Logic Midterm Review

Provide two possible translations of the given sentence into our regimented form of English, and ultimately into our formal system. "Christina eats Cheerios, Sally eats bananas, and Jim drinks juice."

"Both Christina eats Cheerios and both Sally eats bananas and and Jim drinks juice." p.(q.r) "Both both Christina eats Cheerios and Sally eats bananas and Jim drinks juice." (p.q).r

The following pair of sentences is an example of which intuitive logical notion?

"Every Capricorn is happy" and "Lions are ferocious"

Translate the sentence into our regimented form of English. "I will be hungry provided that I don't eat a cheese pizza."

"If I don't eat a cheese pizza, then I will be hungry."

Translate the sentence into our regimented form of English. "I will be hungry only if I don't eat a cheese pizza."

"If I will be hungry, then I don't eat a cheese pizza."

Translate the sentence into our regimented form of English. "The ground will be wet if it rains."

"If it rains then the ground will be wet."

examples of non-truth-functional connectives

"Sarah believes that" "because", "so" "I hope that"

examples of truth-functional connectives

"and" "or" "it is not the case that"

"p unless q" is truth-functionally equivalent to:

"if not-q then p"

What type of a connective is negation?

"not" is a unary truth-functional connective

relationship between "p ⊃ (q r)" and "q ⊃(p⊃r)"

"p ⊃ (q r)" always has the same truth value as "q ⊃ (p⊃r)"

Conventions for dropping parentheses

- We can drop outermost parentheses -> (p.q) can be rewritten as p.q Order of precendence: - . v ⊃ and ≡

effect of uniform substitution

- preserves validity - does NOT preserve satisfiability - preserves unsatisfiability - preserves equivalence - does NOT preserve inequivalence - preserves implication

an argument is in Premise-Conclusion form when:

- the premises are listed, one to a line - then there is a horizontal line - finally, the conclusion is listed The conclusion may optionally be signaled with a "therefore", "so", or "hence"

-(p v q) is equivalent to:

-p . -q

--(p . q) is equivalent to:

-p v -q

the term "logically valid" refers to:

arguments

two properties of disjunction

commutative: A v B is equivalent to B v A associative: (A v B) v C is equivalent to A v (B v C)

two properties of conjunction

commutative: A.B is equivalent to B.A associative: (A.B).C is equivalent to A.(B.C)

recursive definition

definition in which a set of objects is characterized in terms of some basic objects and some general relations - The set is thought of as containing the basic objects and all of the objects that can be constructed from the basic objects using the generation relations. - Here, the basic objects are sentence letters.

disjunctive normal form

disjunction of (conjunctions of sentence letters and negations of sentence letters)

How to show that an argument is logically valid?

find a proof of the conclusion from the premises

Modus Ponens

if P is true, and if P implies Q (P -> Q), then P is true

Logical equivalence refers to:

pairs of sentences

Interchange

replacing one schema with another larger schema; sometimes called "replacement" - interchange of equivalent schemas preserves validity, satisfiability, and unsatisfiability --> useful for allowing us to convert schemas to equivalent schemas with specific, often simpler forms

formal analogue to a logically indeterminate sentence

satisfiable but invalid schema

formal analogue to a consistent set of sentences

satisfiable set of schemas

in Model Theory, validity refers to:

schemas

compound sentences

sentences that contain sentential connectives ex: "It is not the case that Time ate the last cookie"

simple sentences

sentences that do not contain sentential connectives ex: "Tom was hungry"

in Model Theory, satisfiability refers to:

sets of schemas

how to show a schema is not valid

show that the schema is false on at least one of the truth assignments

truth condition

specifies the conditions under which the statement is literally true (and so also tells us the conditions under which it is false)

A schema is valid just in case it is implied by the empty set of premises.

that is, on every truth assignment, A is true

Is the material conditional commutative or associative?

the material conditional is NOT commutative NOT associative

conclusion

the singled out sentence that is (purported to be) supported by the other sentences -- the premises.

formal analogue to a logically false sentence

unsatisfiable schema

formal analogue to inconsistent set of sentences

unsatisfiable set of schemas

equivalent schemas

Two schemas are equivalent if there is not truth assignment on which they have different truth values.

inequivalent schemas

Two schemas are inequivalent if there is a truth assignment on which they have different truth values.

Is unsoundness monotonic?

It's not! dw about this though honestly

Is soundness monotonic?

