Despite the power and rigor of categorical logic, it lacks flexibility. Its methods really only apply to syllogisms in which each of the two premises can be translated into a standard-form categorical claim. For this reason, even an introductory treatment of logic calls for some discussion of modern symbolic logic. Truth-functional or propositional logic is the simplest part of symbolic logic, though you will find it both rigorous enough to let you carry out systematic proofs and broad enough to handle a wide range of ordinary arguments.
This chapter shows how to work with complex arrangements of individual sentences. You will use letters to represent sentences, and a few special symbols to represent the standard relations among sentences: roughly speaking, the relations that correspond to the English words "not," "and," "or," and "if-then." Truth tables and rules of proof show how the truth values of the individual claims determine the truth values of their compounds, and whether or not a given conclusion follows from a given set of premises.
1. Truth-functional logic is a precise and useful method for testing the validity of arguments.
Also called propositional or sentential logic, truth-functional logic is the logic of sentences.
It has applications as wide-ranging as set theory and the fundamental principles of computer science, as well as being useful for the examination of ordinary arguments.
Finally, the precision of truth-functional logic makes it a good introduction to nonmathematical symbolic systems.
2. The vocabulary of truth-functional logic consists of claim variables and truth-functional symbols.
Claim variables are capital letters that stand for claims.
In categorical logic, we sometimes used capital letters to represent terms (nouns and noun phrases). Keep those distinct from the same capital letters that now represent whole sentences.
Each claim variable stands for a complete sentence.
Every claim variable has a truth value.
We use T and F to represent the two possible truth values.
When the truth value of a claim is not known, we use a truth table to indicate all possibilities.
Thus, for a single variable P, we write:
P
T
F
Whatever truth value a claim has, its negation (contradictory claim) has the opposite value.
Using ~P to mean the negation of P, we produce the following truth table:
P ~P
T F
F T
This truth table is a definition of negation.
~P is read "not-P." This is our first truth-functional symbol.
The remaining truth-functional symbols cover relations between two claims.
Each symbol corresponds, more or less, to an ordinary English word; but you will find the symbols clearer and more rigid than their ordinary-language counterparts.
Accordingly, each symbol receives a precise definition with a truth table, and never deviates from that definition.
A conjunction (indicated by "&") is a compound claim asserting both of the simpler claims contained in it.
More precisely, a conjunction is true if and only if both of the simpler claims are true. We write:
P Q P & Q
T T T
T F F
F T F
F F F
Notice that this truth table needs four lines, not two, to capture all the possible truth values of P and Q.
We often say there's only one way for a conjunction to come out true, but many ways for it to come out false.
As the ampersand (&) should indicate, "and" is the most common way to describe a conjunction in English. But:
Other English words translate just as well into &: "but," "although," "while," "even though."
"And" sometimes has connotations that & lacks. "I had dinner and went to bed" suggests that one thing happened before the other; the logical conjunction carries no such suggestion.
A disjunction (indicated by "v") is a compound claim asserting either or both of the simpler claims contained in it.
In rigorous language, we say that a disjunction is false if and only if both of the simpler claims are; and we write:
P Q P v Q
T T T
T F T
F T T
F F F
Aside from the different arrangement of truth values in the final column, this table is set up like the last one.
It's as hard to make a disjunction false as it is to make a conjunction true.
"Or" captures the core meaning of the wedge, "v," but:
"Or" sometimes means that both of the simpler claims can't be true—for example, "You may take the lottery prize in a lump sum or receive payments over twenty years." The logical disjunction never forces us to choose between the disjuncts.
Other English words, like "unless," also get translated into disjunctions.
A conditional claim (indicated by "→") is a compound claim asserting the second simpler claim on the condition that the first simpler claim is true.
To define a conditional more exactly, we first need to define its parts.
The claim before the arrow is the antecedent, the one after it the consequent.
A conditional claim is false if and only if its antecedent is true and its consequent is false. Or:
P Q P → Q
T T T
T F F
F T T
F F T
We read "P → Q" as "If P, then Q." In many cases the logical conditional will strike you as different from the ordinary English "if-then" construction.
The essence of our definition is that the conditional only must be false under one set of circumstances, when the antecedent sets up a promised condition and the consequent does not deliver on it.
In other cases we are not pressed to call the compound claim false. See "Commonly Asked Questions" below for further discussion.
