Download Truth-Functional Propositional Logic

Truth-Functional Propositional Logic
by Sidney Felder
Truth-functional propositional logic is the simplest and expressively weakest member of the class of
deductive systems designed to capture the various valid arguments and patterns of reasoning that are
specifiable in formal terms. The term ‘formal’ in the context of logic has a number of aspects.
(Some of these aspects—those connected with the distinction between form and content—will be
discussed a bit later). The aspect concerning us most right now is that connected with the capacity
to specify meanings and draw inferences by the more or less automatic application of predetermined
rules (rules previously either learned, deduced, or arbitrarily stipulated). This involves far more than
the substitution of simple symbols for words. The examples to have in mind are the rules and operations employed in arithmetic and High School algebra. Once we learn how to add, subtract, multiply, and divide the whole numbers {0,1,2,3,...} in elementary school, we can apply these rules, say,
to calculate the sum of any two numbers, automatically, without thinking, whether we understand
why the rules work or not. The uniformity, simplicity, and regularity of these arithmetical rules, and
their applicability with minimal understanding, is shown by the existence of extremely simple artificial devices for effective arithmetical calculation such as the ancient abacus. Before any system can
be “automated” in this way (that is, before it can be converted into a system for calculation (i.e., a
calculus), its various elements must be represented in an appropriate form. For example, if each
natural number was represented by a symbol possessing no relationship to the symbol representing
any other number, no method could exist for adding two numbers by operations employing paper
and pencil. It is necessary to represent the system of numbers in a systematic form, to formalize
their representation, if such mechanical operations are to be possible. This can be done with various
fragments of logical reasoning also, and hence we speak, for example, of the propositional and firstorder predicate calculus as well as of propositional and first-order predicate logic.
Truth-functional propositional logic, also known as sentential logic, the sentential calculus, the statement calculus, etc. studies the expressive and deductive relationships among certain combinations of
propositions.
The only objects of propositional logic that possess autonomous expressive significance are complete
sentences, statements, or propositions that make assertions possessing definite truth values. (There is
a subtlety here that we will illuminate shortly). All the deductive logics we are examining in this
course are bivalent, meaning that (for the purposes of this course) each sentence is to be considered
either true or false. (There are logics in which more than two truth values are admitted, and logics
in which there are “truth-value gaps”, but unfortunately we will not have time to consider them
here). To reinforce the convention that we are admitting no alternative to a sentence’s truth or falsity (meaning that the values ‘true’ and ‘false’ are exhaustive), it is best to interpret ‘false’ in this
context as equivalent to ‘not true’, meaning that the denial of a sentence’s truth is considered equivalent to the assertion of its falsity.
The symbols A1 , A2 , A3 ,... represent the atomic (elementary) sentence forms from which all others
are formed. Although it is probably best to imagine these symbolic complexes as potentially standing for comparatively simple statements such as “It is snowing”, “It is sunny”, “Albert Einstein published the Special Theory of Relativity in 1905”, and “The moon orbits the earth”, a letter can represent an arbitrarily complex statement: There is nothing in principle to prevent us from letting A1
represent the conjunction of all the assertions made either in the text of Edward Gibbon’s The
Decline And Fall Of The Roman Empire or in the text of Whitehead and Russell’s Principia
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Mathematica. Because in logic we are concerned with the manner in which the truth values of certain compound expressions are related to the truth values of both atomic sentences and other compound expressions, the atomic sentences are best conceived as place-holders for truth values rather
than as ordinary sentences whose truth or falsity is susceptible to evaluation. The decision to represent a sentence by a letter with a subscript commits us to treating it as indivisible or atomic in the
context in which that representation holds. Operationally, this has a two-fold significance. Syntactically, atomicity signifies that an atomic sentence possesses no inner structure, and in particular no
connectives. Semantically, atomicity signifies that truth-values are freely assignable to the class of
atomic sentences, meaning that the truth-values assigned to any set of atomic sentences places no
constraints on the truth-values assignable to any other set of atomic sentences. This implies, for
example, that the truth values freely assigned to an arbitrarily chosen set of atomic sentences S can
never “collide” with the truth values that we choose to assign to any set of elementary sentences
outside S. On the other hand, the fact that the truth values possessed by non-atomic sentences can
in general collide with each other underlies the possibility of deducing one set of sentences from
another.
Truth-Functional Connectives
At the expressive and deductive heart of propositional logic are the truth-functional logical connectives. The application of these connectives permit us in the first instance to construct sentences, formulae, or expressions of arbitrary degrees of formal complexity. Propositional logic is ‘truth-functional’ because the rules of formation for the construction of compound expressions out of simpler
expressions are such, and the semantic properties of the connectives are such, that together they
imply that the truth value of any expression (compound of sentences) composed in a legal way is
uniquely determined by the truth values of the atomic sentences appearing in the expression. (The
proof of this fact is not trivial, but we here take this property, called unique readibility, as given).
Thus a function (an assignment or valuation) that associates a definite truth value to each element of
a set of atomic sentences S uniquely determines the truth value of all compound expressions composed solely of the atomic sentences of S. To elaborate: Consider any set of atomic sentences S and
the set T that includes together with all the elements of S all the non-elementary expressions that
can be formed from the elements of S by applying the truth-functional connectives an arbitrary finite
number of times. (Thus T contains the same atomic sentences as S as well as all compound expressions consisting solely of the atomic sentences of S). It can be proven that any two assignments v
and w that agree on what truth values they associate with the atomic sentences of T (i.e., agree on
S) agree on all sentences of T.1
We will now discuss the intuitively most accessible connectives employed in the construction of
complex formulae from combinations of simpler formulae.
