class: center, middle, inverse, title-slide .title[ # Logic ] .date[ ### MSA 2025 ] --- <script type="text/x-mathjax-config"> MathJax.Hub.Config({ TeX: { Macros: { And: "{\\mathop{\\&}}", Not: "{\\sim}" } } }); </script> # Is/ought distinction In philosophy, we often make a sharp distinction between _is_ and _ought_. - How things *actually are*: **descriptive** project. - How things *should be*: **normative** project. -- Applying this idea to reasoning: - How we *actually* reason - Descriptive project: **The psychology of reasoning** - How we *should* reason - Normative project: **Logic** --- class: center, middle # Logic Logic is the study of _correct reasoning_. --- class: medium-font # Logic: branches Logic has two main branches - Deductive logic - Inductive logic -- Think about these chains of reasoning: .pull-left[ 1. James is in Missouri. 2. If James is in Missouri, then James is in the US. 3. Therefore, James is in the US. ] .pull-right[ 1. In a recent poll, 80% of people said they will vote for James as the next mayor. 2. Therefore, James will be the next mayor. ] --- class: medium-font # Deductive and Inductive Logic .pull-left[ 1. James is in Missouri. 2. If James is in Missouri, then James is in the US. 3. Therefore, James is in the US. ] .pull-right[ 1. In a recent poll, 80% of people said they will vote for James as the next mayor. 2. Therefore, James will be the next mayor. ] The reasoning on the left is an example of **deductive logic**. - The truth of the premises *guarantees* the truth of the conclusion. - If the premises are true, then the conclusion *must* be true. - It's *not possible* for the premises to be true and the conclusion false. The reasoning on the right is an example of **inductive logic**. - The truth of the premises *does not guarantee* the truth of the conclusion. - However, the truth of the premises makes the conclusion *highly probable*. --- class: small-font # Arguments These "chains of reasoning" are called _arguments_: .pull-left[ 1. James is in Missouri. 2. If James is in Missouri, then James is in the US. 3. Therefore, James is in the US. ] .pull-right[ 1. In a recent poll, 80% of people said they will vote for James as the next mayor. 2. Therefore, James will be the next mayor. ] An **argument** is *a set of two or more sentences*, in which some of them are presented as supporting another sentence. - The sentences supporting another sentence are called __premises.__ - The sentence being supported by others is called __conclusion.__ If the premises are true and the argument is a good one, then *you have some reasons to accept* the conclusion. The sentences in an argument can be considered as _steps in reasoning_. An **inference** is a step in reasoning, that goes from premises to conclusion. --- exclude: true class: small-font # Arguments Sentences in an argument express propositions, and the distinctive feature of propositions is that they are the primary bearers of **truth values**. - In deductive logic, sentences are linguistic expressions that can be either _true_ or _false._ - James is in Missouri. - I am not sleepy. - French fries are tasty. - Euthanasia is morally wrong. - 2 + 2 = 5. - If a linguistic expression cannot be true or false, _it's not a sentence_ (in deductive logic), and thus it can't play a role in an argument. - James and Missouri - Are you sleepy yet? - Let's go get french fries! - Is euthanasia morally wrong? - 2 + 2. --- exclude: true class: small-font # Arguments in standard form This is an argument: > 1. James is in Missouri. > 2. If James is in Missouri, then James is in the US. > 3. Therefore, James is in the US. This is also an argument: > James is in the US, because James is in Missouri, and if he is in Missouri, then he definitely is in the US. The first argument is stated in __standard form__. The second argument is stated in __natural language discourse__. Arguments in standard form: - Sentences are organized in an ordered list. - Conclusions start with the word "Therefore" or "Thus". --- exclude: true # Arguments in standard form Each of the following passages contains an argument. Extract each of them and state them in standard form. > There will be a war in the next year. I know this because there has been a massive buildup in weapons. And every time there is a massive buildup in weapons, there is a war. > If Ron went to the store, he’d be home by now, but he isn’t home, and so we may conclude that he didn’t go to the store. > Both Tom and Fred are hardworking, and Tom is as tenacious as a bulldog. So Tom is sure to be a success, for if there is one thing I have learned in life, it is that everyone who is both hardworking and tenacious succeeds. --- class: medium-font # Arguments in standard form > There will be a war in the next year. I know this because there has been a massive buildup in weapons. And every time there is a massive buildup in weapons, there is a war. 1. There has been a massive buildup in weapons. 2. Every time there is a massive buildup in weapons, there is a war. 3. Therefore, there will be a war in the next year In doing this task, the following indicators are sometimes helpful: - Premise indicators: _since_, _for_, _because_, _given that_, _for the reason that_, ... - Conclusion indicators: _therefore_, _thus_, _it follows that_, _so_, _hence_, _consequently_, _as a result_, ... Natural language is _messy_, so sometimes it's more effective to use your linguistic intuitions to identify premises and conclusions. --- exclude: true # Arguments in standard form The following passage contains an argument. Extract it and state it in standard form. > If Ron went to the store, he’d be home by now, but he isn’t home, and so we may conclude that he didn’t go to the store. 1. If Ron went to the store, Ron would be home by now. 2. Ron isn’t home yet. 3. Thus, Ron didn’t go to the store. Usually, it's helpful to start by identifying the conclusion of the argument (the sentence that is supported by the other sentences). --- exclude: true # Arguments in standard form The following passage contains an argument. Extract it and state it in standard form. > Both Tom and Fred are hardworking, and Tom is as tenacious as a bulldog. So Tom is sure to be a success, for if there is one thing I have learned in life, it is that everyone who is both hardworking and tenacious succeeds. 