3. Three rules permit you to construct a truth table for any well-formed combination of claim variables and symbols. Use parentheses when you need them; put enough rows in the table to capture all combinations of truth values; make columns of the parts of the expression.
Parentheses specify where a truth-functional operation is doing its work.
5 + 2 makes no sense in arithmetic. It must be written either (5 + ) 2, in which case it equals 16, or 5 + ( 2), in which case it equals 11.
Similarly, the symbols that link claim variables make no sense when strung together without separation, as in P & Q v R → S. Write (P & Q) v (R → S) or whatever you mean.
The truth table must capture all possible combinations of truth values for the individual sentences contained in the complex expression.
Remember that the truth table is designed to show all the conditions under which a given expression is true or false. Because each of its component claim variables is independent of each other one, the table must reflect every combination.
Make a column at the left of the table for each of the claim variables. These are the reference columns.
If you have n claim variables in an expression, you will need 2^{n} rows: 2 variables require 4 rows, require 8 rows, 4 require 16 rows, and so on.
The rightmost column alternates Ts and Fs; the column just to its left goes T-T-F-F; the column to the left of that, T-T-T-T-F-F-F-F, and so on. The left-hand column is half all Ts and then half all Fs.
Here are the reference columns for a truth table built to handle three variables, P, Q, and R:
P Q R
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
The truth table must contain columns for the parts of the final complex expression, if any of those parts is not a single claim variable.
For example, if you are building a truth table for the expression (P v Q) → R, you should first make a separate column for P v Q and determine its truth values.
You will then refer to those columns in calculating the truth values for the final expression.
4. Truth tables so constructed produce a truth-functional analysis of more complex claims, and show whether two claims are equivalent.
As an example, take the sentence "Either Peleg and Queequeg will both go harpooning, or Queequeg won't."
We first render it, with obvious symbols, (P & Q) v ~Q.
The truth table contains two claim variables and thus needs four rows; and it must have columns for P & Q and ~Q:
P
Q
P & Q
~Q
(P & Q) v ~Q
T
T
T
F
T
T
F
F
T
T
F
T
F
F
F
F
F
F
T
T
Now we can say that the complex expression is false only in row , that is, only when P is false and Q is true.
When two expressions containing the same claim variables have identical columns in truth tables, we call them truth-functionally equivalent (see definition (1)).
You may think of equivalent expressions as claims that mean the same thing.
Consider the truth table for "If Queequeg goes harpooning, so will Peleg," symbolized Q → P:
P Q Q → P
T T T
T F T
F T F
F F T
This final column is identical to the final column for "Either Peleg and Queequeg will both go harpooning, or Queequeg won't." The two claims are truth-functionally equivalent.
5. The first significant work in analyzing and operating on claims with truth-functional logic is the work of translating them into symbolic form.
Ultimately there is no substitute for a careful examination of what the claims are saying.
Translating a compound claim into symbolic form means making its internal logical relations clear and precise.
Because ordinary language often gives us compounds with implied or submerged logical relations, we have to begin by making sure we know what they mean.
Especially with claims involving conditionals, a few rules speed up the process.
When "if" appears by itself, what follows is the antecedent of the conditional.
When "only if" appears as a phrase, what follows is the consequent of the conditional.
The placement of clauses in a sentence is not a reliable guide to their placement in a conditional. (Logical form often departs from grammatical form.)
Thus, "My car will run if you put gas in it" becomes (with "G" and "C" as our symbols) G →C.
"My car will run only if you put gas in it," on the other hand, translates as C → G.
"Provided," or "provided that," often introduces the antecedent of a conditional. "The car will run provided you put gas in it": G → C.
The expression "if and only if" goes peacefully into its logical form if we expand the claim it appears in, into a longer claim.
First, observe that "My car will run if and only if you put gas in it" may be rewritten, "My car will run if you put gas in it, and my car will run only if you put gas in it."
We have already symbolized those two compounds as G → C and C →G, respectively. It is child's play to link the parts with an ampersand and get (G →C) & (C →G).
Other sorts of conditional claims need to be inferred from the statement of necessary and sufficient conditions.
"Literacy is a necessary condition for college graduation" means that you must be literate to be graduated from college, though plenty of other things must be true as well.
