PHIL 240 Midterm Study Guide PDF

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All lectures used up until midterm notes and review...

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PHIL 240 MIDTERM STUDY GUIDE: -

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Logic - Logic studies the laws that govern how people ought to reason - 2 types of logic: inductive and deductive logic Deductive: goal is deductive certainty - Aims to show that a conclusions follows with certainty from a set of premises - Conclusion follows logically from the premises / is a logical consequence of the premise - It’s not possible for the premises to be true and the conclusion to be false - Truth, consequence, validity, entailment, logical form, proof Inductive: the premises support the conclusion, even though the conclusion does not follow logically from the premises - It is perfectly possible for the premises to be true and the conclusion false - Cannot aim for certainty - Probability, risk, uncertainty, likelihood, expectation First Order Logic - Allows you to express things that cannot be expressed in propositional logic - In addition to Boolean logical concepts, FOL includes logical concepts such as ALL and SOME Propositional Logic - The fundamental principles studies in prop logic are principles that govern the behavior of Boolean logical concepts such as AND, OR, and NOT - These logical consequences are fundamental to our reasoning Boolean connectives - AND, OR, and NOT themselves are the Boolean connectives - The functions corresponding to AND, OR, and NOT are called the Boolean functions. Atomic sentences - Simplest kind of sentence that we can construct, simplest logical structure - Constants allow us to pick out objects - Form an atomic sentence by combining name(s) with predicates Validity and soundness - An argument is valid iff the conclusion is a logical consequence of the premises - If a logically valid argument has at least one false premise, then it can have a false conclusion - Not fixed by the truth values that the premises and the conclusions actually have - Valid Arguments - If your premises are true and an argument is valid, then the conclusion is true - The conclusion is a logical consequence of the premises - It is NOT possible for the premises all to be true and the conclusion to be false - Invalid Arguments - The conclusion is NOT a logical consequence of the premises - It is possible for the premises all to be true and the conclusion to be false - What you CAN have - Valid arguments with false premises and a false conclusion - Valid arguments with false premises and a true conclusion - Invalid arguments with true premises and a true conclusion - BUT you CANNOT have …

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- A valid argument with true premises and a false conclusion - An argument is SOUND iff it is valid and all the premises are true Shapes (dodec, etc) - Dodec: Round / 8 sided big hexagon thing - Tetra: Pyramid Proofs with identity - Symbol: = - Principle 1: The Relexivity of Identity - Says that everything is identical to itself; it will always be true - I.E. a = a - Principle 2: The Indiscernibility of Identicals - This says that if a=b, then everything that is true of a must always be true of b - Any properties or features that a has must also be properties of b - Informal Proof of the Symmetry of Identity - Our premise is a=b. - Because of the Reflexivity of Identity, we know that everything is identical to itself, so a=a. - The Indiscernibility of Identicals allows us to substitute the constant b for the constant a. - So, it would be a=b → a=a → b=a - Identity of Introduction - =intro - This rules corresponds to the Reflexivity of Identity. - For any individual constant, a, you can introduce the sentence “a=a” at any line in the proof, without citing any other lines of the proof. - So a=a is =intro. - Identity of Elimination - =elim - This rules corresponds to the Indiscernibility of Identicals - One of the lines is an identity sentence - The other is a sentence saying something about the object named by the constant a - If any of the preceding lines in the proof is the sentence a=b, then you can write down a new sentence that substitutes b for a in any of the preceding sentences in the proof Negation - NOT - --| or ~ Conjunction - AND - ^ Disjunction - OR - V Ana Con rule - Allows you to derive sentences that are analytical consequences of earlier sentences in the proof - A logical consequence that holds in virtue of the meanings of non-logical words Tautologies - A sentence S is a tautology iff the truth for S gives True under S’s main

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connective in every line Tautological equivalence - Sentences A and B are tautologically equivalent iff they have exactly the same pattern of truth values under the main connective - They don’t have to both be tautologies, but the truth values under the main connectives need to match up Tautological equivalence vs logical equivalence - 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 - Crucial difference: Logical equivalence talks about truth values in all possible circumstances, while tautological equivalence talks about truth values under the main connective Tautological consequence - 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 - The joint truth table that assigns True to all the premises also assigns True to the conclusion Logical consequence - 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

You may be given a picture of a world and asked to determine the truth-value of a sentence, or the soundness of an argument, in that world. You may be given a compound sentence, with a partially completed truth table in Boole, and asked how it should be completed. You may be given an incomplete proof in Fitch and asked, for example, what rule needs to be applied in a given step, or which earlier steps need to be cited. Do review exercises marked as review...