## Scatter Plot: Projection of activations on tG and tP ### Overview The image consists of two scatter plots, side by side, showing the projection of activations on tG and tP for affirmative and negated statements. Each plot displays data points colored red (False) or blue (True). The x-axis represents aij^T tG, and the y-axis represents aij^T tP. ### Components/Axes * **Title:** Projection of activations on tG and tP * **Subtitles:** * Left Plot: Affirmative Statements * Right Plot: Negated Statements * **X-axis (both plots):** aij^T tG * Scale: -40 to 40, with tick marks every 10 units. * **Y-axis (both plots):** aij^T tP * Scale: -50 to 20, with tick marks every 10 units. * **Legend (bottom-right):** * Red: False * Blue: True ### Detailed Analysis **Left Plot: Affirmative Statements** * **Red (False) Data Points:** * Trend: The red data points form a somewhat linear cluster, generally sloping upwards from the bottom-left to the top-right. * Range: X values range from approximately -40 to 10. Y values range from approximately -50 to 10. * **Blue (True) Data Points:** * Trend: The blue data points are clustered in the top-right quadrant. * Range: X values range from approximately 0 to 30. Y values range from approximately 0 to 20. **Right Plot: Negated Statements** * **Red (False) Data Points:** * Trend: The red data points are clustered in the top-left quadrant. * Range: X values range from approximately -40 to 10. Y values range from approximately 0 to 20. * **Blue (True) Data Points:** * Trend: The blue data points form a curved cluster, generally sloping downwards from the top-right to the bottom-right. * Range: X values range from approximately 0 to 40. Y values range from approximately -40 to 10. ### Key Observations * In the Affirmative Statements plot, the "True" data points (blue) are clearly separated from the "False" data points (red), indicating a strong correlation between the activation projections and the truth value of the statement. * In the Negated Statements plot, the "True" and "False" data points are more separated than in the Affirmative Statements plot, but still show a distinct clustering pattern. The "False" data points (red) are concentrated in the upper-left, while the "True" data points (blue) are concentrated in the lower-right. ### Interpretation The plots visualize how activations are projected onto tG and tP for affirmative and negated statements, differentiating between true and false statements. The separation of "True" and "False" data points in both plots suggests that the model is able to distinguish between these categories based on the activation projections. The different distributions in the Affirmative and Negated Statements plots indicate that negation affects the activation patterns. The model appears to represent truth and falsehood differently depending on whether the statement is affirmative or negated.

## Scatter Plot: Projection of Activations on τG and τP ### Overview The image presents two scatter plots side-by-side. Both plots visualize the projection of activations onto two axes, denoted as τG and τP. The left plot displays data for "Affirmative Statements," while the right plot shows data for "Negated Statements." Each point in the scatter plots represents a data instance, colored to indicate whether it is classified as "True" or "False." ### Components/Axes * **Title:** "Projection of activations on τG and τP" (centered at the top) * **Subtitles:** "Affirmative Statements" (left plot), "Negated Statements" (right plot) * **X-axis Label (both plots):** a_ij^Tτ_G (ranging approximately from -40 to 40) * **Y-axis Label (both plots):** a_ij^Tτ_P (ranging approximately from -50 to 20) * **Legend (bottom-right of the right plot):** * Red circle: False * Blue circle: True ### Detailed Analysis or Content Details **Left Plot: Affirmative Statements** * **Trend:** The data points exhibit a strong positive linear correlation. As a_ij^Tτ_G increases, a_ij^Tτ_P also tends to increase. * **False (Red):** Points are clustered in the lower-left quadrant, with a_ij^Tτ_G ranging from approximately -40 to -10 and a_ij^Tτ_P ranging from approximately -50 to 0. There is some spread, but the majority of points fall within this region. * **True (Blue):** Points are concentrated along a diagonal line extending from the top-left to the bottom-right, with a_ij^Tτ_G ranging from approximately -20 to 30 and a_ij^Tτ_P ranging from approximately -20 to 20. The points are more densely packed along this line. **Right Plot: Negated Statements** * **Trend:** The data points show a less pronounced positive correlation compared to the left plot. There is more scatter and a wider distribution of points. * **False (Red):** Points are scattered across the plot, with a higher concentration in the lower-right quadrant. a_ij^Tτ_G ranges from approximately -40 to 30, and a_ij^Tτ_P ranges from approximately -40 to 10. * **True (Blue):** Points are also scattered, but with a tendency to cluster in the upper-right quadrant. a_ij^Tτ_G ranges