College notes Introduction to Logic

Michiel de Folter



Introduction to logic ‘22/’23

HC 1: Course start

1. Ferry is a dog Premise
2. All dogs are canines Premise

3. Ferry is a canine Conclusion

Argument: any collection of premises together with a conclusion

Good argument: would let you derive the conclusion based on its premises.

Bad argument:

Content: One or more premises are false

Form: The conclusion does not follow from the premises

Initial thought: The conclusion “follows from” the premises. Given that all the premises are true, the
conclusion therefore must also be true.

Valid argument: An argument is valid when it is 1. impossible for all the premises to be true and 2.
the conclusion to be false.

(!) Remember that validity is NOT about the truth or falsity of the premises, but it is about
whether it is possible for the premises to be true and the conclusion to be false.

1. Ferry is a sausage
2. Ferry is not cute Valid arguments
3. All non-cute sausages are feline

4. Ferry is a feline

Sound argument: An argument is sound when it is valid AND all of its premises are true.

Logic: The science of settling whether an argument is valid or not.



Deductive arguments: deriving a conclusion from premises

Inductive argument: based on past instances and generalizing future instances

Abductive argument: instance of the best explanation, (trying to connect two premises the best way)

Valid deductive arguments give an unbreakable connection between the premises and the
conclusion.

Inductive and abductive arguments do not necessarily express an unbreakable connection
between the truth of the premises and the conclusion.

Assertoric question: A question that can only be true or false.

, Michiel de Folter


Jointly consistent: a sentence is jointly consistent when all the premises can be true together

Necessary true: A sentence that can only be true and never false

Necessary falsehood: A sentence that can only be false and never true

Contingent: A sentence that is possible to be true and possible to be false



Truth functional logic: whatever you put in for A or B, it will always be a valid argument.

1. A 1. A or B 1. Not A and B 1. If A then B
2. If A then B 2. Not A 2. A 2. Not B

3. So B 3. So B 3. So Not B 3. So Not A

The validity is given by the structure of these arguments, the content of the premises and their falsity
or truth does not matter at all.

Limitation of TFL (propositional logic): When given the premises 1. “The pope is a bachelor” and 2.
“Therefore, the pope is a male”, TFL does not recognize this as a valid statement,

As humans we know that “A bachelor is a single male” and when we insert that premise
alongside the other two, the argument above should be valid!, but for that to be true we
would need the knowledge of what a bachelor is which we as humans know, but a computer
by instance wouldn’t.

Atomic sentences: the basic building blocks of which all other sentences are built.

We go from an atomic sentence to an uppercase letter (TFL), so

Dave is hungry  H

Lincoln was shot  S

Ferry is awake or Ferry is snoring  A or N

Dave is hungry and Ferry is snoring  H and N



HC 2: Truth Functional Logic

Atomic sentence: Transcribing English or Dutch or whatever language into capital letters.

Connectives: allows you to construct more difficult sentences.

Symbol Name of symbol Rough meaning
¬ Negation “It is not the case that …”
ꓥ Conjunction “both … and …”
“… but …”
“Although … , …”
ꓦ Disjunction “Neither … nor …”
→ Conditional “if … then …”
“… only if …”

, Michiel de Folter


↔ Biconditional “… if and only if…”


Negation:

C: Rex is a cat

Rex is not a cat  Paraphrasing  It is not the case that Rex is a cat

So: ¬C

Conjunction:

J: Ferry is a dog

V: Vlad is a dog

So: F ꓥ V

Disjunction:

Either Amanda will take a nap or she will come to the party

NVP

Exclusive “or”: excludes the possibility of both disjuncts to be true.

Inclusive “or”: allows both disjuncts to be true.

V always means inclusive or.

Conditional:

If Amanda is in Tilburg, then she is in the Netherlands

T: Amanda is in Tilburg  Antecedent

N Amanda is in the Netherlands  Consequent

TN

Use vs Mention:

“Dave” mentions the word “Dave”

When we talk about Dave we use the name Dave to talk about the person.

 “Dave” is the name of Dave.

Object language: The language we talk in.

Metalanguage: The language we use to talk about the object language.