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- https://human.libretexts.org/Bookshelves/Philosophy/An_Introduction_to_Formal_Logic_(Magnus)/Chapter_1%3A_What_is_logic/Section_5%3A_Other_logical_notionsIt might be raining here and not raining across town, it might be raining now but stop raining even as you read this, but it is impossible for it to be both raining and not raining here at this moment...It might be raining here and not raining across town, it might be raining now but stop raining even as you read this, but it is impossible for it to be both raining and not raining here at this moment. If either of the sentences is true, then they both are; if either of the sentences is false, then they both are.
- https://human.libretexts.org/Bookshelves/Philosophy/An_Introduction_to_Formal_Logic_(Magnus)This books treats symbolization, formal semantics, and proof theory for each language. The discussion of formal semantics is more direct than in many introductory texts. The book is designed to provid...This books treats symbolization, formal semantics, and proof theory for each language. The discussion of formal semantics is more direct than in many introductory texts. The book is designed to provide a semester's worth of material for an introductory college course. It would be possible to use the book only for sentential logic, by skipping chapters 4-5 and parts of chapter 6.
- https://human.libretexts.org/Bookshelves/Philosophy/An_Introduction_to_Formal_Logic_(Magnus)/Chapter_6%3A_Proofs/Section_06%3A_Proof_strategyIf you have ∃xA and intend to use the ∃E rule, then you should assume A[c|x] for some c that is not in use and then derive a conclusion that does not contai...If you have ∃xA and intend to use the ∃E rule, then you should assume A[c|x] for some c that is not in use and then derive a conclusion that does not contain c. Once you have decided how you might be able to get to the conclusion, ask what you might be able to do with the premises.
- https://human.libretexts.org/Bookshelves/Philosophy/An_Introduction_to_Formal_Logic_(Magnus)/Chapter_4%3A_Quantified_logicThis chapter introduces a logical language called QL. It is a version of quantified logic, because it allows for quantifiers like all and some. Quantified logic is also sometimes called predicate logi...This chapter introduces a logical language called QL. It is a version of quantified logic, because it allows for quantifiers like all and some. Quantified logic is also sometimes called predicate logic, because the basic units of the language are predicates and terms.
- https://human.libretexts.org/Bookshelves/Philosophy/An_Introduction_to_Formal_Logic_(Magnus)/Chapter_5%3A_Formal_semantics/Section_5%3A_Truth_in_QLThe formula Px is satisfied in a model M by a variable assignment a if and only if a(x), the object that a assigns to x, is in the the extension of P in \(\mat...The formula Px is satisfied in a model M by a variable assignment a if and only if a(x), the object that a assigns to x, is in the the extension of P in M.
- https://human.libretexts.org/Bookshelves/Philosophy/An_Introduction_to_Formal_Logic_(Magnus)/Chapter_6%3A_Proofs/Section_01%3A_Basic_rules_for_SLIn a natural deduction system, there will be two rules for each logical operator: an introduction rule that allows us to prove a sentence that has it as the main logical operator and an elimination ru...In a natural deduction system, there will be two rules for each logical operator: an introduction rule that allows us to prove a sentence that has it as the main logical operator and an elimination rule that allows us to prove something given a sentence that has it as the main logical operator. When we define the rule, however, we use variables to underscore the point that the rule may be applied to any two lines that are already in the proof.
- https://human.libretexts.org/Bookshelves/Philosophy/An_Introduction_to_Formal_Logic_(Magnus)/Chapter_6%3A_Proofs/Section_09%3A_Soundness_and_completenessDemonstrating that the proof system is sound would require showing that any possible proof is the proof of a valid argument. If using the &E rule on the last line of a proof could never change a valid...Demonstrating that the proof system is sound would require showing that any possible proof is the proof of a valid argument. If using the &E rule on the last line of a proof could never change a valid argument into an invalid one, then using the rule many times could not make an argument invalid. Since the argument so far is valid, A and B are either premises of the argument or valid consequences of the premises.
- https://human.libretexts.org/Bookshelves/Philosophy/An_Introduction_to_Formal_Logic_(Magnus)/Chapter_4%3A_Quantified_logic/Section_5%3A_Sentences_of_QLIn order for ∀xA to be a wff, A must contain the variable x and must not already contain an x-quantifier. ∀xDw will not count as a wff because ‘x’ does not occur in Dw, and...In order for ∀xA to be a wff, A must contain the variable x and must not already contain an x-quantifier. ∀xDw will not count as a wff because ‘x’ does not occur in Dw, and ∀x∃xDx will not count as a wff because ∃xDx contains an x-quantifier
- https://human.libretexts.org/Bookshelves/Philosophy/An_Introduction_to_Formal_Logic_(Magnus)/Chapter_2%3A_Sentential_logic/Section_4%3A_Sentences_of_SLSince the meaningful expressions of SL are the wffs and since every wff of SL is either true or false, the definition for a sentence of SL is the same as the definition for a wff. The sentence ¬¬¬\(D\...Since the meaningful expressions of SL are the wffs and since every wff of SL is either true or false, the definition for a sentence of SL is the same as the definition for a wff. The sentence ¬¬¬D is true if and only if the sentence ¬¬D is false, and so on through the structure of the sentence until we arrive at the atomic components: ¬¬¬D is true if and only if the atomic sentence D is false.
- https://human.libretexts.org/Bookshelves/Philosophy/An_Introduction_to_Formal_Logic_(Magnus)/Chapter_1%3A_What_is_logic/Section_6%3A_Formal_languagesEvery sentence of an argument is then represented as having one of four forms, which medieval logicians labeled in this way: (A) All As are Bs. (E) No As are Bs. (I) Some A is \(B\...Every sentence of an argument is then represented as having one of four forms, which medieval logicians labeled in this way: (A) All As are Bs. (E) No As are Bs. (I) Some A is B. (O) Some A is not B. ~A set of sentences is consistent if it is logically possible for all the members of the set to be true at the same time; it is inconsistent otherwise.
- https://human.libretexts.org/Bookshelves/Philosophy/An_Introduction_to_Formal_Logic_(Magnus)/Chapter_5%3A_Formal_semanticsThe word ‘semantics’ comes from the greek word for ‘mark’ and means ‘related to meaning.’ So a formal semantics will be a mathematical account of meaning in the formal language. In SL, for example, we...The word ‘semantics’ comes from the greek word for ‘mark’ and means ‘related to meaning.’ So a formal semantics will be a mathematical account of meaning in the formal language. In SL, for example, we might say that D means ‘Today is Tuesday’; we might say instead that D means ‘Today is the day after Monday.’ These are two different interpretations, because they use different English sentences for the meaning of D.