1. Tom and Fred are hardworking. 2. Tom is tenacious. 3. Everyone who is both hardworking and tenacious succeeds. 4. Thus, Tom will succeed. Often passages in English (or conversations in natural language) contain more information that is not relevant to the specific argument. When we extract an argument, we leave aside such information, and keep only what's relevant to the argument. --- # Evaluating arguments We also want to get good at identifying good vs. bad arguments. To accomplish this, we will use two properties of arguments. - Validity - Soundness Are these arguments _good_? .pull-left[ 1. James is in Missouri. 2. If James is in Missouri, then James is in France. 3. Therefore, James is in France. ] .pull-right[ 1. If James is in Missouri, then James is in the US 2. James is in the US. 3. Therefore, James is in Missouri. ] --- class: middle, center # Logically valid argument An argument is logically valid if and only if _it is not possible for all the premises to be true and the conclusion false_. An argument is logically invalid if and only if it is not logically valid. When an argument is logically valid, we say that *the truth of the premises guarantees the truth of the conclusion*. --- # Logical validity > An argument is logically valid if and only if _it is not possible for all the premises to be true and the conclusion false_. An argument is logically invalid if and only if it is not logically valid. Steps to determine the validity of an argument: 1. We imagine/assume for the moment that the premises are true. 2. We examine the conclusion and ask: "Assuming that the premises are true, **can the conclusion be false**?" - If no (conclusion can't be false), then the argument is _valid_. - If yes (conclusion can be false), then the argument is _invalid_. An argument is valid when the truth of the premises __guarantees__ the truth of the conclusion. --- # Logical validity Steps to determine the validity of an argument: 1. We imagine/assume for the moment that the premises are true. 2. We examine the conclusion and ask: "Assuming that the premises are true, **can the conclusion be false**?" - If no (conclusion can't be false), then the argument is _valid_. - If yes (conclusion can be false), then the argument is _invalid_. Are these arguments _valid_? .pull-left[ 1. James is in Missouri. 2. If James is in Missouri, then James is in France. 3. Therefore, James is in France. ] .pull-right[ 1. If James is in Missouri, then James is in the US 2. James is in the US. 3. Therefore, James is in Missouri. ] -- .pull-left[ Valid 😊 ] .pull-right[ Invalid 😭 ] --- class: small-font # Logical validity Why is this argument _valid_? 1. James is in Missouri. 2. If James is in Missouri, then James is in France. 3. Therefore, James is in France. Determining validity: 1. We imagine for the moment that the premises are true. 2. We examine the conclusion and ask: "Assuming that the premises are true, **can the conclusion be false**?" - The conclusion can't be false (assuming that the premises are true) - The premises guarantee the truth of the conclusion. - Thus, the argument is **valid**. For validity, it doesn't matter whether the premises are _actually_ true or false. - Premise 2 is false in our world, but when evaluating validity, we *assume* that the premises are true, and then we examine whether the conclusion is guaranteed to be true or not. --- class: small-font # Logical validity Why is this argument _invalid_? 1. If James is in Missouri, then James is in the US 2. James is in the US. 3. Therefore, James is in Missouri. Determining validity: 1. We imagine for the moment that the premises are true. 2. We examine the conclusion and ask: "Assuming that the premises are true, **can the conclusion be false**?" - The conclusion can indeed be false: James could be in another state. - The premises don't guarantee the truth of the conclusion. - Thus, the argument is **invalid**. - For validity, it doesn't matter whether the premises are _actually_ true or false. - The conclusion is true in our world, but this is irrelevant when evaluating validity. --- # Logical validity But what about the premise "If James is in Missouri, then James is in the US"? Would the truth of this premise guarantee that James is in Missouri? - No. Such a premise expresses the logical relation _if/then_, which can be true or false, regardless of whether James is in Missouri. - Consider these examples: - If you plagiarize, you will get an F. (True even if you don't plagiarize) - If you are in Santiago, you are in Chile. (True even if you are not there) --- # Evaluating arguments - Validity ✅ - Soundness ⬅️ --- class: middle, center # Logically sound argument An argument is logically sound if and only if it is _logically valid_ and _all of its premises are true_. An argument is logically unsound if and only if it is not logically sound. --- class: small-font # Logical soundness > An argument is logically sound if and only if it is _logically valid_ and _all of its premises are true_. An argument is logically unsound if and only if it is not logically sound. Sound arguments have two features: - Validity. - Premises are _actually_ true. Are these arguments valid? Are they sound? .pull-left[ 1. Italy is a country that is located in North America. 2. Every country that is located in North America uses the United States dollar as its currency. 3. Thus, Italy uses the United States dollar as its currency. ] .pull-right[ 1. The United States is a country that is located in North America. 2. No country that is located in North America uses the euro as its currency. 3. Thus, the United States does not use the euro as its currency. ] -- Both arguments are valid, but only the one on the right is sound. --- # Validity and soundness - Validity is _only a matter of logical relations_ between sentences. - Soundness is a matter of both (1) _logic_ and (2) _how the world really is_. -- Reasoning can go wrong in *two* distinct ways: - An argument can be logically flawless (valid), but have false premises. - An argument can have true premises, but be logically flawed (invalid).