Introducing Sentential Logic (SL) Part II – Semantics 1. Our refined understanding of entailment invokes the notion of an interpretation. For notions of entailment that center on truth, the purpose of an interpretation is to provide stable and unique assignments of truth values to the various sentences that can participate in sequent expressions. The idea is that we use interpretations to assign either the value true or the value false (but not both!) to each and every sentence of a formal language. The truth and falsity of a sentence is thus relative to an interpretation. 2. In what is called “classical” logic, we assume that each and every sentence will take on exactly one of the following two truth values: True (T) or False (F). There are non-classical truth-functional logics as well, in which other truth-values are introduced, or in which some sentences may be assigned neither of the classical truth values, or even more than one truth-value (for instance, in epistemic logic some sentences may be assigned the value B – standing for both true and false). There are also semantics that don’t employ the notion of a truth-function at all, but instead work with other kinds of primitives (for example, incompatibility semantics). At this stage in the game, we will simply be working with a semantics of the plain-ole vanilla classical type. 3. Basic Properties of |= From the idea that interpretations provide stable and unique assignments of truth values, we may already stipulate the following basic “structural” principles of entailment: Assumptions: For any sentence φ, φ|= φ. That is, any sentence entails itself. Thinning (or Persistence): If Γ|=φ, then Γ,ψ|=φ. Basically, this principle tells us that adding any sentence (or set of sentences) as premises to a valid entailment will not affect its validity. Cutting: If Γ|=φ and φ,Δ|=ψ, then Γ,Δ|=ψ Essentially, this principle captures the “transitivity” of entailment: If some formula is a consequence of some premises and in turn that formula (along with some other possible premises) entails some further formula, then that original set of premises (in concert with the other supporting premises) will entail that further formula, without explicit mention of formula originally entailed. The semantics of SL 4. We now turn specifically to sentences in our logical language SL. SL is what is called “wholly truth- functional.” That means that in giving it an interpretation, all we need to do is to supply truth values for the various capital letters or atomic wffs. The truth-values of compound wffs are then completely determined from the truth-values of simpler (less compound) wffs - indeed, those which compose that very compound wff. That is, the semantics of SL is not just recursive; it is also compositional. 5. You might well already know how to compute the truth values of compound wffs in SL. The operators stand for various familiar truth-functions. A function is simply a “mapping” from inputs (or arguments) to outputs (or values). Specifically, an “n-place function” goes from ordered “n-tuples” of objects from one domain to objects of another domain (which might be one and the same). An “n-place truth function” is thus one that takes n-tuples of truth values as “inputs” and spits out specific truth values as outputs. Individual truth functions are easily represented by means of truth tables. Here, for instance, is a representation of the 3-place truth function (which we can call %) that takes the value F just in case its first argument is T, and there is at least one other T among its other two operators: φ χ Ψ %( φ, χ, ψ) T T T F T T F F T F T F T F F T T F T T F T F T F F T T F F F T In this table, the columns underneath the three stand-alone Greek letters systematically list the various permutations of truth-values that the arguments (or inputs) of a three-place truth function may take. For example, the 5th row of the entire table (which is the 4th row underneath the initial one), represents a condition in which the first argument is T, while the second and third arguments are F. There are 8 distinct permutations of truth values that these three arguments may take, hence 8 rows underneath the initial row. The column of cells underneath the % then display the specific truth value that the % function takes whenever its 3 arguments (or inputs) take on the specific truth values listed to the left of that cell. 6. By convention, the tilde in SL is a one-place operator that represents the negation function. Whenever the sentence it “operates on” (or prefixes) is true, then it is false, and vice-versa. Here, then, is its truth table representation: φ ~φ T F F T These specific functions of the ampersand, wedge, and arrow are easily illustrated by means of the following truth tables: φ ψ (φ&ψ) (φvψ) (φ→ψ) T T T T T T F F T F F T F T T F F F F T We say then that the ampersand stands for the two-place conjunction function: it is true just in case both of its arguments (or conjuncts) are true, and false if either of them (or both) are false. The wedge stands for the disjunction function: it is false just in case both of its arguments (or disjuncts) are false, and true if either (or both) are true. The arrow signifies the material conditional: it is false just in case its first argument (the antecedent) is true and its second argument (or consequent) is false, and it is true just in case either its antecedent is false or its consequent is true. You will need to commit these rules to memory. The ampersand thus bears some resemblance to (some uses of) the English word “and,” while the wedge functions roughly like some uses of the English word “or” (specifically those in which the “or” is used in an inclusive sense). While it is sometimes claimed that the arrow corresponds to the English “if.. then,” this correspondence is tenuous at best. It is better to leave the arrow as it is, and not to claim that it is anything remotely equivalent to the conditional in English. 7. There is a different truth function for each distinct way one can fill out the column of a truth table. That means that there are 16 distinct two-place truth functions (for each of the 4 cells in the column, one can have either a T or an F; 24=16. Other operators may be devised to stand for other truth functions (i.e. the biconditional, or the n/and and n/or operations illustrated below). Indeed, there are many, many truth functions that simply go nameless, especially those with more than 2 arguments. φ ψ (φ↔ψ) (φ↑ψ) (φ↓ψ) T T T F F T F F T F F T F T F F F T T T 8. The truth-functionality of SL also allows for another nice feature: the substitution of equivalent formulas. If two formulas are logically equivalent to one another (they take on the same truth value in every possible situation), then one may freely substitute an appearance of one with the other inside any compound formula of SL to form another formula logical equivalent to that compound. 9. Truth functions may operate upon the outputs of other truth functions. That is what happens with compound or complex formulas. Such compounds can serve to express yet more complex truth functions. Here, for instance, are truth table representations of progressively compound sentence of SL: φ χ ψ ( χ v ψ) (φ&( χ v ψ)) ~ (φ&( χ v ψ)) T T T T T F T T F T T F T F T T T F T F F F F T F T T T F T F T F T F T F F T T F T F F F F F T In this table, the fourth column here represents the disjunction of the second and third columns. The fifth column displays the conjunction of the first and fourth columns. Finally, the fifth column is the negation of the fourth. Notice how this column matches precisely the column above for %. We can thus say that the formula ~ (φ&( χ v ψ)) expresses the very same truth function as %. 10. Truth tables also enable us to put our unpacking of sequent expressions to work. Suppose that we wish to evaluate the following sequent: (P→Q), (~Q→~R), (~P→R) |= Q . What we do is we set up a truth table that jointly displays the truth values of all of these formulas under every permutation of truth and falsity of their component atomic sentence letters. We begin by listing all of the sequent expression’s component atomic sentences along the left of the first row, followed by all of the formulas on the left of the double turnstile, and then the formula(s) on the right: P Q R (P→Q) (~Q→~R) (~P→R) Q We then fill in the columns underneath the component atomics in a way that captures all of the various combinations of truth and falsity these atomics can take. These are typically called a truth table’s base columns. One can most easily and systematically accomplish this task by starting at the right-most base column and alternate between truth and falsity as one goes down the column.