Classical Logic and Its Rabbit-Holes (Lande)
Solutions: https://hackettpublishing.com/pdfs/ClassicalLogic_SelectedExerciseSolutions_ch_1-8_UPDATED.pdf
112 flashcards · Mathematics · by lakesare
Ardberg - trekking guide, Egvalt - evil, North Zedmenistan - country,
Zompadog Pass - mountain, Coco - president of A.N.Z.A. (Association of North Zedmenistan Actuaries), Dagbar - 90 year old dzo-wrestler
Who's a dzo?
A dzo (Tibetan མཛོ་ mdzo) is a hybrid between the yak and domestic cattle.
What does an argument consist of?
Argument is string of sentences: [premise, ..., premise, conclusion]. premise premise premise premise... ___________ conclusion
This is an argument form.
Any argument that has the same form as this argument-form is a valid argument. Validity is purely a formal notion.
What's a sentence?
Something that's either true or false. (== proposition, == statement)
Which of these is a sentence?
(1) sentence
Propositional constant VS propositional variable?
propositional constant P stands for exactly one proposition. Designated with A-O.propositional variable P stands for any proposition whatsoever. Designated with P-Z.
What's Platonic realism? Roman copy of a portrait bust of Plato by Silanion for the Academia in Athens (c. 370 BC).
Platonic realism is the philosophical position that universals or abstract objects exist objectively and outside of human minds. Reality comprises not merely bats and cats and rats, but also numbers and propositions and arguments.
What's a connective?
Connective - word (and, or, not, iff, if-then) that operates on one (or more) sentences to generate a new sentence.
What's a bi-conditional?
Sentence generated with iff (↔) connective.
[Connective names] What's: A B
A - left conjunct B - right conjunct
[Connective names] What's: A B
A - left disjunct B - right disjunct
[Connective names] What's: A B
A - antecedent B - consequent
How can we interpret: a propositional symbol P a proposition "our proposition"
We interpret a propositional symbol:P means "our proposition" We interpret a proposition:"our proposition" is True.
Atomic VS molecular sentence?
Atomic - 0 connectives. Molecular - 1 or more connectives.
What's the difference between ↔ (iff) and ≡ (logical equivalence)?
𝑃 ↔ 𝑄 is a sentence that could be either true or false. 𝑃 ≡ 𝑄 means that 𝑃 ↔ 𝑄 is always a true biconditional. (so, 𝑃 and 𝑄 have the same truth value no matter what).
What's a good analogy for finding the truth table for a → b?
We were just born. we understand logic perfectly well however, we know nothing about the world! Our parent tells us a → b. And then various things happens to us in the world. For example, we find out a is true, but b is false. Then we know our parent is a liar - he told us a false thing (a → b)!
Give an example of a non-truth-functional connective?
Because. "because" connective is not truth-functional.
When is argument valid VS sound? Argument is string of sentences: [premise, ..., premise, conclusion].
valid - "if premises are true → the consclusion is true". sound - valid AND each of its premises is true
When is an argument valid? Argument is string of sentences: [premise, ..., premise, conclusion].
An argument is valid iff "if premises are true → the consclusion is true". This never happens:
What's a counterexample when it comes to argument validity?
It's the truth-value interpretation (A, B, C is T, F, F) under which all premises are T, but the conslusion is F. Counterexample
All sentences can be divided into: tautology - ___________ contradiction - ___________ contingent - ___________
tautology - sentence that is T for every interpretation contradiction - sentence that is F for every interpretation. contingent - T at least once AND F at least once.
How is contingent sentence special? All sentences can be divided into: tautology contradiction contingent
To know the truth-value of a contingent sentence you have to examine the outside world.
When is a sentence consistent? All sentences can be divided into: tautology contradiction contingent
When it's either a tautology, or contingent. It's T for at least one interpretation! tautology contradiction contingent Don't think of a consistent sentence as a fourth kind of sentence: it's nothing more than a sentence that isn't a semantic contradiction.
