## Scatter Plot: Projection of activations on tG and tP ### Overview The image contains two scatter plots, each displaying the projection of activations on tG and tP. The left plot represents "Affirmative Statements," while the right plot represents "Negated Statements." Each plot distinguishes between "False" (red) and "True" (blue) statements. The x-axis represents a_ij^T t_G, and the y-axis represents a_ij^T t_P. ### Components/Axes * **Title:** Projection of activations on tG and tP * **Subtitles:** Affirmative Statements (left plot), Negated Statements (right plot) * **X-axis (both plots):** a_ij^T t_G * Scale: -0.8 to 0.6, with tick marks at -0.6, -0.4, -0.2, 0.0, 0.2, 0.4, 0.6 on the left plot. * Scale: -0.8 to 0.4, with tick marks at -0.8, -0.6, -0.4, -0.2, 0.0, 0.2, 0.4 on the right plot. * **Y-axis (both plots):** a_ij^T t_P * Scale: -0.8 to 0.4, with tick marks at -0.8, -0.6, -0.4, -0.2, 0.0, 0.2, 0.4 on the left plot. * Scale: -1.0 to 0.4, with tick marks at -1.0, -0.8, -0.6, -0.4, -0.2, 0.0, 0.2, 0.4 on the right plot. * **Legend (bottom-right):** * Red: False * Blue: True ### Detailed Analysis **Left Plot (Affirmative Statements):** * **Red (False):** The red data points form a cluster that extends from approximately (-0.6, -0.8) to (0.2, 0.2). The trend is generally upward-sloping. * **Blue (True):** The blue data points form a cluster in the upper-right quadrant, roughly from (0.0, 0.0) to (0.6, 0.4). **Right Plot (Negated Statements):** * **Red (False):** The red data points are clustered in the upper-left quadrant, approximately from (-0.8, 0.0) to (0.0, 0.2). * **Blue (True):** The blue data points are clustered in the lower-right quadrant, approximately from (0.0, -0.8) to (0.4, 0.0). ### Key Observations * In the "Affirmative Statements" plot, there is a clear separation between "False" and "True" statements, with "True" statements generally having higher values on both axes. * In the "Negated Statements" plot, the "False" and "True" statements are more separated, occupying distinct quadrants. * The distribution of points suggests that the model distinguishes between "True" and "False" statements differently for affirmative and negated statements. ### Interpretation The plots visualize how activations are projected onto tG and tP for affirmative and negated statements, categorized by their truth value. The separation between "True" and "False" clusters in both plots indicates that the model is sensitive to the truth value of the statements. The different distributions between the "Affirmative" and "Negated" plots suggest that the model processes these types of statements differently. The model appears to encode truth differently depending on whether the statement is affirmative or negated. The clusters represent the internal representation of these statements within the model.

\n ## Scatter Plot: Projection of Activations on τ_G and τ_P ### Overview The image presents two scatter plots, side-by-side, visualizing the projection of activations onto two time constants, τ_G and τ_P. 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 point, colored to indicate whether the statement is "True" or "False." ### Components/Axes * **Title:** "Projection of activations on τ_G and τ_P" (centered at the top) * **Subtitles:** "Affirmative Statements" (left plot), "Negated Statements" (right plot) * **X-axis Label (both plots):** a_ijτ_G (ranging approximately from -0.8 to 0.6) * **Y-axis Label (both plots):** a_ijτ_P (ranging approximately from -0.8 to 0.4) * **Legend (bottom-right):** * Red circles: "False" * Blue circles: "True" ### Detailed Analysis or Content Details **Left Plot: Affirmative Statements** * **Trend:** The data points form a roughly diagonal band. The points generally slope upwards from the bottom-left to the top-right. * **False (Red):** Points are concentrated in the bottom-left quadrant, with a_ijτ_G ranging from approximately -0.6 to 0.0 and a_ijτ_P ranging from approximately -0.6 to 0.1. * **True (Blue):** Points are concentrated in the top-right quadrant, with a_ijτ_G ranging from approximately 0.0 to 0.6 and a_ijτ_P ranging from approximately 0.0 to 0.4. There is some overlap with the red points. **Right Plot: Negated Statements** * **Trend:** The data points also form a roughly diagonal band, but it is less distinct than in the left plot. The points generally slope upwards from the bottom-left to the top-right. * **False (Red):** Points are scattered across the plot, with a_ijτ_G ranging from approximately -0.8 to 0.4 and a_ijτ_P ranging from approximately -0.8 to 0.3. * **True (Blue):** Points are also scattered, but tend to be more concentrated in the top-right quadrant, with a_ijτ_G ranging from approximately -0.4 to 0.4 and a_ijτ_P ranging from approximately -0.2 to 0.4. There is significant overlap with the red points. ### Key Observations * The separation between "True" and "False" statements is much clearer for affirmative statements than for negated statements. * The distribution of negated statements is more dispersed, suggesting that the projection onto τ_G and τ_P is less effective at distinguishing between true and false negated statements. * There is some overlap between the "True" and "False" classes in both plots, indicating that the projection is not a perfect discriminator. ### Interpretation The plots demonstrate how activations related to statements can be projected onto two time constants, τ_G and τ_P, to differentiate between true and false statements. The clearer separation observed in the "Affirmative Statements" plot suggests that the time constants are more effective at capturing the underlying dynamics of affirmative statements. The more dispersed distribution in the "Negated Statements" plot indicates that negation introduces complexity that is not fully captured by these time constants. The overlap between the classes suggests that the projection is not a perfect representation of truthfulness and that other factors may influence the activations. This could be due to the inherent ambiguity of language or the limitations of the model used to generate the activations. The choice of τ_G and τ_P as projection axes implies these time constants are relevant to the processing of statements and their truth values. Further investigation into the meaning of these time constants could provide insights into the neural mechanisms underlying truth evaluation. The fact that the separation is less clear for negated statements suggests that negation may involve different neural processes or require additional time constants to be accurately represented.