from approximately -20 to 40, and a_ij^Tτ_P ranges from approximately -10 to 20. ### Key Observations * The distinction between "True" and "False" is much clearer in the "Affirmative Statements" plot than in the "Negated Statements" plot. * The positive correlation in the "Affirmative Statements" plot suggests that high values of τG and τP are associated with "True" statements, while low values are associated with "False" statements. * The scatter in the "Negated Statements" plot indicates that the relationship between τG, τP, and truth value is more complex for negated statements. * There are no obvious outliers in either plot. ### Interpretation The plots demonstrate how activations are projected onto two specific vectors (τG and τP) to differentiate between true and false statements. The strong correlation observed in the "Affirmative Statements" plot suggests that these vectors effectively capture the semantic information necessary to identify true affirmative statements. The weaker correlation in the "Negated Statements" plot implies that negation introduces complexity, potentially requiring additional features or a different representation to accurately classify negated statements. The axes τG and τP likely represent some learned features or dimensions in the activation space that are relevant to truth assessment. The projection onto these axes allows for a visualization of how statements are positioned in this space, and how this positioning relates to their truth value. The difference in the patterns between affirmative and negated statements suggests that negation alters the activation patterns in a way that makes it harder to separate true and false statements using these two dimensions alone. This could be due to the need to represent both the original statement and its negation simultaneously, or to the fact that negation introduces a layer of abstraction that is not fully captured by τG and τP.

\n ## Scatter Plot: Projection of Activations on t_G and t_P ### Overview The image displays two side-by-side scatter plots under the main title "Projection of activations on t_G and t_P". The left plot is titled "Affirmative Statements" and the right plot is titled "Negated Statements". Each plot visualizes the relationship between two projected activation values, with data points colored according to a binary "True" or "False" label. ### Components/Axes * **Main Title:** "Projection of activations on t_G and t_P" * **Subplot Titles:** * Left: "Affirmative Statements" * Right: "Negated Statements" * **X-Axis (Both Plots):** Label is `a_ij^T t_G`. The scale runs from -40 to 40, with major tick marks at intervals of 10 (-40, -30, -20, -10, 0, 10, 20, 30, 40). * **Y-Axis (Both Plots):** Label is `a_ij^T t_P`. The scale runs from -50 to 20, with major tick marks at intervals of 10 (-50, -40, -30, -20, -10, 0, 10, 20). * **Legend:** Located in the bottom-left corner of the "Negated Statements" plot. * Red dot: "False" * Blue dot: "True" ### Detailed Analysis **1. Affirmative Statements (Left Plot):** * **Data Series - "False" (Red):** The red points form a dense, elongated cluster that trends diagonally from the bottom-left quadrant (approximately x=-40, y=-50) to the center (approximately x=0, y=0). The trend is strongly positive and linear. * **Data Series - "True" (Blue):** The blue points form a separate, dense cluster located almost exclusively in the top-right quadrant. This cluster is centered roughly around x=25, y=15. The points show a slight positive slope within their cluster but are distinctly separated from the red cluster. * **Spatial Relationship:** There is a clear, wide separation between the two clusters along the diagonal. The "True" (blue) cluster is positioned entirely above and to the right of the "False" (red) cluster. **2. Negated Statements (Right Plot):** * **Data Series - "False" (Red):** The red points are widely scattered across the left and central portions of the plot, primarily in the upper half (y > -10). There is no strong linear trend; the distribution appears somewhat amorphous, centered roughly around x=-20, y=0. * **Data Series - "True" (Blue):** The blue points form a distinct, vertically oriented cluster on the right side of the plot. This cluster spans a wide range of y-values (from approximately y=-50 to y=10) but is confined to a narrow band of x-values (approximately x=5 to x=25). The cluster has a slight negative slope. * **Spatial Relationship:** The two clusters overlap significantly in the central region of the plot (around x=-10 to x=10). The "True" (blue) cluster is positioned to the right of the main mass of "False" (red) points. ### Key Observations 1. **Dichotomy in Affirmative Context:** For affirmative statements, the model's activations (as projected onto t_G and t_P) for "True" and "False" labels are linearly separable into two distinct, non-overlapping regions. 2. **Convergence in Negated Context:** For negated statements, the activations for "True" and "False" labels are not linearly separable. They show significant overlap, with the "True" cluster forming a vertical band that intersects the more scattered "False" distribution. 