And, also designated conjunction, is most commonly represented by an ∧ or an ampersand (&). The
formula A ∧B produced by combining arbitrary sentences A and B, either atomic or composite, by
this connective is true if and only if A is true and B is true; in other words, the sentence A ∧B is
true under the circumstances, and only under the circumstances, that both sentence A and sentence
1
Given a function f defined on a domain (set) S, any function g defined on a superset T of S is called an extension
of f if f and g both agree on their common domain of definition S. In general, functions admit of an infinite number
of distinct extensions, but there is only one extension from truth value assignments defined on any set of atomic
sentences S to truth value assignments defined on expressions consisting solely of occurrences of S.
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B are true. Since A is true under the condition that the state of affairs corresponding to A in fact
holds, and since B is true under the condition that the state of affairs corresponding to B in fact
holds, the sentence A ∧B is true under the (in general more stringent) condition that both the state
of affairs corresponding to A holds and the state of affairs corresponding to B holds. Thus let A be
“Jack is ill” and B be “Joan won the game”. If I say “Jack is ill and Joan won the game”, I will
be making a true statement if and only if both “Jack is ill” is true (that is, Jack is ill) and “Joan
won the game” is true (that is, Joan did win the game). The case in which the conjunction “Jack is
ill and Joan won the game” is true is the case in which True is assigned to “Jack is ill” and True is
assigned to “Joan won the game”, TT for short. On the other hand, “Jack is ill and Joan won the
game” is false under the conditions in which 1) "Jack is ill” is true and “Joan won the game” is
false (TF); 2) "Jack is ill” is false and Joan won the game” is true (FT); and 3) "Jack is ill” is false
and “Joan won the game” is false (FF). Thus, just as in ordinary language, the falsity of a conjunction is entailed by the falsity of even a single one of its conjuncts. This means that if I make a
claim such as A ∧B∧C∧D∧E, the discovery that even a single one of these conjuncts is false—either
A or B or C or D or E—is sufficient to falsify my claim. To make this quantitatively precise, there
is one case (TTTTTTTT) in which A ∧B∧C∧D∧E is true and 255 cases in which A ∧B∧C∧D∧E is
false. Finally, note that the ‘and’ employed in specifying the cases or conditions (1) , (2) , (3) , and
(4) invokes a generic concept of conjointness or combination indispensable to defining any of the
connectives discussed here (and indeed indispensable to defining or considering anything else). The
operations we are now defining connect sentences (and, in more general contexts, symbolic expressions), but they are defined in terms of the relationships among states of affairs.
Not, also designated negation, is most commonly represented nowadays by prefixing a tilde (˜), a ¬ ,
or a minus, to the proposition being negated. Unlike conjunction, negation is a unary rather than a
binary connective, so there are only two cases to consider: 1) that in which A is true and 2) that in
which A is false. The application of a negation operator ¬ to a sentence A produces a sentence ¬ A
called the negation or the contradictory of the sentence A. By definition, ¬ A is true if an only if A
is false, and ¬ A is false if and only if A is true. Because the negation of an arbitrary formula A is
false if and only if A is true and true if and only if A is false, the negation of the negation of A,
¬ (¬ P), possesses the same truth value as A itself. This implies that ¬ (¬ (¬ A)) has the same truth
value as ¬ A, that ¬ (¬ (¬ (¬ A))) has the same truth value as ¬ (¬ A) and hence the same value as A,
etc.
The attempt to formulate the negation of a sentence in ordinary language will help isolate the concept of negation that we regard as logically fundamental. The negation of the sentence “John went
to the dentist” is “John didn’t go to the dentist”—there is hardly room for misunderstanding here.
The caveats now begin: First, the negation of the sentence “It is boiling hot” is not “It is freezing
cold” but rather “It is not boiling hot”; similarly, the negation of “He is very far away” is not “He is
very close by” but rather “He is not very far away”. The pair of sentences “It is boiling hot” and
“It is freezing cold”, which are from one natural (extra-logical) point of view diametrically opposed,
are called contraries, as are the sentences “He is very far away” and “He is very near”. By our
definition, the negation of a sentence is true if and only if the original statement is false, but this
property does not hold for the pair of sentences “It is boiling hot” and “It is freezing cold”: Because
these two extreme temperatures do not exhaust the range of possible temperature conditions (that is,
because there are many conditions of temperature between the two extremes of boiling heat and
freezing cold that could not be classified as either), the falsity of the proposition that it is boiling
hot is entirely consistent with the falsity of the proposition that it is freezing cold. Thus while it is
obviously impossible for the two contraries “It is boiling hot” and “It is freezing cold” to both be
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true (these statements are mutually inconsistent), it is possible for both of them to be false. In the
standard nomenclature, contraries are mutually exclusive (mutually inconsistent) not collectively
exhaustive. In contrast, contradictories are both mutually exclusive and collectively exhaustive.
Given any proposition A, there exists an infinite number of logically non-inequivalent propositions
whose falsity is implied by A’s truth (and whose truth is implied by A’s falsity). At the level of
propositonal logic, the notion of a contrary possesses no natural expression. (The logical concept of
a contrary possesses a degree of correspondence to the intuitive concept of an opposite, a concept
that is not in general definable in any metaphysically natural way).
At least at the level of propositional logic, the presence of contraries is connected with our capacity
to identify opposites: hot-cold, dry-wet, heavy-light, dark-light, full-empty, high-low, large-small, etc.
In many cases, there is no natural (non-contradictory) contrary to a proposition because there is no
unique plausible opposite or combination of opposites to the terms of the proposition. In the case
“John went to the dentist”, even the clumsiest attempt at the formulation of a negation produces a
contradictory rather than merely a contrary because there are no obvious ‘anti-dentists’. Even in the
case of sentences in which all the salient terms do have more or less natural opposites, it may be
difficult to define a natural contrary to the sentence. For example, what is the contrary of the sentence “There is a tremendous concentration of positive electric charge near me”? Is it 1) "There is a
minute concentration of positive charge near me”; 2) "There is a tremendous concentration of negative charge near me”; 3) "There is a minute concentration of negative charge far away from me”;
4) "There is no net concentration of positive or negative charge near me”; 5) "There is nothing near
me”, etc.? (This is without even mentioning anti-matter!).
