Logical CONNECTIVES
Logical connectives articulate simple or complex well-formed propositions so as to construct well-formed complex propositions, or formulas. Propositional calculus studies logical syntax, that is the rules for constructing well-formed formulas. It determines, among these formulas, which are valid formulas (logical laws, tautologies).
Propositions are denoted by the capital letters P, Q, R… They are said to be unanalyzed, that is, taken as a whole, in contrast to the analyzed propositions “[Subject] is [Predicate]” considered in the predicate calculus.
A binary logical connective combines two propositions (simple or complex) P and Q into a new complex proposition “P [connective] Q”. Logical connectives (or connectors) are also called functors, function words or logical operators
The most common connectives are denoted and read as follows:
↔ equivalence, “P is equivalent to Q”,
→ implication, “if P then Q”
& conjunction, “P and Q”
V disjunction, “P or Q”
W exclusive disjunction, “either P or Q (not both)”
Logical connectives are defined on the basis of the possible truth values given to the propositions they combine. A particular logical connective is defined by the kind of combination it accepts between the truth values of the component propositions.
1. The truth table approach to binary connectives
A logical connective is defined by its associated truth table. The truth table of a “P connec Q” binary connective is a table with three columns and five rows.
— The letters P, Q … denote the propositions; the letters T and F denote their truth values: true (T) or false (F). P and Q are propositions, while truth and falsity are said of propositions, “P is True”, “P is False”; so, the corresponding abbreviating letters use a different typographical character.
| P | Q | P connective Q |
| T | T | depending on the connective |
| T | F | depending on the connective |
| F | T | depending on the connective |
| F | F | depending on the connective |
— Columns:
The truth values of the proposition P are expressed in the first column
The truth values of the proposition Q are expressed in the second column
The corresponding truth values of the complex formula “P connec Q” are expressed in the third column.
— Lines:
The first line mentions all the propositions to be considered, P, Q and “P connec Q”.
The next four lines express the truth values of these propositions. Since each proposition can be T or F, there are four combinations to consider, each corresponding to a line.
1.1 Conjunction “&”
By definition, the conjunction “P & Q”
— is true if both P and Q are simultaneously true: line 2
— is false if one of them is false: line 3, 4; or both of them: 5.
This is expressed in the following truth table:
|
|
Q | P & Q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
This truth table reads:
line 2: “when P is true and Q is true, then ‘P & Q’ is true”
line 3: “when P is true and Q is false, then ‘P & Q’ is false”
line 4: “when P is false and Q is true, then ‘P & Q’ is false ”
line 5: “when P is false and Q is false, then ‘P & Q’ is false ”
1.2 Equivalence, “ ↔ ”
The logical equivalence “P ↔ Q” reads “P is equivalent to Q”. This resulting proposition is true if and only if the original propositions have the same truth values.
Truth table of logical equivalence:
| P | Q | P ↔ Q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
By this definition, all true propositions are mutually equivalent, and all false propositions are mutually equivalent, regardless of their meaning.
1.3 Disjunctions: Inclusive “V”; Exclusive, “W”
The inclusive disjunction “P ∨ Q” is false if and only if P and Q are simultaneously false; otherwise, it is true.
Truth table of the inclusive disjunction:
| P | Q | P V Q |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
The exclusive disjunction <P W Q> is true if and only if only one of the two propositions it connects is true. In all other cases, it is false.
Truth table of the exclusive disjunction:
| P | Q | P W Q |
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | F |
1.4 Implication: “→”
The logical implication symbol “→” reads “P implies Q”. P is the antecedent of the implication and Q, its consequent.
Truth table of the logical implication:
| P | Q | P → Q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
This table reads:
line 2: The true implies the true
line 3: The true does not imply the false
line 4: The false implies the true
line 5: The false implies the false
Only truth can be logically derived from truth (line 1), whereas, everything can follow from a false proposition, a truth as well as a falsehood.
The connectives for equivalence, conjunction, inclusive disjunction and exclusive disjunction are symmetric, that is, for these connectives, “P connective Q” and “Q connective P are equivalent (convertible):
P ↔ Q ↔ Q ↔ P
P & Q ↔ Q & P
P V Q ↔ Q ∨ P
P W Q ↔ Q W P
The implication connective is not convertible; that is, “P → Q” and “Q → P” have different truth tables.
The laws of implication express the notions of necessary and sufficient conditions:
A → B (is true)
A is a sufficient condition for B
B is a necessary condition for A
A causal relationship can be expressed as an implication. To say that if it rains, the road is wet, means that rain is a sufficient condition for the road to be wet, and that the road is necessarily wet when it rains.
The implication thus defined is called a material implication; it has nothing to do with Toulmin’s substantial logic.
The implication “P → Q” is false only when P is true and Q false (line 2). In other words, “P → Q” is true if and only if “not-(P & not-Q)” is true.