We express this relationship by saying: If you are graduated from college, you are literate (G → L). Necessary conditions become the consequents of conditionals.
"Erudition is a sufficient condition for college graduation" means that if you have become erudite, you are guaranteed your graduation. That condition suffices.
We can say: If you are erudite, you will be graduated from college (E → G).
The word "unless," for all its subtleties, translates as "v."
In other complex English claims, the location of words like "either" and "if" shows how to group logical relations, and hence where to place parentheses.
"Either I will dance and sing or I will juggle" goes into logic as (D & S) v J, because the "either" and "or" tell you to put parentheses there.
By comparison, "I will dance and either sing or juggle" becomes D & (S v J). Not the same thing at all.
Along similar lines, "if" and "then" disambiguate claims that might otherwise be mistaken for one another. In "If I sing or yodel, then I'll get booed," we know to enclose the disjunction within parentheses: (S v Y) → B.
Compare: "I'll sing, or else, if I yodel, I'll get booed." This is written S v (Y → B).
6. Truth tables offer one method for testing an argument for validity. (We will also look at two others.)
This method builds from a single principle, the definition of validity.
Recall that for an argument to be valid, it must have a true conclusion whenever all its premises are true.
So, we enter all the premises of the argument, and its conclusion, in a truth table, and examine the rows in which all premises are true.
If the conclusion is true in all such rows, the argument is valid. If even one row exists in which all premises are true and the conclusion is false, the argument is invalid.
Sometimes all the premises of an argument cannot be true at once. (They contain a contradiction.) In that case the argument is still valid, for we have no rows in which all premises are true and the conclusion false.
When using this method, number the columns of your truth table and keep three sorts of columns distinct from one another.
First, on the left, are the reference columns, headed by single-letter claim variables. You use those to calculate all truth values.
Scattered among the other columns, you are likely to have columns for the parts of complex expressions. Do not confuse these with the premises and conclusions.
The columns for premises and conclusions are the only ones that matter.
7. A second method is known as the short truth-table method.
The short truth-table method is practically necessary.
Complete truth tables are tedious to fill out.
Moreover, the number of calculations of truth values means greater opportunities for small errors that can lead to a wrong answer.
The short truth-table method is a kind of indirect proof.
Rather than go directly to demonstrate an argument's validity, you work to see if it can possibly be invalid. If not, of course, it is valid.
An argument is invalid in case any circumstance exists in which all its premises are true and its conclusion false.
So the short method consists in trying to make the conclusion come out false and the premises true.
It is usually quickest to begin with the argument's conclusion, assigning the claim variables values that make that conclusion false and seeing what the values of the other variables must be.
Here is an argument:
~A v B
C → A ~B & D
~C
For the conclusion to be false, C must be true. We begin:
A B C D
T
We want all the premises to be true. If C is true, then the second premise can be true only if A is true; thus:
A B C D
T T
When A is true, naturally ~A is false. So the first premise can be true only on the assumption that B is true:
A B C D
T T T
What about the third premise, ~B & D? We want it to be true. But no matter what truth value we give to D, ~B is false and makes the third premise false.
We have failed to produce a set of circumstances that make the premises all true and the conclusion false; the argument is valid.
Some examples make it hard to use this method.
There might be too many ways to make premises come out all true and the conclusion false. Sometimes we simply have to consider several possibilities.
At other times, it is easier to begin with one premise. Assume that the premise is true, and carry out the argument to make all the other premises true as well, and the conclusion false.
8. The method of deduction is the third and most sophisticated way of demonstrating an argument's validity.
The method has disadvantages.
It can be cumbersome as a test of invalidity, because failing to arrive at a conclusion from a set of premises can mean either that the argument is invalid, or merely that we have not found a good proof for a valid argument.
Deduction also presupposes familiarity with a set of rules that guide you through the proof; these must be learned until they feel automatic, and learning the rules takes some time and energy.
However, deduction possesses the great advantage of exposing the logical relations at work in an argument.
Doing such a proof or derivation resembles thinking through an argument.
When using this process you learn not only that a conclusion is true when all the premises are, but also why.
Because deduction brings out the actual logical connections in an argument, it also makes excellent training in critical thinking.
A truth table is a (very slow) computer program that delivers an answer for any argument you put in: a machine.