Even after acquiring a complete grasp of the distinction between the contradictory (negation) of a
sentence and its contrary, one must be careful in approaching the task of negating an ordinary
English sentence, particularly when the main predicate of a sentence appears in a modified form.
The negation of “He is definitely coming to the meeting” is “He is not definitely coming to the
meeting” and not “He is definitely not coming to the meeting”. Similarly, the negation of “He is
becoming increasingly enlightened in his attitudes” is “He is not becoming increasingly enlightened
in his attitudes” and not “He is becoming increasingly unenlightened in his attitudes”. If one really
wants to be on the safe side, one can form the negation of a sentence such as “He is definitely coming to the meeting” by prefacing the entire sentence by the phrase “It is not the case that”, meaning
that the negation of “He is definitely coming to the meeting” is “It is not the case that he is definitely coming to the meeting”. Of course, no linguistic device is a substitute for the understanding
that the negation of a sentence is to be interpreted in such a way that any condition under which A
is not true is necessarily a condition under which the negation of A should be assigned the value
‘True’. Alternatively stated, there is no possible condition that makes both A and its negation true,
and no possible condition that makes both A and its negation false.
Before going on to the definition of our next truth-functional connective, a consideration of the
result of applying the negation operator to a truth-functional conjunction of two propositions should
help further nail down the interpretations of the connectives that we are employing here. The
negation of the conjunction of A and B, ¬ (A ∧B), is 1) true if and only if the conjunction of A and
B is false and is 2) false if and only if the conjunction of A and B is true. Because A ∧B is true if
both A is true and B is true, and is false otherwise, it follows that the negation of A ∧B is false if
both A is true and B is true, and is true otherwise. Let’s go through an example. Consider the two
sentences “Jack is ill” and “Joan won the game”. If I assert the conjunction of these two sentences,
which is the sentence “Jack is ill and Joan won the game”, I am making a claim that is only true if
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both Jack is ill and Joan won the game. I’ve committed myself to the truth of both of these statements, and the falsity of either one of them (that is, of at least one of them) is sufficient to make
my statement false. Because the negation of “Jack is ill and Joan won the game” is false under the
condition that both “Jack is ill” is true and “Joan won the game” are true, this negation is true if
either Jack is not ill or Joan did not win the game. This takes us to the “official” truth-functional
definition of the connective “Or”.
Or, also designated Disjunction, is represented by a ∨ (vel or wedge). The components of a disjunction A∨B are called its disjuncts, and the disjunction is stipulated to be true if and only if either one
of its disjuncts are true, that is, if and only if at least one of its disjuncts is true. Going back to
our two stock sentences A = “Jack is ill” and B = “Joan won the game”, the claim “Jack is ill or
Joan won the game” is true in three of the four salient cases: 1) the case in which Jack is ill and
Joan won the game (TT); 2) the case in which Jack is not ill and Joan won the game (FT); and
3) the case in which Jack is ill and Joan did not win the game (TF). The disjunction is false only in
the case in which both the statements “Jack is ill” and “Joan won the game” are false (FF), that is,
the case in which Jack is not ill and Joan did not win the game.
The variant of “or” we are defining here (which is the only variant of “or” that is honored with its
own symbol) is often called the inclusive or, to distinguish it from the also frequently employed
exclusive or (abbreviated xor by computer programmers). The application of the exclusive or to sentences A and B, which can be rendered in terms of the symbolism developed thus far as
(A∨B)&¬ (A&B) (or as (A&¬ B)∨(¬ A&B)), defines a truth function that is true if either A is true or
B is true, but not both, that is, if exactly one of the disjuncts A and B is true, regardless of whether
that is A or B. The employment of the exclusive or is most prominent in the context of choices,
“Your money or your life!” being a vivid example.
There is a certain relationship between conjunction, negation, and disjunction, critical in both informal and formalized contexts, (Augustus) de Morgan’s Laws. First, the negation of the conjunction
of A and B [¬ (A ∧B)] is equivalent to the disjunction of the negations of A and B [¬ A∨ ¬ B].
Consider again the two propositions “Jack is ill” and “Joan won the game”. The conjunction of
these two propositions is true in the single case that Jack is ill and Joan won the game, and hence
the negation of their conjunction is true if either Jack is not ill or Joan did not win the game, this
condition corresponding to the three cases in which either “Jack is not ill” (¬ A) or “Joan did not
win the game” (¬ B) is true. Second, the negation of the disjunction of A and B [¬ (A∨B)] is
equivalent to the conjunction of the negations of A and B [¬ A ∧¬ B]. Employing the same examples, the disjunction of the two propositions “Jack is ill” and Joan won the game” is true if either
Jack is ill or Joan won the game and hence the negation of their disjunction is true only under the
condition that Jack is not ill and Joan did not win the game, the case in which both “Jack is not ill”
(¬ A) and “Joan did not win the game” (¬ B) are true.