[Proving argument validity: long method] How would you prove that this argument is valid (long method)?
1. Translate it into the logical notation 2. Contstruct the truth-table Observe that in all rows where both premises are true; the conclusion is true too.
[Proving argument validity: short method] How would you prove that this argument is valid (short method)?
1. Assign premises: T, conclusion: F 2. Fill in all other truth values. 3. If there is no contradiction - we found the interpretation in which premises: T, conclusion: F - this is a counterexample, and the argument is invalid!
[Formal systems: Propositional Logic] A formal system consists of: formal language { symbols } { formation rules } deductive apparatus { axioms } { derivation rules }
[Formal systems] Formula VS Well-Formed Formula? A formal system consists of:formal language: { symbols }, { formation rules }deductive apparatus: { axioms }, { derivation rules }
Formula - any string of symbols in this language. Well-Formed Formula - a string of symbols that follows the formation rules.
What's wrong with this logical translation of the argument?
The argument is valid, however it's invalid in our translation - we can effortlessly produce a counterexample! This happens because our translation is insufficiently detailed.
Suppose there exists the ⇑ connective. How to express it in our 5 basic connectives?
Find its disjunctive normal form.
[Formal systems] Why do we use △ and ◯ when we describe the formation rules instead of, e.g., P and Q? A formal system consists of:formal language: { symbols }, { formation rules }deductive apparatus: { axioms }, { derivation rules }
We need to use the metalanguage in order to describe the object language.
Give a good real-life example for object language and metalanguage.
In the Introductory French course: object language - French metalanguage - English
[Formal systems] When is a formal language expressively complete? A formal system consists of:formal language: { symbols }, { formation rules }deductive apparatus: { axioms }, { derivation rules }
When: any truth-functional connective we just made up can be translated into a wff
[Formal systems: Propositional Logic] What's the name of this connective?
Sheffer Stroke (NAND).
[Formal systems: Propositional Logic] How would you show that Sheffer Stroke (| or NAND) is the only connective we need in Propositional Logic?
We should show that all 3 dnf (disjunctive normal form) connectives (–, ∧, ∨) can be represented by the Sheffer Stroke (|). – ∧ ∨
[Formal systems: Propositional Logic] What 2 types of assumption derivation rules (A) are there?
Assumptions can be: simply given introduced by us
[Formal systems] Argument VS sequent?
Argument is semantic Double turnstile is simply another way of expressing ∴Sequent is syntactic
[Formal systems] What's this formula called?
A sequent.
[Formal systems] How would this sequent look in a vertical notation?
[Formal systems] How would this sequent look in a horizontal notation?
[Formal systems: Propositional Logic] How many derivation rules are there in Propositional Logic deductive apparatus? A formal system consists of:formal language: { symbols }, { formation rules }deductive apparatus: { axioms }, { derivation rules }
11 derivation rules: 2 per each connective, and One allows you to eliminate a molecular, and one allows you to introduce a molecular.Assumption
[Formal systems: Propositional Logic] What do these stand for: ∧E ∧I A formal system consists of:formal language: { symbols }, { formation rules }deductive apparatus: { axioms }, { derivation rules }
∧E - conjunction Elimination ∧I - conjunction Introduction
[Formal systems: Propositional Logic] When the derivation is truth-preserving? A formal system consists of:formal language: { symbols }, { formation rules }deductive apparatus: { axioms }, { derivation rules }
When there is no interpretation that makes each wff T, and the conclusion F.
[Formal systems: Propositional Logic] What's another name for the →E derivation rule? Conditional elimination
→E is Modus Ponens.
[Formal systems] How to show that an argument is valid VS a sequent is provable?
an argument is valid - build the Truth-Table. Or use the Short Method.a sequent is provable - use the derivation rules to syntactically derive the last wff.
[Formal systems] What do first-column numbers mean?
The A (Assumption) line numbers the line is ultimately derived from.