It's not. If you take a sound argument and add a false claim to the premises, it will no longer be false.

example of an argument that is NOT cogent

Joe Biden is President. Therefore, Joe Biden is President.

Logical validity has something to do with the _________ of an argument, rather than its __________________________.

Logical validity has something to do with the _________ of an argument, rather than its __________________________. FORM, SUBSTANTIVE CONTENT

how we will use statements

Most (almost all) sentences used in ordinary discourse are not statements. If we restricted logic to only apply to statements, logic wouldn't be very useful outside a few specialized circumstances -maybe mathematics and fundamental physics. So, a better idea is to say that we're going to focus on cases where we treat ordinary sentences as statements, by imagining them "to have been uttered by a single speaker at a single time in a conversational setting that uniformly resolves any ambiguities." (Goldfarb, section 1)

Our formal language will directly model _________________________.

Our formal language will directly model _________________________. the behavior of truth-functional connectives.

logically equivalent

Two sentences are logically equivalent just in case it is not logically possible for the sentences to differ in their truth values.

logically inequivalent

Two sentences are logically inequivalent just in case they are not logically equivalent.

How does one show that an argument is logically invalid?

We can show an argument is logically invalid by finding a coherent possibility in which the premises are true and the conclusion is false.

ancestors

We can think of schema as having "ancestors" -- all of the simpler schemas that it was constructed out of

We focus on the __________________, rather than the ___________________ conditional.

We focus on the indicative, rather than the subjunctive/counterfactual conditional.

material conditional

a connective with this truth table

expression

a finite sequence of symbols ex: "(p.q..(","p.q"

interpretation

a function that maps each sentence letter to a statement of English

schema

a grammatical expression, defined as follows: - any sentence letter is a schema - if α is a schema, so is "-α" - If α and β are schemata, so are "(α.B)", "(αvB)", and "(α≡B)" - nothing else is a schema

logical truth refers to:

a sentence

statement

a sentence that is determinately true or determinately false, independent of the circumstances in which it is used, and independent of the speaker, audience, time, place, and conversational context - must be a declarative sentence - must be grammatical - must function as an assertion - cannot be a question or a command - cannot be ambiguous - cannot be vague - cannot contain indexicals or demonstratives (e.g. "I", "now", "here") - cannot contain tensed language (e.g. "later", "earlier", "did") - cannot contain other context-dependent language (e.g. "Timmy is tall") * Most sentences that are used in ordinary discourse are NOT statements.

Describe the sentence below using intuitive logical notions. Brown University is located in Providence, Rhode Island.

a sentence that is true, but not logically true

Logical consistency refers to:

a set of sentences

argument

a set of sentences, one of which is singled out to be the conclusion. The sentences that AREN'T the conclusion are the premises.

sentential connective

a word or phrase that can be used to take a sentence or sentences of English and connect them to form a new sentence ex: "and", "or", if... then", "it is not the case that", "because"

Soundness refers to:

an argument

parse tree

an upside down family tree for a schema

Explain why logical validity has the property of monotonicity.

The property of monotonicity says that given any logically valid argument, if we add more premises to the argument, the resulting argument will remain logically valid. Why? If it's not logically possible for the premises of the original argument to be true and the conclusion to be false, then it is not logically possible for the premises of the original premises plus additional premises to be true and the conclusion to be false.

Logical equivalence, characterized in terms of logical validity

A pair of sentences is logically equivalent just in case any argument with one sentence as the premise and the other premise as the conclusion or vice versa is logically valid.

how can we show a formal analogue of logical truth?

A schema has the formal analogue of logical truth just in case it is true on every interpretation. --> no matter what the sentence letters are mapped to, the schema is mapped to a sentence that is true

formal analog of logical truth

A schema is valid if it is true on all truth assignments.

valid schema

A schema is valid if there is not truth assignment on which it is false.

logically false

A sentence is logically false if and only if it is not logically possible for the sentence to be true.

logical truth

A sentence is logically true if and only if it is not logically possible for the sentence to be false.

logical truth, characterized in terms of logical validity

A sentence is logically true just in case any argument with the sentence as its conclusion is logically valid.

truth-functionally

A sentential connective is used truth-functionally just in case it is used to generate a compound sentence from one or more sentences in such a way that the truth value of the compound sentence is wholly determined by the truth value(s) of the sentence(s) out of which it is composed.

implies

A set of schemas P implies another schema B if there is not truth assignment on which all of the schemas in P are true and A is false.

satisfiable

A set of schemas is satisfiable if there is a truth assignment on which all of the schemas are true.

