Video Lecture Key Points

Question 1

Video 3.1: Logical Languages (4)

Answer 1

• First order logic is a framework for building logical languages.
• A logical language is a tool for describing a particular domain precisely and unambiguously.
• Logicians study the logical relations between sentences in logical languages.
• These logical relations are determined by the logical structure of the sentences and the logical concepts they contain.

Question 2

Video 3.2: Building Atomic Sentences (3)

Answer 2

• Atomic sentences are formed by putting a predicate of arity n in front of n names (enclosed in parentheses and separated by commas).
• Atomic sentences are built from the identity predicate, =, using infix notation: the arguments are placed on either side of the predicate.
• The order of the names is crucial in forming atomic sentences.

Question 3

Video 5.1: Introducing Arguments (3)

Answer 3

• An argument is logically valid iff the conclusion is a logical consequence of the premises.
• The conclusion of an argument is a logical consequence of the premises iff it is not possible for the premises to be true and the conclusion to be false.
• An argument is sound iff it is valid and all of its premises are true.

Question 4

Video 5.2: Proofs: Basic Principles (4)

Answer 4

• A proof is a series of truth-preserving steps that lead from a set of premises to a conclusion.
• A correct proof demonstrates that the conclusion is a logical consequence of the premises.
• Proofs can be formal or informal.
• Each step in a formal proof is explicitly justified by rules governing the behavior of logical symbols in our formal language.

Question 5

Video 5.3: Nonconsequence & Counter-Examples (4)

Answer 5

• In an invalid argument, it is possible for the premises all to be true and the conclusion to be false.
• And so, in order to show that an argument is invalid you need to construct a scenario or possible world in which the premises are all true and the conclusion is false.
• This scenario or world is called a counterexample, or a counter-model.
• Except in the limited environment of Tarski’s World, proofs of nonconsequence are informal, rather than formal.

Question 6

Video 6.1: Introducing the Identity Symbol (3)

Answer 6

• Each step in a formal proof has to take place in accordance with logical rules that govern the behavior of logical symbols in our formal language.
• The logical rules that govern the behavior of logical symbols in our formal system are rigorous versions of basic principles that we employ in everyday, informal reasoning.
• Those basic principles in everyday reasoning govern the behavior of the logical expressions corresponding to our logical symbols.

Question 7

Video 6.3: Analytical Consequence & the Ana Con Rule (4)

Answer 7

• Analytic consequence is a special kind of logical consequence.
• Analytic consequence is logical consequence that holds in virtue of the meanings of non-logical words.
• The Ana Con rule in Fitch allows you to derive sentences that are analytical consequences of earlier sentences in the proof.
• The Ana Con rule only works for predicates in the blocks language.

Question 8

Video 7.1: Compound Sentences & Truth Tables (4)

Answer 8

• There are three Boolean connectives, corresponding to the English logical expressions, AND, OR, and NOT.
• We can apply these Boolean connectives to atomic sentences to construct compound sentences.
• Compound sentences constructed in this way are truth-functional. This means that their truth value is completely determined by (is a function of) the truth values of the atomic sentences from which they are built up.
• The meaning of each Boolean connective is given by its truth table.

Question 9

Video 7.2: Conjunction & Disjunction (4)

Answer 9

• Conjunction and disjunction are both binary logical connectives, which means that they connect two sentences.
• The conjunction A ∧ B is True only when the two conjuncts, A and B, are both True.
• The disjunction A ∨ B is False only when the two disjuncts are both False.
• Unlike “or” in English, the disjunction A ∨ B in FOL is True when both A and B are true.

Question 10

Video 7.3: Logical Equivalence (3)

Answer 10

• Two sentences are logically equivalent when there is no logically possible scenario on which they have different truth values.
• The two DeMorgan Laws show that ¬(A ∧ B) is logically equivalent to (¬A ∨ ¬B) and ¬(A ∨ B) is logically equivalent to (¬A ∧ ¬B)
• Both DeMorgan Laws can be informally proved from the truth tables for conjunction and disjunction

Question 11

Video 9.1: Introducing Necessity & Possibility (4)

Answer 11

• A sentence is logically possible iff it is not contradictory to suppose that it might be true (i.e. iff its truth would be consistent with the laws of logic)
• A sentence is logically necessary iff it is contradictory to suppose that it might be false (i.e. iff its falsity would be inconsistent with the laws of logic)
• A sentence is TW-possible iff there is a world in Tarski’s World in which it is true.
• A sentence is TW-necessary iff it is true in every world in Tarski’s World.

Question 12

Video 9.2: Building Truth Tables (4)

Answer 12

• Every compound sentence is built up from atomic sentences with Boolean connectives has a main connective that defines the sentence’s basic logical category.
• Truth tables are used to explain what truth value a sentence will have for each possible assignment of truth values to its atomic sentences.
• If a compound sentence is built up from n atomic sentences, then its truth table will have 2^n lines.
• There is an algorithm (a recipe) for writing down the lines of a truth table.

Question 13

Video 10.1: Tautologies & Logical Truths (3)

Answer 13

• Every tautology is a logical truth, but not every logical truth is a tautology.
• A tautology is logically true in virtue of the meanings of the Boolean connectives.
• This means that there are logical truths that are not tautologies (e.g. a = a ∧ b = b, which is true in virtue of the meaning of the identity symbol).

Question 14

Video 10.2: Tautological Equivalence (4)

Answer 14

• Sentences A and B are logically equivalent iff they have the same truth values in all possible circumstances.
• Sentences A and B are tautologically equivalent iff they have exactly the same pattern of truth values under the main connectives.
• If A and B are tautologically equivalent then they are most certainly logically equivalent.
• But two sentences can be logically equivalent without being tautologically equivalent (e.g. a = b ∧ Smaller (a,c) and a = b ∧ Smaller (b,c)

Question 15

Video 10.3: Tautological Consequence (2)

Answer 15

• A sentence (conclusion) is a tautological consequence of a set of sentences (premises) iff there is no line of the joint truth table that assigns True to all the premises and False to the conclusion.
• A conclusion can be a logical consequence of a set of sentences without being a tautological consequence of those sentences.