PHIL 279

Arguments

composed of sentences, specifically STATEMENTS - True or false

Statements

Premises meant to support a conclusion

Validity

An argument is valid if it is impossible for the premises to be true and the conclusion false

see examples

invalidity/counterexample

when the premises of an argument can be true while the conclusion is false

Soundness

A sound argument is a valid argument whose premises are true

we are interested in the conditions of the argument and whether it is valid, not if the statements leading to the conclusion themselves are true

see examples

Contingency

A statement where it is possible for it to be true and false at the same time.

Joint possibility

it is possible for all of the statements to be true simultaneously

Joint impossibility

It is impossible for the set of statements to be true simultaneously

Necessary falsehoods

some statements that are always false

Necessary truths

Some statements that are always true

Equivalence

Two sentences are equivalent when, in all situations, they are both true or both false.

Counterexample of this is when one sentence is true and the other is false.

Five phrases for propositional logic

• and

• or

• not

• if..then

• if, and only if

also known as logical constants, connectives, or operators.

Logical Symbols (connectives/constants)

⊥ (The False)
¬ (Not; it is not the case that)
∨ (Or; either or)
∧ (And)
→ (If, then; only if; if)
↔ (If, and only if; by definition)

Expressions

Strings of symbols - A((B¬ → C

Not necessarily a formula of the language

Formulas

Expressions are formulas/sentences of our object language when, by definition:

• It is an atom

• It is ⊥

• If A is a formula, so is ¬A

• If A, B are a formulas, so is (A∨ B)

• If A, B are a formulas, so is (A∧ B)

• If A, B are a formulas, so is (A → B)

• If A, B are a formulas, so is (A ↔ B)

Main connective/logical operator

Last connective of the formula in the construction

Subformula: Any formula from the expression

operator/connective scope

The subformulas of the formula of which the connective is the main operator