3. **Cluster Shape Difference:** The "True" cluster shape changes dramatically between plots: a tight, diagonal blob for affirmative statements versus a tall, narrow vertical band for negated statements. 4. **Axis Range Utilization:** Both plots use a similar range on the x-axis (-40 to 40). The y-axis range is also similar, but the data in the "Affirmative" plot utilizes the lower negative values more extensively for the "False" series. ### Interpretation This visualization likely comes from an analysis of a neural network's internal representations, specifically examining how it processes affirmative versus negated statements. The projections `a_ij^T t_G` and `a_ij^T t_P` represent activations projected onto two specific directions (vectors t_G and t_P) in the model's activation space. * **Affirmative Statements:** The clear separation suggests that for simple affirmative claims, the model's internal state (as captured by these projections) cleanly distinguishes between true and false propositions. The positive linear correlation for "False" statements indicates that as one projected component (`a_ij^T t_G`) increases, the other (`a_ij^T t_P`) also increases proportionally for false claims. * **Negated Statements:** The overlapping clusters indicate that negation introduces complexity or ambiguity into the model's internal representation. The model does not map "True" and "Negated False" (or similar) to distinctly separable regions in this projected space. The vertical "True" cluster suggests that for negated statements labeled true, the `a_ij^T t_G` value is relatively consistent (positive), while the `a_ij^T t_P` value varies widely. This could imply that handling negation relies on a different or more distributed mechanism within the model compared to processing affirmative statements. **In summary, the data suggests a fundamental difference in how the model represents truth value for affirmative versus negated linguistic constructs. Affirmative truth appears to be encoded in a simple, separable manner within this subspace, while negated truth results in a more entangled and complex representation.**

## Scatter Plots: Projection of activations on t_G and t_P ### Overview Two scatter plots compare activation projections for affirmative and negated statements. Each plot uses red (False) and blue (True) markers to distinguish truth values. The left plot shows affirmative statements, while the right shows negated statements. Both axes represent transformed activation values (a^T_ij t_G and a^T_ij t_P). ### Components/Axes - **X-axis (both plots)**: `a^T_ij t_G` (range: -40 to 40) - **Y-axis (both plots)**: `a^T_ij t_P` (range: -50 to 20) - **Legend**: Located at bottom-right of each plot - Red: False - Blue: True - **Plot Titles**: - Left: "Affirmative Statements" - Right: "Negated Statements" ### Detailed Analysis #### Affirmative Statements (Left Plot) - **False (Red)**: Clustered in lower-left quadrant (x: -40 to 0, y: -50 to -10) - **True (Blue)**: Concentrated in upper-right quadrant (x: 0 to 40, y: 0 to 20) - **Trend**: Positive correlation between `a^T_ij t_G` and `a^T_ij t_P` for True statements. False statements show minimal correlation. #### Negated Statements (Right Plot) - **False (Red)**: Scattered across upper-left quadrant (x: -40 to 0, y: -10 to 20) - **True (Blue)**: Clustered in lower-right quadrant (x: 0 to 40, y: -50 to 0) - **Trend**: Negative correlation between `a^T_ij t_G` and `a^T_ij t_P` for True statements. False statements show no clear pattern. ### Key Observations 1. **Affirmative Statements**: - True statements consistently show higher values for both `a^T_ij t_G` and `a^T_ij t_P`. - False statements cluster in regions of low activation for both axes. 2. **Negated Statements**: - True statements exhibit higher `a^T_ij t_G` but lower `a^T_ij t_P` compared to False statements. - False statements are more dispersed, with some outliers in the upper-right quadrant. 3. **Divergence in Patterns**: - Affirmative True/False clusters are spatially distinct. - Negated True/False clusters overlap more but maintain directional separation. ### Interpretation The data suggests that: - **Affirmative Statements**: The model's activations for True statements align strongly with both `t_G` and `t_P`, while False statements lack this alignment. - **Negated Statements**: True statements show a trade-off between `t_G` and `t_P` activations, whereas False statements lack consistent patterns. - **Implications**: The separation in activation projections may reflect how the model processes truth values differently based on statement polarity (affirmative vs. negated). The negated statements' dispersed False cluster could indicate ambiguity or contextual variability in negation handling. No textual content in other languages was detected. All axis labels, legends, and plot titles were transcribed directly from the image.