If-then, also designated the material conditional, is commonly represented either by ⊃ (“horseshoe”)
or --> (“arrow”). The sentence preceding the horseshoe is called the antecedent and the sentence
succeeding it is called the consequent. A⊃B is stipulated to be false in the case that A is true and
B is false (TF), and stipulated to be true otherwise. A⊃B is the closest rendering of “if-then” that
we can obtain in purely truth functional terms. Informally, A⊃B should be taken to mean that the
truth of A is a sufficient condition for the truth of B, a proposition that will be considered to be
false only when the truth values possessed by A and B are inconsistent with such a claim. Consider
the statement “If George is at the party, Susan is also”. The presence of both George and Susan at
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the party is obviously consistent with this claim, but no less consistent with it is 1) the absence of
George and the presence of Susan or 2) the absence of both George and Susan. So long as the truth
value of A⊃B is a function solely of the truth values of A and B, there are no facts beyond the
presence or absence of George and Susan to appeal to in determining whether the presence of
George at the party is a sufficient condition for the presence of Susan there. Thus just as the
absence of both George and Susan from the party does not indicate that Susan would have been
absent whether or not George was present (and hence is not an indication that the claim “George’s
presence is a sufficient condition for Susan’s presence” is false), the presence of both George and
Susan at the party does not indicate that Susan was present because George was present (and hence
is not an indication that “George’s presence is a sufficient condition for Susan’s presence” is true).
The presence of George at the party and the absence of Susan is the only combination of events that
yield a definite verdict concerning the truth or falsity of the claim that the presence of George is a
sufficient condition for the presence of Susan. In particular, this combination falsifies the claim, and
hence “If George goes to the party, Susan will also” is assigned the value false. All the other combinations, none of which are inconsistent with the claim and none of which confirm it, are assigned
the value true. Despite all the frequently discussed oddities of associating the material conditional
with the informal idea of “if-then” and “implies”, there is a certain logic to evaluating all the cases
in which the claim that A is a sufficient condition for B is neither confirmed nor disconfirmed by
the actually realized combination of events identically, and to evaluating all these cases in a distinct
way from the unique disconfirming case. Specifically, it is difficult to see how one could justify not
assigning the three combinations of truth values TT, FT, and FF the same truth value, and how one
could justify assigning any of these combinations the same truth value (False) that is assigned to the
combination TF.
The relationship between the conditional and negation raises the general matter of necessary and sufficient conditions and their relationships. Causation is the paradigmatic sufficient condition: We say
that event A causes event B when the occurrence of event A ensures (we are grossly simplifying
here) the occurrence of event B. Although we expect any instance of the event-type of cause A to
be invariably accompanied by an instance of the event-type B, this expectation is not symmetrical.
The unbroken fall of a thin glass from a four-foot high table to a hard floor will cause the glass to
break, by which we mean (very roughly speaking), that the unbroken fall of the glass from the table
(event A) will almost certainly be accompanied by the subsequent appearance of multiple fragments
of this glass on the floor (event B). The occurrence of A is a sufficient condition for the occurrence
of B in the sense that the occurrence of A more or less assures the occurrence of B: If I discover
that A occurred and learned nothing else, I would have sufficient information to make the inference
that B occurred. (Of course, I am not under the delusion that highly improbable circumstances can
come into play that prevent the glass from breaking). However, if A and B are defined as here, the
occurrence of B is by no means even plausibly a sufficient condition for the occurrence of A. The
appearance of broken glass on the floor does not permit me to infer that A occurred—that the glass
fell off the four-foot table. The glass may have fallen from someone’s hand; it may have been
thrown down; there may have been an explosion that tossed the glass into the air; the glass may
have been knocked off the table, and then bounced off something else before it fell to the floor and
broke; the glass may have been knocked off a nearby table; the glass may have been carefully
placed on the floor and accidently stepped on; etc. The appearance of the broken glass on the floor
is not a sufficient condition for the unbroken fall of the glass from the four-foot table. And more
generally, the fact that the occurrence of A-type events are reliable indicators for the occurrence of
B-type events does not mean that the occurrence of B-type events are reliable indicators for the
occurrence of A-type events.
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The truth of A is a sufficient condition for the truth of B if B is true whenever A is true. Commitment to the truth that B is true whenever A is true is not commitment to the truth that A is true
whenever B is true. Thus if I say “If George is at the party, Susan is also”, I am not denying the
possibility, or even the likelihood, that Susan will be at the party without George. In other words, I
am not committed to the converse of “If George is at the party, Susan is also” (A⊃B), which is “If
Susan is at the party, George is also” (B⊃A). This is a very realistic example. To take one obvious
possibility, Susan may love parties, but George may find parties tolerable only if Susan attends. I
am also not committed to the inverse of “If George is at the party, Susan is also”, which is “If
George is not at the party, Susan is not at the party either” (¬ A⊃¬ B). The same combination of
sentiments we ascribed to George and Susan that explains why A⊃B is true and B⊃A is false simultaneously explains why ¬ A⊃¬ B is false. This is no coincidence: B⊃A is logically equivalent to
¬ A⊃¬ B in the sense that B⊃A and ¬ A⊃¬ B are both true under the same circumstances and hence
(given our assumptions about bivalence) are both false under the same circumstances. Given all this,
it should be clear that A⊃B and its contrapositive ¬ B⊃¬ A are logically equivalent. If George’s
presence at the party is sufficient to establish that Susan is present at the party, certainly Susan’s
absence from the party is sufficient to establish that George is absent. (A truth-table analysis could
establish this formally: The only case that falsifies A⊃B is the assignment of T to A and F to B,
and the only case that falsifies ¬ B⊃¬ A is the assignment T to ¬ B and F to ¬ A, which is equivalent to the assignment of F to B and T to A, the same case that falsifies A⊃B).
Birth in the territorial confines of the United States is a sufficient condition for citizenship in the
U.S. (you’re then a citizen, period), but it is not a necessary condition for citizenship. Many people
born in other parts of the globe come to the United States each year, and there are various mechanisms by which they can acquire citizenship. Oxygen is a necessary condition for human life,
because without it human life cannot be sustained, but it is not a sufficient condition. Food and
water, most notably, are also necessary. In general, if the presence of A is a necessary condition for
the presence of B, the absence of A is a sufficient condition for the absence of B. So, for example,
since (as we know) oxygen is a necessary condition for human life, the absence of oxygen is a sufficient condition for the absence of human life.