## 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 plots visualize the relationship between two projection values, `a_ij^T t_G` (x-axis) and `a_ij^T t_P` (y-axis), for two different conditions: "Affirmative Statements" (left plot) and "Negated Statements" (right plot). Data points are colored based on a binary label: "False" (red) or "True" (blue). ### Components/Axes * **Main Title:** "Projection of activations on t_G and t_P" (centered at the top). * **Subplot Titles:** * Left Plot: "Affirmative Statements" (centered above the left chart). * Right Plot: "Negated Statements" (centered above the right chart). * **Axes Labels (Identical for both plots):** * X-axis: `a_ij^T t_G` (centered below the x-axis). * Y-axis: `a_ij^T t_P` (rotated 90 degrees, centered to the left of the y-axis). * **Axes Scales (Approximate):** * X-axis Range (both plots): -0.8 to 0.6. * Y-axis Range (both plots): -1.0 to 0.4. * **Legend:** Located in the bottom-right corner of the *right* subplot ("Negated Statements"). * Red circle: "False" * Blue circle: "True" ### Detailed Analysis **Left Plot: Affirmative Statements** * **Trend Verification:** Both data series show a clear positive correlation. The cloud of points slopes upward from the bottom-left to the top-right. * **Data Distribution:** * **"False" (Red):** Points are densely clustered in the lower-left quadrant. Approximate centroid: x ≈ -0.3, y ≈ -0.4. The cluster extends roughly from x = -0.7 to 0.1 and y = -0.8 to 0.0. * **"True" (Blue):** Points are densely clustered in the upper-right quadrant. Approximate centroid: x ≈ 0.4, y ≈ 0.3. The cluster extends roughly from x = 0.0 to 0.6 and y = 0.0 to 0.5. * **Relationship:** There is a clear separation between the two clusters along the diagonal. "True" statements have significantly higher values for both `a_ij^T t_G` and `a_ij^T t_P` projections compared to "False" statements. **Right Plot: Negated Statements** * **Trend Verification:** The two data series show an inverse relationship. The "False" cluster is in the upper-left, and the "True" cluster is in the lower-right. * **Data Distribution:** * **"False" (Red):** Points are clustered in the upper-left quadrant. Approximate centroid: x ≈ -0.4, y ≈ 0.1. The cluster extends roughly from x = -0.8 to 0.0 and y = -0.2 to 0.4. * **"True" (Blue):** Points are clustered in the lower-right quadrant. Approximate centroid: x ≈ 0.2, y ≈ -0.4. The cluster extends roughly from x = -0.2 to 0.5 and y = -1.0 to 0.0. * **Relationship:** There is a clear separation between the two clusters. "True" statements have higher `a_ij^T t_G` values but lower `a_ij^T t_P` values compared to "False" statements. ### Key Observations 1. **Distinct Cluster Separation:** In both plots, the "True" and "False" data points form distinct, non-overlapping clusters, indicating the projections are highly discriminative for the given task. 2. **Opposite Patterns:** The spatial relationship between "True" and "False" clusters is inverted between the two conditions. Affirmative statements show a positive correlation for both classes, while negated statements show a negative correlation between the classes. 3. **Axis Range Utilization:** The "Affirmative Statements" plot uses the positive x-axis range more heavily, while the "Negated Statements" plot uses the negative y-axis range more heavily for its "True" cluster. ### Interpretation This visualization demonstrates how a model's internal activations (projected onto two directions, `t_G` and `t_P`) encode truth value differently depending on linguistic context (affirmation vs. negation). * **For Affirmative Statements:** The model's representation of a "True" statement involves high activation along both the `t_G` and `t_P` directions. A "False" statement involves low activation along both. This suggests `t_G` and `t_P` may be correlated features for affirming truth in simple statements. * **For Negated Statements:** The pattern flips. A "True" negated statement (e.g., "It is not raining") is characterized by high `t_G` but low `t_P` activation. A "False" negated statement shows the opposite. This indicates that the `t_P` direction may be sensitive to the *presence* of a concept, while `t_G` is sensitive to its *factual validity*, and negation decouples these two signals. * **Underlying Mechanism:** The plots provide visual evidence that the model uses a compositional representation where truth (`t_G`) and a secondary property (`t_P`, possibly related to grammatical polarity or surface-level assertion) are processed by distinct but interacting neural pathways. The clear separation suggests these projections are effective probes for understanding the model's reasoning about truth and negation.

