School & Chat

Sentential Logic (SL)

Sentences are things that are capable of being True or False

Truth Value {T, F}

Pittsburgh is the capital of PA. ✔
Where is Pittsburgh? ✘

An argument is a set of 2 or more sentences, one of which is designated to be the conclusion, and the others are the premises.

Bad:
Joe is smart
Joe is silly

Good:
All humans are mortal.
Socrates is a human.
Socrates is mortal.

An argument is deductively valid iff it is not possible for the premises to all be true and the conclusion false.
An argument is sound iff (i) it is valid and (ii) its premises are all true.

An argument is logically consistent iff it is possible for every member of the set to be true (together).

{Joe is silly, Joe is smart}

A sentence is logically true iff it is not possible for the sentence to be false.
A sentence is logically false iff it is not possible for the sentence to be true.
A sentence is logically indeterminate iff it is neither logically true nor logically false.
* Any argument that has a logically true conclusion is valid. Any argument that has a logically false premise is valid.
Two sentences are logically equivalent iff it is not possibel for one to be true and the other to be false.

Sentence Logic

basic components are simple or atomic sentences

Socrates is silly.
Aristotle is smart.

A sentence connective is used truth-functionally iff it is used to separate a compound sentence from one or more sentences in such a way that the truth value of the compound sentence is completely determined by the truth values of the compound.

Truth-value assignment is the assignment of elements of {T, F} to every atomic sentence of SL.

Conjunction ("and") -- ^ or &




Disjunction ("or") -- ˅




Negation ("it is not the case that") -- ~ or ¬




Material conditional ("If, then") -- → or




Biconditional ("if and only if") -- ↔