[Formal systems] What do second-column numbers mean?
Just a line number.
[Formal systems] How to decode the 4––E line?
Eliminate––(line 4) This line is generated by Eliminating –– from line 4.
[Formal systems] What's a good analogy for: top-down derivation bottom-up derivation
top-down derivation - simplification bottom-up derivation - wishful thinking
[Formal systems: Propositional Logic] Name 2 example derivation rules. A formal system consists of:formal language: { symbols }, { formation rules }deductive apparatus: { axioms }, { derivation rules }
[Formal systems: Propositional Logic] What's the difference between these derivation rules? A∧E ∧I∨E ∨I→E →I↔E ↔I––E ––I Propositional Logic has 11 derivation rules.
green - we don't need an assumption purple - we need to introduce our own assumption Though we discharge those assumptions in the end!
[Formal systems: Propositional Logic] The introduction of a conditional (→I) is one of the derivation rules that requires us to introduce our own assumption. What's a good analogy for what this rule is doing (line 5)?
→I(2, 4) records horizontally what line 2 and line 4 record vertically.
[Formal systems: Propositional Logic] Explain what constitutes wishful thinking (bottom-up approach) for this sequent.
assume → wish We want Q→R, therefore we temporarily assume P.Assume the antecedent, aim for the consequent.
[Formal systems: Propositional Logic] What's the gist of ∨E (disjunction elimination)?
a ∨ b a → bullshit b → bullshit bullshit ✨
[Formal systems: Propositional Logic] All purple rules require A (Assumption) - which rules do they all conduce to though? A∧E ∧I∨E ∨I→E →I↔E ↔I––E –I Propositional Logic has 11 derivation rules.
To →I (if-then introduction), and Vertical to horizontal.–I (not introduction) Suppose A, and show that this leads to a contradiction.
[Formal systems: Propositional Logic] When is the set of wffs satisfiable?
When there is at least one interpretation that makes each wff T.
[Formal systems: Propositional Logic] What does no line number (to the left) mean?
This is a tautology - wff that's true no matter what our environment is.
[Formal systems: Propositional Logic] What's another name for the –I rule? A∧E ∧I∨E ∨I→E →I↔E ↔I––E –I
Reductio ad Absurdum
[Formal systems: Propositional Logic] What's the form of the syntactic contradiction? A∧E ∧I∨E ∨I→E →I↔E ↔I––E –I
△ ∧ –△
[Formal systems: Propositional Logic] How would you perform the reductio ad absurdum (–I) derivation of this sequent. A∧E ∧I∨E ∨I→E →I↔E ↔I––E –I
[Formal systems] Theorem VS sequent?
Theorem is a sequent resting on no assumptions.
[Formal systems] What's the law of excluded middle in logic?
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true (there is no middle value).
[Formal systems] What does S.I. stand for in S.I.-generator?
Substitution-Instance
[Inserting previously proved sequents] What does TI stand for?
Theorem Introduction
[Inserting previously proved sequents] What does SI stand for?
Sequent Introduction
[Formal systems] Given this wff and this Substitution-Instance-generator (S.I.-generator) - what's the Substitution Instance (S.I.)?
[Inserting previously proved sequents] What does TISI stand for?
Theorem Introduction Substitution-Instance
[Inserting previously proved sequents] What does SISI stand for?
Sequent Introduction Substitution-Instance
[Formal systems] What would it mean to have a formal system with not enough of derivation rules?
There would be arguments which are valid (which can be shown with the Truth Table) but impossible to derive (with the derivation rules).
What's the main operator in a wff?
The main operator is the last operator added to a wff according to the formation rules. Every wff has one main operator.
Transformation VS derivation?
Transformation - each wff is logically equivalent. Derivation - each wff is simply implied by the predecessor.
Name 3 forms the quantified wffs can take.
prenex - the rightmost quantifier scopes entire wff purified - none of the quantifiers falls within the scope of another quantifier mixed - neither prenex nor purified.
