Truth-Functional Propositional Logic

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1 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 A 1,A 2,A 3,... 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 A 1 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 Course Notes Page 1

2 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 Bistrue; 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 afunction 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. Ingeneral, 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. Course Notes Page 2

3 Bare 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 Isay 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 asingle 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 AorBorCorDorE is sufficient to falsify my claim. To make this quantitatively precise, there is one case (TTTTTTTT) in which A B C D E istrue 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 ageneric concept of conjointness or combination indispensable to defining any ofthe 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, sothere 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, Aistrue if an only if A is false, and Aisfalse 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, asare 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 Course Notes Page 3

4 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 ofpropositional 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 neg ative 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 Bisfalse and is 2) false if and only if the conjunction of A and B is true. Because A B istrue if both A is true and B is true, and is false otherwise, it follows that the negation of A B isfalse if both A is true and B is true, and is true otherwise. Let s gothrough 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 Course Notes Page 4

5 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 Bistrue, 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 ismost 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 isstipulated 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 Course Notes Page 5

6 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 isafunction 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 Aand 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 anearby 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. Course Notes Page 6

7 The truth of A is a sufficient condition for the truth of B if B is true whenever Aistrue. Commitment to the truth that B is true whenever Aistrue is not commitment to the truth that A is true whenever Bistrue. 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 isfalse. This is no coincidence: B A islogically equivalent to A B inthe 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 isthe assignment of T to A and F to B, and the only case that falsifies B A isthe assignment T to Band 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. Ingeneral, 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 asufficient 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 Course Notes Page 7

8 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 Chappened, and not be the case that B happened because C happened. tab(@); c c c c c & B@A or B@A > B@A iff BA@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 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, wecan 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 ofthe 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 Course Notes Page 8

9 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. Inthis 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 Course Notes Page 9

10 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),Ican 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 anequation 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 Xand 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 Course Notes Page 10

11 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 A 1,B 1,C 1,..., A 2,B 2, B 2,..., 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) isacontradiction: 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. Course Notes Page 11

12 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 P 1,P 2,P 3,...P n 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 P 1 is A&B&C, P 2 is (A D) (A&C), P 3 is D (B C), and Q is E. The premises P 1,P 2,P 3 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 P 1,P 2,and P 3 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 asingle 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, Course Notes Page 12