unsatisfiable

A set of schemas is unsatisfiable if there is no truth assignment on which all of the schemas are true.

logically consistent

A set of sentences is logically consistent if and only if it is logically possible for the members in the set to be jointly true. Each sentence doesn't have to be guaranteed to be true; rather, they just need to be able to be JOINTLY true.

logical consistency, characterized in terms of logical validity

A set of sentences is logically consistent just in case there is an argument with the set of sentences as premises that is not logically valid.

cogent

An argument is cogent when it can be used to rationally convince someone of its conclusion

logically invalid

An argument is logically invalid if and only if it is not logically valid.

sound

An argument is logically sound just in case both: (i) the argument is valid, and (ii) the premises are all true

logically valid

An argument is logically valid just in case it is not logically possible for the premises to be true and the conclusion to be false. That is, an argument is logically valid just in case the truth of the premises logically guarantees the truth of the conclusion.

logical validity, characterized in terms of logical inconsistency

An argument is logically valid just in case the set of sentences containing the premises and the negation of the conclusion is logically inconsistent.

alternate conception of logical validity:

An argument is logically valid just in case there is a proof of the conclusion from the premises.

unsound

An argument is unsound if and only if it is not sound. That is, an unsound argument is either invalid or at least one premise is false

Given a valid argument, any other argument with _____________________ will be valid.

Given a valid argument, any other argument with _____________________ will be valid. THE SAME LOGICAL FORM

monotonicity

Given any logically valid argument, if we add more premises to the argument, the resulting argument will remain logically valid.

proof by induction

Given any recursive definition of a set, there is a corresponding method of proof. If we can show that both: (i) all the basic objects have a property, and (ii) the generating relations preserve the property, then we can conclude that (iii) all members of the set have the property.

Translate the sentence into our regimented form of English. "I will be hungry unless I eat a cheese pizza."

If I don't eat a cheese pizza, then I will be hungry."

Steps for translating from English into our formal system

1. Take an ordinary English sentence. 2. Paraphrase with a sentence that is less ambiguous, less vague, and less context-dependent. 3. Paraphrase this sentence with a sentence in a more regimented form of English that can be directly handled by our logical system. 4. Translate into symbols.

2 special cases of logically valid arguments

1.) Any argument whose premises are logically inconsistent is logically valid. In this case, it is not logically possible for the premises of this argument to be true. Thus the argument is valid because it is not logically possible for the premises of the argument to be true and the conclusion to be false. 2.) Any argument whose conclusion is a logical truth is logically valid. In this case, it is not logically possible for the conclusion to be false. Thus the argument is valid because it is not logically possible for the premises of the argument to be true and the conclusion to be false.

3 ways to disprove expressive adequacy

1.) show that all connectives are true on the first line of the truth table 2.) show that all connectives are false on the last line of the truth table 3.) show that all connectives are true on an even number of rows of the truth table

falsity condition for the conditional

A conditional is false just in case its antecedent is true and its consequent is false, and it is true otherwise.

truth condition for the conditional

A conditional is true just in case either its antecedent is false or its consequent is true.

truth condition for disjunction

A disjunction is true just in case at least one of the disjuncts are true. It is false otherwise.

inclusive vs exclusive or

inclusive: A v B is true just in case either A is true or B is true, or both. exclusive: A v B is true just in case either A is true or B is true, but not both.

deductions

intended to formalize a psychologically natural kind of reasoning -reasoning with assumptions or suppositions = a finite sequence such that every element of the sequence is a pair containing a finite set of schemas (the set of operative suppositions) and a single schema (the schema taken to hold given the operative suppositions)

Which logical notion does the set of sentences below demonstrate? {"PHIL 640 is taught in Room 130", "Tom Cruise is famous", "People have walked on the moon"}

logical consistency

Which logical notion does the set of sentences below demonstrate? {"John is very happy", "John is not at all happy", "Texas is bigger than Rhode Island"}

logical inconsistency

The following pair of sentences is an example of which intuitive logical notion? "Every Capricorn is happy" and "There is no Capricorn that is not happy"

logically equivalent sentences

The following sentence is an example of which intuitive logical notion? I went to the park yesterday and I didn't go to the park yesterday.

logically false sentence

The following sentence is an example of which intuitive logical notion? Either the sky is blue or the sky is not blue.

logically true sentence

The following sentence is an example of which intuitive logical notion? The sky is the sky.

logically true sentence