Introducing Sentential Logic (SL) Part II

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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. Then as we shift to the next column on the left, we double the period of alternation (1T followed by 1F, then 2T’s by 2F’s, 4T’s by 4F’s, and so on). In this case, the result will look as follows:

P Q R (P→Q) (~Q→~R) (~P→R) Q T T T T T F T F T T F F F T T F T F F F T F F F

Note that when there are n component atomics in a sequent, there will be 2n rows underneath the first in its corresponding truth table representation. Each of these rows will correspond to a distinct possible interpretation of those atomic sentences. The fifth row (underneath the first), for example, represents the interpretation in which P is assigned the truth value of False, while Q and R are both assigned the truth value of True.

We then fill out the columns underneath the other formulas according to the truth values that they would take according to the assignments of truth values indicated in that row:

P Q R (P→Q) (~Q→~R) (~P→R) Q T T T T T T T T T F T T T T T F T F F T F T F F F T T F F T T T T T T F T F T T F T F F T T F T F F F F T T F F

We are now in a position to put our understanding of sequent expressions to work. Recall that the double turnstile tells us that there are no interpretations in which every sentence on the left is true and some sentence (perhaps the only sentence) on the right is false. Is this correct in this case? What we are looking for is a row in the above truth table for which the cells underneath the formulas on the left of the original sequent are all true, and for which the cell (or some cells) underneath the formula(s) on the right is false. Looking underneath the columns of the formulas on the left, we can see that there are three rows in which all of those formulas are true (the first, second, and fifth rows beneath the top). But in each of these rows, the truth value of formula on the right of the double turnstile (represented in the final column) turns out also to be true. Hence there are no rows (or interpretations) in which all of the formulas on the left of the double turnstile are true and the formula on the right is false. The sequent expression is thus correct!

11. Now in some circumstances, we can argue for the correctness or incorrectness of a certain sequent expression in an informal, but nevertheless rigorous fashion in more or less natural language. Consider once more the example above. We are wondering whether there could be any interpretations that make (P→Q), (~Q→~R), and (~P→R) all true and Q false. Now for Q to be false and (P→Q) to be true, P would have to false as well. Similarly, for Q to be false and (~Q→~R) to be true, ~R would have to true (why?), and thus R would have to be false. But if both P and R have to be false, then (~P→R) would also have to be false. And so there is no way in which one can make Q false and also (P→Q), (~Q→~R), and (~P→R) all true. Hence we can say once again that the sequent in question is correct.

12. To take another example, here’s a truth table representation for evaluating the following sequent: ~(A → B), (~BvC), ~C |= (A→ C) .

A B C ~(A → B) (~BvC) ~C (A→ C) T T T F T F T T T F F F T T T F T T T F F T F F T T T F F T T F T F T F T F F F T T F F T F T F F F F F F T T F

Now look at the fourth row following the top (the one corresponding to a truth-value assignment of True to A and False to both B and C). Note that on this assignment of truth values all of the formulas on the left of the double turnstile are true, and the formula on the right of the double turnstile is false. Hence we do have an interpretation in which everything on the left is true and something on the right is false. This interpretation is called a counterexample to the sequent, which thereby demonstrates its incorrectness. Hence we may write ~(A → B), (~BvC), ~C |≠ (A→ C) .

13. In conclusion, the take-home message is this: an interpretation in SL consists of 1) an assignment of truth values to the various capital letters plus an assignment of an n-place truth-function to each n-place operator (usually the various operators will be given the interpretation that they are conventionally understood to symbolize). That way, one can use an interpretation to calculate the truth values of all the sentences of S.L., which we can in turn use to evaluate the correctness of sequent expressions. COMMIT THIS TO MEMORY!!

14. Exercises

1. Write out truth tables representing the following truth functions:

(a) The three-place function which takes the value T just in case (j.i.c.) exactly one of its arguments or sentential inputs takes the value T

(b) The three-place function which takes the value T j.i.c. at least two of its arguments take the value T.

(d) The three-place function that takes the value F j.i.c its first argument is T and the second and third arguments have opposite truth values.

2. For each of the truth functions specified above, find a formula in SL (using the standard operators) that expresses it.

3. How many total three-place truth functions are there? (Though you don’t need to list them all, it would be nice for you to give some sense of how you arrived at your answer.)

4. Use the truth table method to determine whether the following sequents are correct:

(a) ~(P→Q) |= (P→~Q)

(b) (P&~Q) v (~P&Q) |= (P v ~Q)

(d) ((P&Q) & (~P&~Q)) |=

(e) |= ((P&Q) v ~Q)

(f) |= ((P→Q) v (R→P))

(g) ((Q v R) →P) =||= ((Q→P) v (R→P))

[Think of =||= as asserting that the entailment goes “both ways.” That means you’ll have to provide separate determinations for each direction.]

5. Without using a truth table, determine whether the following sequents are correct. Be sure to indicate your reasoning.

(a) (P→Q), (Q→R), (R→S) |= (P→S)

(b) (PvQ), ~(P&R), ~(Q&S) |= ~(R&S)

(d) (P→(QvR)), (R→(P→S)), ~(S&P) |= (Q→P)

6. Show why the following “basic principles” of the standard operators hold:

(a) The basic property of Negation

NEG: Г|=φ just in case Г (or if and only if) Г,~φ |= .

(b) The basic property of Conjunction

CONJ: Г |= (φ & ψ) just in case Г |= φ and Г|= ψ.

DISJ: Г, (φ v ψ) |= just in case (or if and only if) Г, φ|= and Г, ψ|=.

(d) The basic property of the Material Conditional

COND: Г |= (φ → ψ) just in case Г ,φ |= ψ