Small intro
On Mondays, the prof will give a brief recap of the last lecture. Not on Tuesdays, since it’s been 24 hours since the last lesson.
Check your course schedule!
NO CLASS ON MONDAY NEXT WEEK
What is deductive logic?
The argument is valid if there is no situation where, given that the premises are true, the conclusion is not. The premises do not have to be “relevant” to the argument.
Fields like Relevance logic exist, but that complicates the mathematics significantly, so in the class we simply go by a convention that simplifies our workload significantly.
is drinking a coffee a process or an act? If he dies while he’s sitting down, drinking coffee, from a ceiling fan dropping on his head, will he die from “this”?
Nobody has ever died from drinking coffee at Mustache,
Pete is currently drinking a coffee at Mustache,
so Pete will not die from this.
The premises do not guarantee the conclusion.
They do, however, render it highly unlikely.
All logicians are smart, Albert is smart, so Albert is a logician.
The premises could be true, with the conclusion being false. As Albert belongs to the larger set of “smart” people, it is possible for him to be “smart” but not a “logician”.
Deductive reasoning
Inductive reasoning
- from past to future
- from a number of specific cases to all cases
- not deductively valid
- (usually) very reliable
Scientific division of labor
Deductive reasoning is generally up to logic (valid / invalid / possible / impossible)
Inductive reasoning is usually in the realm of statistics (highly likely / unlikely)
Whether or not you can imagine a situation is not the same as whether a situation exists. Imagining is subjective, while the existence or lack thereof of a situation is objective.
Formal logic
validity has to do with formal structure, not with concrete contents
all F are G, n is F, so n is G
- the argument form itself is deductively valid
- every concrete argument that instantiates this form, is deductively valid
most G are H, most H are K, so there exist G that are K
- there exist concrete arguments that instantiate the form and that are deductively invalid (cf. Irish, Catholic, against abortion)
- not every concrete argument that instantiates this form, is deductively valid
- the argument form itself is deductively invalid
Chapter 2
an argument is (deductively) valid if and only if there is no possible situation in which all premises are true and in which the conclusion is false
Jo jumped down from the 20th floor and landed on the concrete sidewalk, so Jo is badly injured
This is not a valid argument, as there is a situation where this is possible, albeit by breaking the laws of physics. The situation is logically possible, not physically possible.
Logical possibility
- what can be thought coherently, without contradicting oneself
- the weakest, most inclusive kind of possibility
- whatever is physically possible, is logically possible as well
- whatever is financially possible, is logically possible as well
necessary = not possible not
- it is necessary that 2 + 2 is equal to 4
- it is not possible that 2 + 2 is not equal to 4
logical necessity is the strongest kind of necessity.
Very little is logically necessary, but many things are logically possible.
Validity and necessary truth preservation
- an argument is valid if and only if it is necessary that if the premises are true, then the conclusion is true as well
- validity = necessary truth preservation
- from the premises
- from the conclusion
validity is independent of the de fact truth
validity = necessary truth preservation
PL (propositional logic)
- the premises and conclusion in an argument are propositions
- a proposition is true or false in a given situation
- this is exactly what we need to make (and evaluate) arguments!
- validity = necessary truth preservation
- some people may argue that propositions are only expressed by sentences, but prof doesn’t care and a sentence is good enough