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Video 3.1: Logical Languages (4)

- 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.

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Video 3.2: Building Atomic Sentences (3)

- 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.

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Video 5.1: Introducing Arguments (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.

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Video 5.2: Proofs: Basic Principles (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.

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Video 5.3: Nonconsequence & Counter-Examples (4)

- 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.

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Video 6.1: Introducing the Identity Symbol (3)

- 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.

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

- 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.

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Video 7.1: Compound Sentences & Truth Tables (4)

- 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.

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Video 7.2: Conjunction & Disjunction (4)

- 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.

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Video 7.3: Logical Equivalence (3)

- 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

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Video 9.1: Introducing Necessity & Possibility (4)

- 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.

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Video 9.2: Building Truth Tables (4)

- 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.

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Video 10.1: Tautologies & Logical Truths (3)

- 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).

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Video 10.2: Tautological Equivalence (4)

- 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)

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Video 10.3: Tautological Consequence (2)

- 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.