Name: Benjamin Meixner Course: 5156
1a. What Is a Valid Inference (or a Valid Argument)?
• A valid inference is one in which, if the premises are true, the conclusion must necessarily be true.
– In other words, the logical form is such that no possible situation exists in which all premises hold but the conclusion does not. (See also [0])
1b. How Does a Valid Inference Differ from a Sound Inference (Proof)?
• A valid inference concerns the form of the argument only—it is structurally correct.
• A sound inference (or proof) is a valid inference in which the premises are, in fact, true.
– Thus, soundness requires both logical validity and factual (or conceptual) truth of the premises.
2a. What Are the Subject and the Predicate in the Given Assertoric Sentences?
For each sentence, the subject is the term about which something is asserted, and the predicate is the attribute or property that is affirmed or denied of that subject.
• “No humans can fly”
– Subject: “humans”
– Predicate: “can fly” (expressed in a negative form: it denies that flying is a property of humans)
• “Some flowers blossom”
– Subject: “flowers”
– Predicate: “blossom” (affirming that blossoming is a property of some flowers)
• “All whales have 3 ear-bones.”
– Subject: “whales”
– Predicate: “have 3 ear-bones” (affirming that having three ear-bones belongs to the class of whales)
2b. Examples of an Affirmative Sentence and of a Sentence Where the Predicate Is Denied
• Example of an affirmative sentence:
– “All birds have wings.”
▪ Here the predicate “have wings” is affirmed of the subject “birds.”
• Example of a negative sentence:
– “No dogs are cats.”
▪ Here the predicate “are cats” is denied to belong to the subject “dogs.”
3a. Identifying the Middle Term in a Syllogistic Argument and Another Example
• In the argument “If all humans are mortal, and all Greeks are humans, then all Greeks are mortal,” the middle term is “humans” because it appears in both premises but not in the conclusion.
• Another example of a valid syllogism with a middle term:
– “All mammals are warm-blooded. All dogs are mammals. Therefore, all dogs are warm-blooded.”
▪ Here the middle term is “mammals.”
3b. Two Everyday-Life Discussions Where an Appeal to a Middle Term Functions as the (Often Unstated) Reason
Below are two examples along with the premises that are assumed though not explicitly mentioned.
• Example 1:
– Conversation:
▪ “Why is this smartphone reliable?”
▪ “Because it is made by a trusted brand.”
– Implicit premises:
▪ Premise 1: “All smartphones made by trusted brands are reliable.”
▪ Premise 2: “This smartphone is made by a trusted brand.”
▪ Conclusion: “Therefore, this smartphone is reliable.”
– The middle term here is “smartphones made by trusted brands.”
• Example 2:
– Conversation:
▪ “Why is this restaurant popular?”
▪ “Because it’s highly rated.”
– Implicit premises:
▪ Premise 1: “All highly rated restaurants are popular.”
▪ Premise 2: “This restaurant is highly rated.”
▪ Conclusion: “Therefore, this restaurant is popular.”
– The middle term is “highly rated restaurants.”
• Note: I am not entirely sure these examples exactly match the style discussed in the lecture, but they illustrate the point of an unstated middle term replacing explicitly stated premises.
4a. Which Syllogistic Form Corresponds to “Barbara”?
• “Barbara” is the name given to the syllogistic form featuring three universal affirmative propositions.
– Standard form:
▪ Major premise: “All B are C.”
▪ Minor premise: “All A are B.”
▪ Conclusion: “All A are C.”
4b. Which Syllogistic Form Corresponds to “Celarent”?
• “Celarent” is the form containing a universal negative and two universal affirmatives (an EAE form in the first figure).
– Standard form:
▪ Major premise: “No B are C.”
▪ Minor premise: “All A are B.”
▪ Conclusion: “No A are C.”
5a. Instance of Darii, Its Name, and Explanation of the Invalid Example Provided
• An instance of Darii (an AII form in the first figure) is:
– “All mammals are animals. Some cats are mammals. Therefore, some cats are animals.”
– Here, the form is labeled Darii.
• Explanation of why “All Viennese are Austrians. Some Austrians are musicians. Some Viennese are musicians” is NOT valid:
– The intended argument fails because the middle term (“Austrians”) is not properly distributed.
▪ In the first premise (“All Viennese are Austrians”), “Austrians” is used in a universal affirmative, but in the second premise (“Some Austrians are musicians”) the middle term is used in a particular context where it is not distributed—that is, it does not refer to all Austrians.
– Because the connection via the middle term is insufficiently established, the conclusion cannot be guaranteed, and the syllogism does not conform to the valid AII (Darii) form.
5b. Meaning of “Distribution” of the Terminus Mēdius and the Term “Terminus Major”
• Distribution of the middle term (terminus medius):
– Refers to whether a term in a premise is used in such a way that it covers or refers to all members of its class.
– In an Euler diagram, distribution means that the area representing the term is entirely encompassed when making an assertion.
– If the middle term is undistributed (i.e., not referring to all of its members), it cannot guarantee the necessary link between the subject and predicate of the conclusion.
• The terminus major:
– This is the term that serves as the predicate of the conclusion in a syllogism.
– It is “major” in that it appears as the predicate in the final conclusion, and its distribution is crucial when the conclusion is universal.