Three statements are given followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements. Statements: No file is a folder. Some files are PDFs. All files are Excel sheets. Conclusions: I. Some PDFs are folders. II. Some Excel sheets are folders. III. Some PDFs are Excel sheets.
Only conclusion III follows
Analyzing Syllogism Statements and Conclusions
To determine which conclusions logically follow from the given statements, we need to analyze the relationships between the categories mentioned: files, folders, PDFs, and Excel sheets.
Understanding the Given Statements
- Statement 1: No file is a folder. This establishes a complete separation between the set of files and the set of folders. Their intersection is empty.
- Statement 2: Some files are PDFs. This means there is an overlap between the set of files and the set of PDFs. At least one element exists that is both a file and a PDF.
- Statement 3: All files are Excel sheets. This means the entire set of files is a subset of the set of Excel sheets. Everything that belongs to the category file also belongs to the category Excel sheet.
Evaluating Conclusion I: Some PDFs are folders.
Conclusion I proposes that there is an overlap between PDFs and folders. From Statement 2, we know there are files that are PDFs. From Statement 1, we know no files are folders. While the part of PDFs that are files cannot be folders, the statements give us no information about the relationship between PDFs (specifically those that are not files, if any exist) and folders. There is no information provided that forces an overlap between the general set of PDFs and the set of folders. Therefore, this conclusion does not logically follow from the given statements alone.
Evaluating Conclusion II: Some Excel sheets are folders.
Conclusion II suggests an overlap between Excel sheets and folders. Statement 3 tells us All files are Excel sheets, meaning the set of files is entirely contained within the set of Excel sheets. Statement 1 tells us No file is a folder, meaning the set of files and the set of folders are disjoint.
Combining these two statements using logical deduction: If All files are Excel sheets (All M are P) and No file is a folder (No M is S, which is equivalent to No S is M - No folder is file), a standard syllogistic form (Figure 1, EAE mood - Celarent) yields the valid conclusion: No folder is an Excel sheet (No S is P).
If it is logically true that No folder is an Excel sheet, then there cannot be any overlap between the set of folders and the set of Excel sheets. Conclusion II, stating that Some Excel sheets are folders (meaning there is an overlap), directly contradicts this logical deduction. Therefore, Conclusion II does not logically follow from the given statements.
Evaluating Conclusion III: Some PDFs are Excel sheets.
Conclusion III claims there is an overlap between PDFs and Excel sheets. Statement 2 states that Some files are PDFs. This guarantees the existence of at least one entity that is both a file and a PDF.
Statement 3 states that All files are Excel sheets. This means any entity that is a file must also be an Excel sheet.
Since there is at least one entity that is a file and a PDF (from Statement 2), and everything that is a file is also an Excel sheet (from Statement 3), that same entity must also be an Excel sheet. Thus, there is at least one entity that is both a PDF and an Excel sheet.
Therefore, the conclusion Some PDFs are Excel sheets logically follows from the given statements.
Summary of Logical Conclusions
Based on the analysis of each conclusion:
| Conclusion | Logically Follows? |
|---|---|
| I. Some PDFs are folders. | No |
| II. Some Excel sheets are folders. | No |
| III. Some PDFs are Excel sheets. | Yes |
The analysis shows that only conclusion III is a logical consequence of the given statements.
Revision Table: Syllogism Proposition Types
Understanding the different types of categorical propositions is key to analyzing syllogisms:
| Type | Quantity | Quality | Standard Form | Relationship |
|---|---|---|---|---|
| A | Universal | Affirmative | All S are P | S ⊂ P (S is a subset of P) |
| E | Universal | Negative | No S is P | S ∩ P = ∅ (S and P are disjoint) |
| I | Particular | Affirmative | Some S are P | S ∩ P ≠ ∅ (S and P intersect) |
| O | Particular | Negative | Some S are not P | S \ P ≠ ∅ (Some S are outside P) |
Additional Information: Categorical Syllogisms
A categorical syllogism is a deductive argument consisting of three parts: a major premise, a minor premise, and a conclusion. Each part is a categorical proposition, and together they contain exactly three terms, each appearing in exactly two of the propositions.
The terms are:
- The Major Term: The predicate of the conclusion.
- The Minor Term: The subject of the conclusion.
- The Middle Term: Appears in both premises but not in the conclusion, connecting the major and minor terms.
In this problem:
- Major Term: Excel sheets (predicate of Conclusion II & III)
- Minor Term: PDFs (subject of Conclusion I & III) and Excel sheets (subject of Conclusion II), Folders (predicate of Conclusion I & II)
- Middle Term: Files (appears in all statements)
The validity of a syllogism depends on its logical form, not the truth of the statements themselves. A conclusion logically follows if its truth is guaranteed whenever the premises are true.
Important Questions from Conventional Syllogism
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
No bank is an office.
All offices are stalls.
Conclusions:
I. No bank is a stall.
II. No stall is a bank.
III. Some stalls are offices.
IV. All the stalls are offices
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All flowers are beautiful.
Vaidehi is beautiful.
Conclusions:
I. Vaidehi is a flower.
II. Some beautiful are flowers.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
1. All rugs are blankets.
2. All blankets are pillows.
3. Some blankets are frames.
Conclusions:
I. All pillows are rugs.
II. Some pillows are rugs.
III. All rugs are frames
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
Some fingers are toes.
Some toes are rings.
Some rings are hands.
Conclusions:
I. Some hands are toes.
II. Some rings are fingers.
III. Some hands are fingers.
V. Some fingers are rings.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All polygons are angles.
All angles are diagonals.
All cones are cubes.
All cubes are decagons.
No diagonal is a cube.
Conclusions:
I. Some diagonals are polygons.
II. All diagonals are decagons.
III. No polygon is a cone.
IV. Some cubes are angles.