Deduction, on the other hand, works like a tool and requires craft. So it leaves you more skilled than the truth tables can.
In every deduction, certain basic principles apply.
You begin with the set of premises and apply rules from Group I and Group II (see below) to them.
If applying a rule produces the conclusion, the deduction is complete. If not, the result becomes another line in the proof, which you can use as you go as if it were a premise.
When you produce a new line for the deduction, write (to the right of it) the lines you used in producing it, and the abbreviation for the rule you used. This is called the annotation for the deduction.
9. Elementary valid argument patterns constitute the first set of rules you must learn before carrying out a deduction (Group I rules). These apply only to whole lines of a deduction, not to the individual parts of lines.
Modus ponens (MP; rule 1) says that, given a conditional claim in one line of a deduction, and the antecedent to that conditional in another line, you can deduce the consequent:
A →B
(A & B) →C
~C → (A v ~B)
A
A & B
~C
B
C
A v ~B
Note, in the above examples, that either antecedent or consequent may be more complex than a single letter.
Modus tollens (MT; rule 2) functions similarly. Given a conditional claim and the denial to its consequent, you can deduce the denial of the antecedent:
A → B ~B
~A
The chain argument(CA; rule ), one of the easiest to remember, applies when the consequent of one conditional is the antecedent of another:
A →B B → C
A → C
Disjunctive arguments (DA; rule 4) let you infer one of two disjuncts in a disjunction, when you are given the negation of the other disjunct:
A v B ~A
B
The motivating idea is simple: Given two alternatives, and the denial of one of them, you take the other alternative.
Remember that an unnegated expression is itself the negation of the same expression with "~" in front of it. So, given ~(A & B) v C, and (A & B), you can conclude that C.
Because a conjunction asserts that both of its conjuncts are true, you can begin with any conjunction and derive either conjunct: This is simplification (SIM; rule 5).
A & B
A (or B)
Conjunction (CONJ; rule 6), as its name implies, takes any two separate lines of a deduction and joins them:
A B
A & B
It is worth restating the obvious: The parts of this conjunction may be as complex as you like:
A → B C v D
(A → B) & (C v D)
Superficially similar to these rules about conjunction is the rule of addition (ADD; rule 7). Given any line in a deduction, you may create a disjunction that contains that line as one of its elements, and anything at all as the other one:
A
A
A
A v B
A v (B & C)
A v (C → ~B)
The constructive dilemma (CD; rule 8) begins with two conditional claims and the disjunction of their antecedents, and moves to the disjunction of their consequents:
A → B
C → D A v C
B v D
If at least one of the antecedents is given, at least one of the consequents can be inferred. That's all the rule says.
This rule of course relies on modus ponens (rule 1).
In a destructive dilemma (DD; rule 9) we have two conditionals again, but the disjunction of the negations of their consequents; we derive the disjunction of the negations of the antecedents:
A → B
C → D ~B v ~D
~A v ~C
As the constructive dilemma relied on modus ponens, the destructive dilemma relies on modus tollens (rule 2).
If at least one of the consequents is being denied, then at least one of the antecedents is being denied as well.
10. Truth-functional equivalences form the Group II set of rules (see definition (2)).
These rules work somewhat differently from the argument patterns that make up Group I rules.
Truth-functional equivalence means that two claims say exactly the same thing. We can therefore replace them with one another without changing the meaning of a claim.
Unlike Group I rules, which are rules of inference, rules of equivalence work equally well in both directions.
Also unlike those rules, which can only be applied to complete lines of a deduction, these let us replace any part of a line (part of a claim) with its equivalent.
Remember that you will still make annotations: Indicate the line you went to, and what you did to it.
Double negation (DN; rule 10) lets you remove two consecutive negation signs, or insert two such signs, anywhere:
A ↔ ~~A
Commutation(COM; rule 11) applies to conjunctions and disjunctions. The order of their elements does not matter:
(A & B) ↔ (B & A)
(A v B) ↔ (B v A)
This should remind you of the commutative laws of addition and multiplication in arithmetic (7 + 5 = 5 + 7).
Just as those laws do not apply to subtraction or division, so commutation here does not apply to conditionals.
According to the rule called implication (IMPL; rule 12), conditionals can be turned into disjunctions, and vice versa:
(A → B) ↔ (~A v B)
People sometimes find this one hard to remember, or even hard to believe. You may want to construct truth tables for A →B and ~A v B and see their equivalence.
