Logic Semantics

Prenex Normal Form

When all quantifiers occur at the beginning of the sentence. Any sentence can be put into prenex normal form.

A |- C <-> A |= C

While provability (sound) and entailment (complete) are different notions, they are EXTENSIONALLY equivalent

Identity

a=b Not that a and b are indistinguishable but that they are the very same object.

How to determine when a conclusion follows from premises?

check if it's valid!

Definite descriptions

descriptions intended to designate UNIQUE objects

2 ways an argument might go wrong (invalid argument)

i) one or more of the premises might be false ii) the conclusion might not follow from the premises!!

Jointly Possible

sentences are jointly possible iff it's possible for them all to be true together (Jointly Satisfiable) There is some valuation which makes them all true. There is some interpretation which makes them all true.

Equivalent Sentences

2 sentences are necessarily equivalent iff their truth values agree on EVERY valuation. The columns under their main connectives are identical. (Provably Equivalent) The sentences can be derived from each other. A and B are true in exactly the same interpretations.

Contradiction

A is a contradiction iff it is false on every valuation. A sentence whose truth table only has Fs under the main connective. A sentence whose negation can be derived without any premises.

Modal Truth

A is true in a model M IFF for each world in M, A is true.

Empty Predicate

A predicate need not apply to anything in the domain. When F is an empty predicate, Ax(F(x) -> ...) is vacuosly true.

Necessary falsehood

A sentence is a necessary falsehood iff the sentence must be false. It could not possibly be true.

Necessary truth

A sentence is a necessary truth iff the sentence must be true. It could not possibly be false.

Contingent

A sentence is contingent iff it is not a necessary truth (tautology/theorem) or necessary false (contradiction). A sentence whose truth table contains both Ts and Fs under the main connective. It could be true or false upon different interpretations.

Tautology

A sentence whose truth table only has Ts under the main connective. It is true on every valuation. (Theorem) A sentence that can be derived without any premises. (Validity) A sentence is a validity iff it is true in every interpretation

Possible World

A world is only a possible world IFF anything logically necessary here is true in that world. If p is necessary, then in all worlds accessible to this one, p is true.

Sound

An argument is sound iff it is VALID and all of its premises are true.

Valid

An argument is valid IFF: - it is IMPOSSIBLE for all premises to be true and the conclusion false. - There's no valuation which makes the premises true and the conclusion false. - Truth table has no lines where there are all Ts under main connectives for the premises and an F under the main connective for the conclusion. - No world in any interpretation at which premises are T and conclusion is F. - One can derive the conclusion from the premises. (Validity of an argument is not about truth or falsity of the sentences in the argument)

Counter Interpretation

An interpretation in which all premises are true but the conclusion is false.

Bound variable

An occurrence of a variable x within scope of Ax or Ex

Sentence of FOL

Any formula of FOL that contains no free variables

Free variable

Any occurrence of a variable not bound

Names

Each name exactly one thing from the domain. A single member of the domain may be picked out by one, many or no names.

2 necessarily equivalent sentences, one of which is a necessary truth and one of which is contingent (T/F)

F

A jointly possible collection of sentences that contains a necessary falsehood (T/F)

F

A necessary truth that is contingent (T/F)

F

A valid argument that can be made invalid by addition of a new premise (T/F)

F

An invalid argument, the conclusion of which is a necessary truth (T/F)

F

Necessity

IFF A is true at EVERY world possible relative to w (NOT truth functional)

Possibility

IFF A is true at SOME world possible relative to w (NOT truth functional)

Semantic System K

N (p -> q), N p |= N q

Domain

Quantifiers range over domains. Must always include AT LEAST 1 thing.

Unsatisfiable/Inconsistent Sentences

Sentences which do not have a single line in their truth table where they are all true. (Provably inconsistent) Sentences from which one can derive a contradiction.

Satisfiable/Consistent Sentences

Sentences which have at least one line in their truth table where they are all true. Sentences from which one cannot derive a contradiction.

2 necessarily equivalent sentences that together are jointly impossible (T/F)

T

2 necessarily equivalent sentences, both of which are necessary truths (T/F)

T

A jointly impossible set of sentences that contains a necessary truth (T/F)

T

A valid argument that has 1 false premise and 1 true (T/F)

T

A valid argument that has a false conclusion (T/F)

T

A valid argument that has only false premises (T/F)

T

A valid argument with only false premises and a false conclusion (T/F)

T

A valid argument, the conclusion of which is a necessary falsehood (T/F)

T

An invalid argument that can be made valid by addition of new premises (T/F)

T (We can ALWAYS make an invalid argument valid by adding a contradiction into the premises. If premises contradict, it's impossible for all the premises to be true!)

Extension of a Predicate

The things a predicate is true of. FOL is an extensional language because it doesn't represent differences of meaning between predicates that have the same extension.