If-and-only-if (often abbreviated iff both in logic and in philosophical writing), also designated the
material biconditional or material equivalence, is most commonly represented by an equal sign = or
a double-headed arrow <-->. A=B means that A is true under precisely the same conditions that B
is true. In other words, A is true whenever B is true, and A is false whenever B is false. In other
language, A=B means that the truth of A is both a necessary and a sufficient condition for the truth
of B (which implies, as can be verified, 1) that the truth of B is both a necessary and sufficient condition for the truth of A and 2) that the falsity of A is both a necessary and sufficient condition for
the falsity of B (as well as vice-versa)). Truth functionally, it is equivalent to the conjunction of
A⊃B and B⊃A, that is (A⊃B)&(B⊃A).
Finally, some examples of non-truth-functional connectives should be mentioned. Particularly important unary non-truth-functional connectives are modal operators, which involve the concepts of possibility and necessity, and epistemic operators, which involve the concepts of knowledge and belief.
For example, the propositions “2+2=4” and “The Sun is larger than the Earth” are both true, but
while the statement “It is necessarily the case that 2+2=4” is true (2+2=4 being true of logical
necessity), the statement “It is necessarily the case that the Sun is larger than the Earth” is false (the
sun being larger than the earth being a merely contingent truth, something that could conceivably
have been false). “X believes P” is also obviously not truth-functional, because X may believe one
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true statement and disbelieve another true statement, and may disbelieve one false statement and
believe another. The most familiar binary non-truth-functional connective is because. Although the
statements that A, B and C happened may all be true, it may be the case that A happened because
C happened, and not be the case that B happened because C happened.
tab(@); c c || c | c | c | c. @@A & B@A or B@A –> B@A iff B A@B@A ∧ B@A ∨ B@A ⊃
B@A <–> B _ T@T@T@T@T@T T@F@F@T@F@F F@T@F@T@T@F F@F@F@F@T@T
@@Conj.@Disj.@M. Implic.@Bi-Cond.
Truth-Functional Logical Validity
Although each single propositional symbol A, B, C, etc. formally represents the kind of object—a
sentence—that must be either true or false, in the context of propositional logic these letters are to
be treated as variables for which the values True and False are both admissible substitutions. This
assertion requires a bit more explication than might be obvious on the surface. Let’s begin with a
simple mathematical example. Consider the variable ‘X’, defined as ranging over the natural numbers N = {0,1,2,3,...}. What is the numerical value of X? Obviously, there can be no definite
answer to this question until a definite number is substituted for X. Analogously, we can’t say
whether, for example, the sentence represented by B is true until a definite proposition is substituted
for B. Thus B may stand for any of the following propositions: 1) "Isaac Newton is identical to
himself”, 2) "2+2=5”, 3) "The star Alpha Centauri is more distant from the Earth than is the Sun”,
4) "Abraham Lincoln was born in the year 2000”, 5) "It snowed in Manhattan, on January 1, 1500”,
6) "Joan won the game”, and 7) "All mimsy were the borogoves”. Sentence (1) is true of logical
necessity (though we don’t officially know that), and sentence (2) is false of logical necessity. Sentence (3) is a contingent statement (a statement whose truth or falsity cannot be established by reasoning about abstract ideas alone) that is known to be true, and sentence (4) is a contingent statement that is known to be false. (5) is also a contingent statement, perhaps true, perhaps false, but
not known to be either by any person living in the twenty-first century. Sentences (6) and (7) do
not acquire their truth values from truths of logic or facts about the world, either discoverable or
undiscoverable; it really makes no sense to speak of these two sentences possessing definite truth
values other than those we arbitrarily choose to assign them.
Our single atomic letters are permitted to stand for any sentence, of whatever character, whether
simple or arbitrarily complex. The representational indivisibility of these elemental symbols (letters)
carries with it a certain semantic impenetrability, which we now explain. As noted before, any
proposition treated as an elementary sentence in a given context may be analyzable as a truth-functional combination of other sentences (“Fred and Karen exchanged cards at time t” might in a certain context be usefully rendered as “Fred gave Karen his card at time t” and “Karen gave Fred her
card at time t”); indeed, a sentence that it is useful to treat as an elementary proposition in a given
context may even already be in explicit truth-functional compound form (D may be “George has just
returned from the year 802,701 and George has some exceedingly interesting stories to tell”). (The
decision to represent a certain complex proposition as an elementary sentence, though not strictly
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Truth-Functional Propositional Logic
speaking factually inaccurate, may of course be unwise if it results in discarding information of
transparent formal deductive significance). The fundamental notational conventions of propositional
logic imply that no structure internal to any sentence represented by a single letter is available to
play any expressive or deductive role in the logical system: In particular, all truth-functional connectives located within any proposition that we choose to represent as an elementary sentence perform
no logical function. All logical “action” is external to (takes place between) the elementary sentences, where the truth-functional connectives such as &, ¬ , ∨, and ⊃ are operational (i.e., effective).
Because any truth-functional connectives lying within the confines of a sentence are made inert (and
hence logically inconsequential) by the choice to treat the sentence as an elementary proposition, and
because no other logical structure is being considered in this context, nothing strictly logical determines the truth value of any elementary sentence. (Sometimes it is convenient to allow the character ‘T’ (“verum”) to represent all logically true statements and an upside-down ‘T’ (“falsum”) to
represent all logically false statements). And, because the truth-functional connectives provide the
only admissible means to derive the truth values of formulae from other formulae, the only logically
salient thing there is to know about elementary sentences is whether they are identical or distinct. If
they are distinct, there can be no non-trivial logical relationships between them. (They do have a
trivial logical relationship: Neither implies the other). In making our next statement, we invoke
what may be called the Fundamental Principle of Symbolism: Distinct symbol types may represent
either distinct objects or the same object, but distinct tokens (in the context of symbolic logic and
mathematics, usually called occurrences) of the same type invariably represent the same object.