What's the free variable in this wff?
x.
What's the free variable in this wff?
z.
What's another name for Predicate Logic?
Quantification Theory. First formulated by Gottlob Frege (1848-1925)
In what way is Predicate Logic an extension of Propositional Logic?
Formal language of Predicate Logic includes the formal language of Propositional Logic. Deductive apparatus of Predicate Logic includes the deductive apparatus of Propositional Logic.
What's meant by Predicate in Predicate Logic?
Predicates designate: properties assigned to the individual objects relations that individual objects stand in with respect to each other
[Predicate Logic] How does Ta read?
a has property T
[Predicate Logic] Give 2 examples of quantifiers.
∀ - universal quantifier ∃ - existential quantifier
[Predicate Logic] In what way is this algebraic identity a quantified statement?
Predicate VS proposition VS quantifier?
predicate - a function taking some value proposition - sentence that's either true or false quantifier - ∀ (universal quantifier), ∃ (existential quantifier)
"Some tyrants are clowns" is numerically imprecise. How many exactly do we take "some" to mean?
At least 1. Numerical precision is important!
[Predicate Logic] What is this?
The Square of Opposition. The origin of the square can be traced back to Aristotle making the distinction between two oppositions: contradiction and contrariety. However, Aristotle did not draw any diagram. This was done several centuries later by Apuleius and Boethius. Depiction from the 15th century
[Predicate Logic] Free VS bound variable?
[Predicate Logic] Open VS closed wff?
Open wffs - at least one variable is free (neither true nor false). Closed wffs - all variables are bound (do possess truth-value)
[Predicate Logic] Which of these wffs has a truth-value?
Has truth-value: This wff is closed, all x are bound to the universal quantifier. Doesn't have truth-value: The last x is not bound to any quantifier, and thus this wff is open.
[Predicate Logic] Monadic VS polyadic predicates?
monadic - Px polyadic - Pxyz
[Predicate Logic] How would you translate "Someone loves everyone"?
Tackle the leftmost quantifier, and gradually translate into Loglish:
[Predicate Logic] Why are proper names NOT like quantifiers?
(11) == (12), however (13) != (14).
[Predicate Logic] Name an example of an open wff that bears truth-value.
Tx ∨ –Tx, Tx → Tx, etc.
[Predicate Logic] What's a good approach to translate the complex sentence, such as "everyone loves at least one person who loves everyone"?
Top-down, and use Loglish!
[Predicate Logic] What does L2⭑ mean?
L accepts 2 variables (polyadic predicate).
[Predicate Logic] What does ≠ denote in predicate logic?
It denotes the I2⭑ predicate.
[Predicate Logic] What predicate can helps us express these sentences? Everyone except Dagbar shrieked. Ardberg slew exactly two dragons. The dog that bit Bobo bit Coco.
Identity predicate (I2⭑).
[Predicate Logic] Name 2 possible interepretations of (1). Which interpretation do we choose in Predicate Logic?
Every trekker except Ardberg is a Platonist, and Ardberg is NOT a Platonist. This is the wrong meaning.Every trekker except Ardverg is a Platonist, and we DON'T KNOW whether Ardberg is a Platonist. This is the right meaning. Minimalism!
[Predicate Logic] How would you approach translating this? Tx - x is a tyrant
[Predicate Logic] How would you approach translating this?
[Predicate Logic] What's meant by an unrestricted VS restricted domain?
Unrestricted domain - all things in the universe Restricted domain - a particular set of objects (e.g. "eels, worms, and snakes")
[Predicate Logic] In a 3-object domain (Ardverg a, Bobo b, Coco c), what are these equivalent too: (∀x)Sx (∃x)Sx
(∀x)Sx - Sa ∧ Sb ∧ Sc (∃x)Sx - Sa ∨ Sb ∨ Sc
[Predicate Logic] What does QN stand for?
Quantifier Negation