Again, bear in mind that these equivalence rules work for more complicated expressions and for parts of claims:
A & (~B → C) ↔ A & (B v C)
(A & ~B) R C ↔~(A & ~B) v C
Contraposition (CONTR; rule 1) switches the antecedent and consequent of a conditional with the negations of one another:
A → B ↔ ~B → ~A
~(A & B) → (C v ~D) ↔ ~(C v ~D) → (A & B)
DeMorgan's Laws (DEM; rule 14) govern the negations of conjunctions and disjunctions:
~(A & B) ↔ (~A v ~B)
~(A v B) ↔ (~A & ~B)
Always remember to change the & to a v, or vice versa, when moving the ~ inside the parentheses.
One rule you might find hard to memorize, but that is worth keeping in mind, is exportation (EXP; rule 15):
(A → (B → C)) ↔ ((A & B) → C)
This is not as strange as it looks. When you say, "If I get invited to the party, then if it's on Friday night, I can go," you're naming two conditions that both must be met before you can go. You may as well say, "If I get invited to the party and it's on Friday night, I can go."
Note: This rule does not apply to a complex conditional of the form (A → B) → C.
Association (ASSOC; rule 16) tells you that strings of conjuncts or disjuncts may be grouped in any way:
(A & (B & C)) ↔ ((A & B) & C)
(A v (B v C)) ↔ ((A v B) v C)
Like commutation, this rule should remind you of the comparable rules of association in arithmetic.
All the signs must be the same for association to apply: You must have a string of letters all joined by & or v. (And it does not work for the conditional.)
The two versions of distribution (DIST; rule 17) let us handle combinations of conjunction and disjunction, as follows:
(A & (B v C)) ↔ ((A & B) v (A & C))
(A v (B & C)) ↔ ((A v B) & (A v C))
Note the symmetry of the two rules. Once you have learned one, you can turn it into the other by replacing every & with a v, every v with an &.
Also note that the thing outside the parentheses, which has a sign next to it, keeps that sign next to itself.
Finally, trivially, and obviously, we have the rules of tautology (TAUT; rule 18). No comment is necessary:
A ↔ (A v A)
A ↔ (A & A)
11. Along with these rules of deduction, the method of conditional proof (CP) offers a strategy for showing the truth of conditional claims.
The idea behind the strategy is this: If a set of premises supports the moves from A to B, then those premises show the truth of A → B.
The essence of the strategy is that A is not a premise given in the argument.
We add A as a hypothetical assumption.
Once we derive B (the consequent of the desired conditional) within the deduction, we may conclude that B follows from A, that is, that A → B.
The complications of conditional proof follow from the fact that this additional premise is not really given. It must be used to derive the needed conditional claim and then eliminated, or discharged.
Though the method may strike you as needlessly complex, it actually turns very hard arguments into manageable ones.
The method of conditional proof consists of a few new steps:
We begin by writing down the assumed premise, the antecedent of the desired conditional.
We circle the number of that step.
The annotation reads, "CP Premise."
We continue through the deduction until we reach the consequent of the desired conditional.
In the next line, we state the conditional that unites the CP premise with the consequent.
We draw a line to the left, connecting the CP premise with the consequent.
The annotation lists all the steps bracketed by that line (e.g., "2–5"), and gives CP as the rule.
In addition to the steps just listed, be sure to follow certain rules for conditional proofs:
They only prove the truth of a conditional claim.
If you use more than one CP premise to reach a single final conditional, discharge the premises in reverse order from their assumption. (Lines on the left don't cross.)
Once a premise has been discharged, none of the steps bracketed by the line may appear again in the deduction.
Discharge all CP premises.
12. Truth-functional logic as defined in this chapter is a formal system with two properties of great interest to philosophers and logicians.
Truth-functional logic is sound.
The soundness of a system means that all proofs following the rules of that system will be valid arguments.
If you apply this chapter's rules properly in your deductions, they will all produce valid arguments; from true premises you are guaranteed to reach true conclusions.
Even more remarkably, truth-functional logic is complete.
The completeness of a system means that every valid inference in the system can be produced within that system.
If conclusion C follows from a set of premises P1, P2, etc., then it can be shown to follow.
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