Everything of purely logical significance that follows from this principle so far as propositional logic
is concerned is summed up in two facts: 1) the truth value assigned to one elementary sentence A
imposes no constraints on the truth value that can be assigned to any distinct elementary sentence B
and 2) all occurrences of an elementary sentence must be assigned the same truth value. (Henceforth, when we wish to refer to two distinct types of sentences, we will speak simply of distinct
sentences; when we wish to refer to distinct tokens of the same sentence type, we will speak of two
distinct occurrences or tokens of the sentence in question). By the way, our leap to the assignment
of truth values rather than concrete sentences to abstract sentences is not an oversight. It reflects the
fact that propositional logic is truth-functional, meaning that only the truth values of a formula’s
component sentences are relevant to the truth value of the formula, and that the logical relationships
among propositions are determined by the relationships among the truth values consistently
assignable to them.
Let’s now return to our simple algebraic examples, and consider the expression X+Y, where, once
and for all, X and Y are distinct variables ranging over the natural numbers N. We already know
that we cannot assign a definite numerical identity to the sum X+Y until definite numbers are substituted for X and Y. Letting X=0 and Y=0, X+Y=0; letting X=0 and Y=1, X+Y=1; letting X=1 and
Y=0, X+Y=1, etc. There is a slightly more compact way to express this: We speak of assigning the
ordered pair of numbers a, b (written (a,b)) to the ordered pair of variables (X,Y), meaning that the
first element belonging to the pair (a,b) is assigned to X and the second element belonging to the
pair (a,b) is assigned to Y. In this notation, the substitution of (2,3) in X+Y assigns 2 to X and 3
to Y and yields the sum 5 and the assignment of values (11,14) to X+Y yields 25. The specification of a numerical value for each ordered pair of natural numbers, which can be exhibited in an
infinite table, may be regarded as the definition of the function ‘the sum of two natural numbers’;
we certainly take the specification of a truth value for each of the four possible ordered pairs of
truth values (TT, TF, FT, FF), which can be exhibited in a four-rowed Truth Table, as the definition
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Truth-Functional Propositional Logic
of the various binary truth-functional connectives. The function ‘+’, which is best represented by
the expression ‘X+Y’, is a two-variable function that assigns numbers to ordered pairs of numbers;
the function ‘&’, best represented by ‘A&B’, is a two-variable function that assigns truth values
(either T or F) to ordered pairs of truth values.
At one level, X+Y is simply a number (once particular numbers are substituted for X and Y) and
A&B is simply a truth value (once truth values are substituted for A and B). We now wish to consider A&B as an object—a sentential or propositional form—that can be be employed to assert
something, and from this perspective a perhaps more instructive mathematical parallel is a propositional expression such as X+Y=5, a propositional expression that acquires a definite truth value once
particular natural numbers are assigned to X and Y. X+Y=5 may be viewed as a function from
pairs of numbers to truth values: Just as A&B is true for some values of A and B and false for others, X+Y=5 is true for some numerical values of X and Y and false for others. In language that
becomes particularly illuminating in the context of quantificational logic, we say that the formula
A&B is satisfied by some pairs of truth values and not satisfied by others, and that X+Y=5 is satisfied by some pairs of natural numbers and not satisfied by others. Roughly speaking, X+Y=5 is satisfied by those pairs of numbers that make X+Y=5 true, and A&B is satisfied by those pairs of truth
values that make A&B true. A more accurate way of speaking (whose greater power will become
apparent as we move further) is that X+Y=5 is satisfied by those pairs of numbers whose substitution in the formula X+Y=5 produce a true sentence, and that A&B is satisfied by those pairs of
truth values whose substitution in the formula A&B produce the value T.
If I learn that X+Y=5 is true (i.e., if I learn that two natural numbers sum to 5) , I can deduce from
that fact alone that one of the six ordered pairs (O,5) ,(1,4) ,(2,3) ,(3,2) ,(4,1) ,(5,0) is associated with
(X,Y); if I learn that A&B is true, I can deduce from that fact alone that both A and B are true.
When we are asked to solve an equation in algebra, we are asked to find the value or values that
make the equation a true statement—we are asked, in other words, to isolate the values under which
the condition of equality holds. Algebraic statements such as X+Y=Y+X (again, the domain of both
X and Y is the set of natural numbers), which are true for all assignments of natural numbers to X
and Y, are called identities. It follows from the fundamental convention of symbolism referred to
above (“whatever is substituted for one occurrence of a variable must be substituted for all occurrences of that variable”) that whatever natural number is substituted for the variable X on one side
of the equation must be substituted for the variable X on the other side of the equation, and whatever natural number is substituted for Y on the left must be substituted for Y on the right. Thus the
same numerical sum must appear on both sides of the equality sign, whatever values are assigned to
X and Y, and hence this equality, this assertion, is true under all admissible conditions (i.e. is unconditionally true). The equation X+X=5, on the other hand, possesses no solution in the domain of
natural numbers. The same number must be substituted for both occurrences of the variable X,
which means that this equation becomes a true proposition when for X we substitute a natural number that when added to itself equals 5. Obviously, no natural number satisfies this condition (2.5 is
not a natural number!)—this equality is thus false under all admissible conditions.
This tripartite division carries over to propositional logic, and is characteristic of what interests us in
our inquiries into logic in general. Each formula F of propositional logic falls into one of three
classes. First, F may be true for all possible combinations of assignments of truth values to the elementary sentences of F, in which case F is said to be truth-functionally or tautologically valid, or a
tautology for short. (It is also correct to say that F is logically valid, but it must be understood that
the class of logically valid formulae—which we have yet to define—is far wider than the class of
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Truth-Functional Propositional Logic
logically valid formulae of the propositional calculus). Second, F may be true for some combinations of assignments of truth values to the elementary sentences of F and false for some combinations, in which case F is said to be truth-functionally contingent. Finally, F may be false for all
possible combinations of assignments of truth values to the elementary sentences of F, in which case
F is said to be truth-functionally unsatisfiable or inconsistent, or a contradiction for short. Logic, as
is appropriate to its essential character, possesses nothing within its legitimate scope that permits us
to distinguish between contingently true and contingently false propositions.
