y, k is X, so k is Y
these schemas represent the same argument form.
it doesn’t matter which symbols we use as schematic variables
- empty placeholders
- no content
- indicates structure
a concrete argument exemplifies many argument forms
there is no unique argument form that is exemplified by the argument
valid argument form
→ all of its instances are valid
→ none of its instances is invalid
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Proofs
fairly simple arguments with not too many premises so far, valid or invalid, does there exist a situation…
validity has to do with necessary truth preservation, metaphorically the conclusion is guaranteed by the premises
now what about more complex arguments?
all babies are illogical; nobody is despised who can manage a crocodile; illogical persons are despised; so babies cannot manage a crocodile
not a good example, doesnt illustrate the point it wants to
applying the definition of validity,
| derivation | hmm2 |
|---|---|
| 1. babies are illogical | premise |
| 2. nobody is despised who can manage a crocodile | premise |
| 3. illogical people are despised | premise |
| 4. babies are despised | from 1,3 |
| 5. babies cannot manage a crocodile | from 2,4 |
derivation is a sequence of annotated inference steps, the validity of each inference step is absolutely clear. all individual inference steps are necessarily truth-preserving → the entire sequence is as well.
guaranteed so hard no extra premise can change it
deductive validity means that the conlcusion is guaranteed by the premises.
an argument can be:
- valid (necessarily truth-preserving)
- sound (valid + true premises)
a derivation / proof can be
- valid (necessarily truth-preserving)
- sound (valid + true premises)
in everday language, a proof is supposed to estabhlish the truth of its conclusion
it should not merely be valid, but also have true premises
it should not merely be valid, but also sound
but we’re not interested in soundness, in the language of logic, a proof does not have to be sound
the RHS of derivations
each step in a derivation is annotated
- which previous steps does the current step depend on?
- which basic argument form is being invoked?
hypothetical syllogism (HS): if A then B, if B then C, so if A then C
modus tollents (MT): if A then B, not B, so not A
if Lorenz is a logician, then he’s smart; if Lorenz is smart, then he knows Greek; Lorenz doesn’t know Greek; therefore he’s not a logician
| derivation | rhs |
|---|---|
| 1. if Lorenz is a logician, then he’s smart | premise |
| 2. if Lorenz is smart, he knows Greek | premise |
| 3. Lorenz doesn’t know Greek | premise |
| 4. if Lorenz is a logician, he knows Greek | from 1,2 by HS |
| 5. Lorenz is not a logician, | from 3,4 by MT |
another one
| derivation | rhs |
|---|---|
| 1. if Lorenz is a logician, then he’s smart | premise |
| 2. if Lorenz is smart, he knows Greek | premise |
| 3. Lorenz doesn’t know Greek | premise |
| 4. Lorenz is not smart | from 2,3 by MT |
| 5. Lorenz is not a logician, | from 1,4 by MT |
smart → A, knows Greek → B
if A then B, not B, so not A
Validity of Argument Forms
valid argument form: if A then B, if B then C, not C, so not A
the validity of follows HS, MT
such derivations are an ideal, arguments in everyday life are not like this, it’s a good exercise to massage them into this idealized format. this way, you can discover redundancies (not needed premises), discover gaps (suppressed premises that are needed).
tbc 2026-10-13