## Scatter Plots: Projection of activations on t_G and t_P ### Overview The image contains two scatter plots comparing the projection of activations on two variables, **t_G** (likely a ground truth or target variable) and **t_P** (possibly a predicted or processed variable). The plots are split into **Affirmative Statements** (left) and **Negated Statements** (right). Points are color-coded: **red** for "False" and **blue** for "True". --- ### Components/Axes - **X-axis**: Labeled **a^T_ij t_G** (projection of activations on t_G), ranging from **-0.8 to 0.6**. - **Y-axis**: Labeled **a^T_ij t_P** (projection of activations on t_P), ranging from **-0.8 to 0.4**. - **Legend**: Located in the lower-right corner of the **Negated Statements** plot. - **Red**: "False" - **Blue**: "True" --- ### Detailed Analysis #### Affirmative Statements (Left Plot) - **Red points ("False")**: - Clustered in the **lower-left quadrant** (x ≈ -0.6 to 0.0, y ≈ -0.8 to -0.2). - Some outliers extend toward the upper-right quadrant (x ≈ 0.2–0.4, y ≈ 0.0–0.2). - **Blue points ("True")**: - Concentrated in the **upper-right quadrant** (x ≈ 0.0–0.6, y ≈ 0.0–0.4). - A few outliers appear in the lower-left quadrant (x ≈ -0.4–0.0, y ≈ -0.4–-0.2). #### Negated Statements (Right Plot) - **Red points ("False")**: - Dominant in the **upper-left quadrant** (x ≈ -0.8 to -0.2, y ≈ -0.2 to 0.2). - Fewer points in the lower-right quadrant (x ≈ 0.0–0.4, y ≈ -0.6 to -0.2). - **Blue points ("True")**: - Clustered in the **lower-right quadrant** (x ≈ 0.0–0.4, y ≈ -0.8 to -0.2). - Some outliers in the upper-left quadrant (x ≈ -0.6–-0.2, y ≈ -0.4–-0.2). --- ### Key Observations 1. **Affirmative Statements**: - "True" statements (blue) show a stronger positive correlation between **t_G** and **t_P** (higher x and y values). - "False" statements (red) are more dispersed, with a tendency toward negative projections on **t_P**. 2. **Negated Statements**: - "False" statements (red) are tightly clustered in the upper-left quadrant, suggesting a consistent negative projection on **t_G** and near-zero projection on **t_P**. - "True" statements (blue) are concentrated in the lower-right quadrant, indicating a positive projection on **t_G** and negative projection on **t_P**. 3. **Outliers**: - A small number of red points in the affirmative plot appear in the upper-right quadrant, suggesting rare cases where "False" statements align with positive projections. - In the negated plot, blue points in the upper-left quadrant may represent anomalies where "True" statements exhibit negative projections on both variables. --- ### Interpretation - **Affirmative Statements**: The separation between "True" and "False" suggests the model distinguishes between valid and invalid affirmative claims based on their projections. "True" statements align more closely with the expected relationship between **t_G** and **t_P**, while "False" statements deviate, particularly in their negative **t_P** values. - **Negated Statements**: The clustering of "False" statements in the upper-left quadrant implies that negated claims are often associated with negative **t_G** projections and neutral **t_P** values. "True" negated statements (blue) show a paradoxical pattern: positive **t_G** but negative **t_P**, which may indicate a model bias or error in handling negated truths. - **Model Behavior**: The distinct groupings suggest the model processes affirmative and negated statements differently. Affirmative "True" statements are more consistent with the target variables, while negated "True" statements exhibit conflicting projections, potentially highlighting a need for improved handling of negation in the model. - **Uncertainties**: The exact numerical values of individual points are not labeled, so precise trends (e.g., slopes) cannot be quantified. The clustering patterns are approximate and based on visual inspection.