A very brief elaboration of compositional and combinatorial matters, systematic in spirit though not
in detail (the latter can be found in the assigned reading), should intensify the reader’s sense of the
significance of these ideas. Our elementary sentences are symbolized by simple capital letters A, B,
C, D, etc. There are an arbitrary number of elementary sentences, and if more than twenty-six are
required in a given context, these can be handled by adding symbols such as A1 , B1 , C1 ,..., A2 , B2 ,
B2 ,..., etc. Our formulae, represented by italicized letters such as F, G, H, etc. are certain “wellformed” finite combinations of elementary sentences and truth-functional connectives: Examples are
1) A; 2) A&B; 3) A&(B∨C); 4) D∨D∨D∨D; 5) ¬ ((A⊃B)∨ ¬ E)&(B⊃¬ F); and 6) ¬ (¬ (¬ (¬ A))).
Examples of strings composed of connectives and elementary sentences that are not “well-formed”
are 1) A&; 2) B¬&; 3) ∨B∨C∨; and 4) A¬&B. The syntactic “formation” rules that define the class
of well-formed formulae are such as to ensure the unique readability of all formulae composed
according to these rules. This means, specifically, that these rules are formulated in such a way as
to remove all ambiguity concerning which truth-functional operations are to be performed and in
which order, and formulated in such a way as to ensure that every possible comprehensive assignment of truth values to the sentence letters belonging to any formula F uniquely determines the truth
value of F. (Just to make sure there is no misunderstanding, a comprehensive assignment is an
assignment of a definite truth value to each elementary sentence belonging to F).
Without wandering in any substantial way into the quantitative, a few points about the combinatorial
logic of all this should be highlighted here. There are a class of constructible formula—indeed an
infinite class—in which all and only the sentence letters A, B and C appear. The shortest are as
short as (A&B)&C, and there is no longest: ((A&B)&C)&A, (((A&B)&C)&A)&B,
((((A&B)&C)&A)&B)&C, (((((A&B)&C)&A)&B)&C)&A, etc. The truth of any formula F containing only the sentences A, B, and C is determined by the substitution of either True or False for each
occurrence of a sentence in F, and the Truth Table for a very long formula will correspondingly be
very large. By the fundamental principle of symbolism, however, there are only eight independent
combinations of truth values that can be substituted for A, B, and C (TTT, TTF, TFT, TFF, FTT,
FTF, FFT, FFF) because whatever truth value is assigned to one occurrence of an arbitrary sentence
A must be assigned to all occurrences of A. To emphasize this constraint, we will sometimes speak
of all possible consistent assignments or consistent combinations of assignments of values to variables. In these terms, a formula F containing only the sentence letters A, B, and C is a tautology if
all eight consistent combinations of assignments of truth values to A, B, and C yield the value
true—equivalently, if there is no possible combination of assignments of truth values to the variables
A, B, and C under which the formula F is false. This means, in general, that F is a tautology if
and only if the negation of F (¬ F) is a contradiction: F is satisfied by all possible assignments of
truth values to the elementary sentences of F if and only if ¬ F is satisfied by no assignment of
truth values to the elementary sentences of F. More generally, the negation of a tautology is a contradiction; the negation of a contradiction is a tautology; and the negation of a contingent formula is
another contingent formula.
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Truth-Functional Propositional Logic
We now come to the definition of the most elementary variant of the central idea of deductive validity, truth-functional (tautological) implication or consequence. First, we say that one formula F
truth-functionally (or tautologically) implies or entails formula G, or that formula G is a truth-functional (or tautological) consequence of formula F, if and only if there exists no completely comprehensive assignment of truth values to the elementary sentences of both F and G that makes F true
and G false. Let F be A&B and let G be A. There exists one combination of assignments of truth
values to sentences A and B, TT, that makes both F and G true; there exist two combinations of
truth-value assignments to sentences A and B, FF and FT, that make both F and G false; there
exists one combination of truth-value assignments to A and B, TF, that makes F false and G true;
but there exists no combination of truth-value assignments to A and B that makes F true and G
false. Given the definition of truth-functional implication and the analysis just given, we have
shown that A&B implies A, and that A does not imply A&B.
Typically, when we consider implication, whether truth-functional or otherwise, it is in a context in
which arguments proceeding from something to something else are formulated and evaluated. If we
are employing A&B as an argument for A, A&B is the premise of the argument and A is the conclusion. Note that by the conception of truth-functional deductive validity presented here—and the
property we are noting that is attached to this conception is shared by all the more general variants
of deductive validity—the truth or falsity of either the premise or the conclusion in themselves is
irrelevant to the issue of the argument’s validity. Arguments with false premises may be valid, and
arguments with false conclusions may be valid, but an argument with a true premise and a false
conclusion is invalid. (The fact that P implies Q is not yet a reason to believe that Q is true—it is,
however, a conclusive reason to believe that Q is true if P is true). In the general case, we are dealing with arguments with a finite multiplicity of premises P1 , P2 , P3 ,...Pn and a single conclusion Q.
This more general form of argument is defined to be valid if there is no way for all of its premises
to be true without its conclusion also being true. More precisely, an argument of this general form
is valid if and only if there exists no combination of assignments of truth values to every elementary
sentence appearing in any of the premises or in the conclusion that makes all the premises true and
the conclusion false. In yet other words, an argument is valid iff the class of complete assignments
that make the conclusion false is a (proper or improper) superset of (is no less comprehensive than)
the class of complete assignments that make the conjunction of premises false. Or equivalently, the
argument is valid iff the class of complete assignments that make the conclusion true is a (proper or
improper) subset of (is no more comprehensive than) the class of complete assignments that make
the conjunction of premises true. A valid argument whose premises are true is called sound; clearly,
given the definition of validity, the conclusion of a sound argument must be true.
