# Satisfaction and assignments

When you define satisfaction for quantified formulas, e.g., $$\forall x\, A$$, you have to have a way to make $$x$$ range over all elements of the domain. Here are the common options:

A. Tarski-style: use variable assignments $$s\colon V \to D$$ (where $$V$$ is the set of variables and $$D$$ the domain), then define $\mathfrak{M}, s \models \forall x \, A \iff \mathfrak{M}, s’ \models A \text{ for all x-variants s’ of s}.$ This requires a definition of “$$x$$-variant” but is otherwise very clean. Its drawback is that it obscures how we let $$x$$ range over all elements of $$D$$, and my students have a hard time understanding it and an even harder time working with it.

B. Alternative Tarski-style: we use variable assignments as above, but avoid talking about $$x$$-variants. Instead, we define the notation $$s[m/x]$$, the variable assignment just like $$s$$ except it assigns $$m \in D$$ to $$x$$. Then we have $\mathfrak{M}, s \models \forall x \, A \iff \mathfrak{M}, s[m/x] \models A \text{ for all } m \in D.$

C. Model theory style: instead of introducing variable assignments that provide the interpretation for variables, we define directly when a formula is satisfied by a sequence of objects: if the variables of $$A$$ are among $$y_1, \dots, y_k$$ then $$\mathfrak{M} \models A[n_1,\dots, n_k]$$ means what $$\mathfrak{M}, s \models A$$ means Tarski-style for the assignment $$s$$ that maps each $$y_i$$ to $$n_i$$. Then the clause for the universal quantifier becomes $\mathfrak{M} \models \forall x \, A[n_1, \dots, n_k] \iff \mathfrak{M} \models A[m, n_1, \dots, n_k] \text{ for all } m \in D.$ This is simple in that it avoids an intermediary function, but can easily be confusing for beginning students because it is neither clean nor precise. We have to understand that $$A$$ is a formula with the free variables $$x, y_1\, \dots, y_k$$. But what determines the order? Or, put another way, which object interprets which variable?

D. In logic textbooks for philosophers you often see semantics developed for sentences only (i.e., formulas with free variables are avoided). Given a structure $$\mathfrak{M}$$ we can define $$\mathfrak{M}[m/a]$$ as the structure that’s just like $$\mathfrak{M}$$ except the constant $$a$$ is interpreted by $$m\in D$$. Then we can define truth (not satisfaction) using $\mathfrak{M} \models \forall x \, A \iff \mathfrak{M}[m/a] \models A[a/x] \text{ for all } m \in D,$ where $$A[a/x]$$ is the substitution operation and $$a$$ is a constant not already occurring in $$A$$.

E. Finally, there’s Robinson-style: we treat every $$m\in D$$ as a constant symbol that names itself. Then substituting $$m$$ for $$x$$ in $$A$$ is possible, since $$m$$ belongs to both the domain of the structure and to the language, and we can write $\mathfrak{M} \models \forall x \, A \iff \mathfrak{M} \models A[m/x] \text{ for all } m \in D.$ Naturally, this is not something philosophers like to do because it just seems confused to allow domain elements to function as linguistic symbols naming themselves.

Maybe I’ll find the time to write a paper tracing the origins of all of these at some point. But for now, I wonder: which way is best, pedagogically? Specifically, the Open Logic Project uses Tarski-style, but I’d like to replace it with a definition that is easier to understand and avoids the use of $$x$$-variants. Which would you prefer for the OLP? Which do you prefer in your own teaching or research and why?

# Letter grades in Brightspace/D2L (or other LMS)

To set up the grade scheme in the first place, and to solve problem b), we have to do some math. D2L does grade schemes in percentages, but we think more naturally of grade point values: an A+ is 4.3, and A is 4, and A- is 3.7, etc. To convert these into percentages, just divide by the maximum score (i.e., 4.3): A+ is 100%, A is 4/4.3 = 93.02%, A- is 3.7/4.3 = 86.04% etc. Do the same for any other grade scheme you want to use. E.g., if you want slash grades, you’d assign 3.5 to A/B, and 3.5/4.3 = 81.50%.

Note that whatever the maximum score is in your grade scheme here should also be the “maximum score” in any individual grade item that uses this score.

The grade scheme in D2L asks for not just an “assigned value %” for each letter grade (or other scale item), but also for a “start %”. If you average the grades (or rather, their “assigned %” values), you may get a value that does not correspond precisely to a letter grade. You have to decide where to start to “round up”. Say, you have three papers. If you want a student to get an A overall only if they get three As, then the “start %” should be the same as the “assigned value %” for an A (93.02%). But maybe you want to give them an A if they turn in two As and an A- (or some other threshold, e.g., 3 As and 2 A- on 5 papers, etc.). If so, compute the average of the grade point values of the threshold pattern, e.g., if A/A/A- earns an A, (4.0+4.0+3.7)/3 = 3.9. Then convert that to a percentage: 3.9/4.3 = 90.70. You probably want to be careful with the start % value of a D: If you have three papers and want to pass a student with a D if they have turned in 2/3 papers with Ds, the start % is (1+1+0)/3/4.3, i.e., 15.50%. But if you have many assignments, a low percentage like that will make it possible to earn a D with an A on a single assignment (if you have 6 papers, and a student gets an A on the first and then never submits another paper, they will earn 15.5% overall. But probably you don’t want to pass that student). For this reason, I like to make the start value of a D the percentage equivalent of 0.9/4.3, or 20.93).

If you’re using some other LMS, you will have to figure out how to do all this there. E.g., Moodle has grading scales (corresponding to D2L’s “assigned %” scale) and also a course-wide system of converting percentages to letter grades, which corresponds to D2L’s “start %” scale.