Thus let’s say P1 is A&B&C, P2 is (A⊃D)∨ ¬ (A&C), P3 is ¬ D⊃(B⊃C), and Q is E. The premises
P1 , P2 , P3 will imply Q if there is no possible combination of assignments of truth values to the elementary sentences A, B, C, D, and E that make all the premises P1 , P2 , and P3 true and the conclusion false. In this case, one doesn’t have to look very carefully to notice that the sole elementary
sentence in the conclusion doesn’t even appear in any of the premises; this means that no combination of assignments of truth values to the elementary sentences of the premises can place any constraint whatsoever on the truth value of the conclusion, which in turns means that if there exists
even a single assignment of truth values to A, B, C, and D that makes all the premises true (there
is), this argument is invalid. The assignment of truth values T, T, T, T, F to A, B, C, D, and E
respectively simultaneously makes all the premises true and the conclusion false—any such falsifying
assignment is called a counterexample to the argument in question. (Note that the same premises
also do not provide a valid argument for the negation of the above conclusion: The assignment of T,
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Truth-Functional Propositional Logic
T, T, T, T to A, B, C, D, and ¬ E respectively simultaneously makes all the premises true and the
conclusion (¬ E) false). The existence of a counterexample can be viewed as an assignment of truth
values that simultaneously satisfies the totality of premises and the negation of the conclusion. Thus
if an argument is deductively valid, the conjunction of the premises is inconsistent with the negation
of the conclusion.
Given the fact that a valid deduction is one in which every assignment that makes all the premises
true also makes the conclusion true, it should be clear that any argument with premises P1 , P2 ,...Pn
is equivalent to an argument consisting of a single premise corresponding to the formula that is the
conjunction of the premises P1 , P2 ,...Pn . (By definition, the conjunction of P1 , P2 ,...Pn is true under
the same assignments of truth values that make all of these individual conjuncts true). This observation is connected with the property of deductive logic called monotonicity, which means that if an
argument is deductively valid, the addition of further premises must enable us to validly deduce (that
is, must imply) at least what was validly deduced before—perhaps more, but never less. Utilizing
the implications of the equivalence between the augmentation of a set of premises {P1 , P2 , P3 } by a
new premise P4 (producing the set of premises {P1 , P2 , P3 , P4 } and the expansion of a conjunction
of premises P1 &P2 &P3 by forming its conjunction with the new premise P4 (producing the conjunction P1 &P2 &P3 &P4 ), it can be seen that the addition of a premise to an argument expands the
opportunities for assignments of truth values that make the conjunction of premises false but does
not expand the opportunities for assignments of truth values that make the conjunction of premises
true. (Again, a consideration of the definition of conjunction will verify this). Perhaps the relevance of this can be more clearly seen if one considers two particular consequences of the definition
of truth-functional deductive validity: 1) that a set of premises that no truth-value assignment makes
true provides a valid argument for any conclusion and 2) that a conclusion that no assignment makes
false is validly implied by any set of premises. More generally, consider an arbitrary valid argument
for a conclusion Q: Any modification of the premises of that argument that converts the original set
of premises {P} into another set of premises {R} whose conjunction is false under a more comprehensive set of conditions necessarily yields a valid argument for Q. Because the addition of new
premises can never contract, and can only expand, the set of conditions that falsify the conjunction
of any argument’s premises, truth-functional logic is monotonic. (As we will see later, probabilistic
arguments—at least as standardly interpreted—are non-monotonic).
The standard condition for an argument’s truth-functional validity presented above is equivalent to
the condition that the set of assignments of combinations of truth values to elementary sentences that
make the conjunction of premises true is a subset (proper or improper) of the set of assignments of
combinations of truth values to elementary sentences that make the conclusion true. We will employ
this characterization of logical implication in our official definition of the essentially equivalent but
somewhat more portable notion (or rather terminology) of logical strength. To say that A/(ANB is
logically stronger than A is to say that the set of combinations of truth values that falsify A/(ANB
is a superset of the set of combinations of truth values that falsify A alone. We say that two formulae F and G are of equal logical strength—‘equal’ here has to be interpreted in a strong sense—if
and on if they are logically equivalent, meaning that F and G are true under precisely the same conditions. When F and G are related in this way, when the same set of assignments of truth values to
the elementary sentences of F and G make F and G true, and (hence) when the same set of assignments of truth values to the elementary sentences of F and G make F and G false, F logically
implies G and G logically implies F. (The fact that F and G are logically equivalent is consistent
with F and G both being tautologies, with F and G both being contingent statements (i.e., with both
being sometimes true, sometimes false, but both being true (and false) in the same cases), and with
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Truth-Functional Propositional Logic
F and G both being contradictions). Finally, it must be understood that the relation of logical
strength forms just a partial ordering, not a total ordering, meaning that two arbitrarily chosen formulae may be incomparable in regard to the relation of logical strength. Thus formula F may be
neither weaker nor stronger than formula G, which is the same as to say that neither implies the
other. (The relation of logical strength forms an equivalence relation among truth-functional propositions that partitions the set of propositional formulae into an infinite number of equivalence classes,
each consisting of an infinite number of logically equivalent formulae). A particular confusion that
it is vital to avoid is that between logical strength and the essentially probabilistic idea of informativeness. More will be said about this later in this course, but for the present it will suffice to note
that it is certainly not correct to characterize the statement “F is logically stronger than G” as either
meaning “G is less probable than F” or as meaning “G is false under fewer conditions than F ”.
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