Line (3) asserts the truth of the implication “If the moon is a soft cheese (false proposition), then Napoleon died in St. Helena (true proposition)”. Like the other logical connectives, the implication is indifferent to the meaning of the propositions it connects. It takes into consideration only their truth-values. The strict implication of Lewis tries to elminate this paradox by requiring that for “P → Q” to be true, Q must be deducible from P. This new definition introduces semantic conditions, in addition to the truth values. This explains why the word “implication” is sometimes used in the sense of “deductive inference”.
Systems of natural deduction are defined in logic (Vax 1982, Deduction). They have nothing to do with Grize’s Natural Logic.
2 Logical laws
Using connectors and simple or complex propositions, one is able to construct complex propositional expressions, for example “(P & Q) → R”. The truth value of such a complex expression is only a function of the truth of its component propositions. Truth tables can be used to evaluate these expressions. Some of them are always true, they correspond to logical laws.
2.1 “Laws of thought”
Binary connectors combine in equivalences known as De Morgan’s laws, which are considered to be laws of thought. For example, the connectives “&” and “V” enter into the following equivalences:
The negation of an inclusive disjunction is equivalent to the conjunction of the negations of its components:
¬ (P V Q) ↔ (¬P & ¬Q)
The negation of a conjunction is equivalent to the disjunction of the negations of its components:
¬ (P & Q) ↔ (¬P V ¬Q)
Case-by-case argumentation is based on inclusive disjunction.
2.2 Hypothetical (or conditional) syllogism S. Deduction
2.3 Conjunctive syllogism
The following statement expresses a logical law:
If a conjunction is false and one of its components is true, then the other component is false
[¬(P & Q) & P] → ¬Q
The corresponding three-step deduction is called a conjunctive syllogism:
¬(P & Q) the major proposition denies a conjunction
P the minor affirms one of the two propositions
————
¬Q the conclusion excludes the other
An adaptation to ordinary reasoning:
No one can be in two places at the same time
Peter was seen in Bordeaux yesterday at 6:30pm (UT)
So, he was not in London yesterday at 6:30 pm. (UT)
Knowing that Peter is a suspect; that his interest is to hide that he was really in Bordeaux, and that the witness is more reliable than the suspect, we can conclude that Peter lied when he pretended to be in London at 6:30pm yesterday.
In the following example, the major of the disjunctive syllogism is the negation of an exclusive disjunction:
¬(P W Q) a candidate cannot be admitted and rejected
¬P my name is not on the list of successful candidates
————
¬Q I am rejected
All these deductions are common in ordinary language, where their self-evidence ensures that they go unnoticed. It would be a mistake to ignore them on the pretext that, since these arguments are valid, they are not arguments.
3. Connectives in logic and in language
Introductory courses in logic make consistent use of ordinary language to illustrate both the capabilities and the peculiarities of logical languages. In general, logic can be “applied to ordinary language” (Kleene 1967: p. 67-73) as a tool for expressing, analyzing and evaluating ordinary arguments as valid or invalid reasoning. These translation exercises are as follows (id. p. 59):
I will only pay you for your TV installation only if it works — translated as P → W
Your installation does not work — translated as ¬W
So I will not pay you — translated as ¬P
Using the truth table method this reasoning is then tested for validity, and declared valid.
To identify similarities and differences, natural language components and properties can be compared with their logical language counterparts. This allows us to better understand both types of languages. Such exercises are helpful in gaining a better understanding of logical or linguistic systems, and may also be useful in teaching argumentation. However, there are some additional facts which should be taken into one consideration when using this methodology.
(i) The previous exercise did not focus on the correct combination of the truth values of semantically independent propositions, as was the case of the logical argument about the moon and Napoleon (see §1.4 supra). The exercise introduces a strong condition on semantic coherence between the linked propositions, they belong to the same domain of practical action, in this case, TV installation.
(ii) Natural language connectives do not connect propositions in the way that logical connectives do. The former can be said to be between the two propositions, whereas the latter are syntactically attached to the second proposition. Logical connectives and natural language connectives have two different syntaxes.
As a consequence, the right scope of a linguistic connective is essentially defined by the sentence to which it belongs, whereas its left scope can be much larger, and may include an entire narrative, with various twists and turns:
So, the prince married the princess — The End —
Connectors are classically thought of as connecting two statements in a complete discourse, such as yet in:
the path was dark, yet I slowly found my way (google)
Still, in:
It is good, yet it could be improved (d.c, Yet)
yet introduces a more complex scenario, and the previous example is not a complete discourse. Yet announces that more hints will follow, specifying the weaknesses of the task being evaluated.
(iii) In many cases, the logical reconstruction of ordinary reasoning must introduce new propositions that are said to be present but are left implicit in the discursive string under consideration. This string is then said to contain an “incomplete argument”, see Enthymeme.
(iv) Elementary logical reasoning does not cover all ordinary reasoning:
I ate three apples and two oranges, so I have had my five-fruit diet today
First, this seemingly crystal-clear reasoning is loaded with implicit knowledge, such as “apples are fruits,” “oranges are fruits,” and that “no orange is an apple”. “Three citrus fruits and two oranges” sum up as five fruits only if none of the mentioned three citrus fruits is an orange.
Second, the critical fact here is that the conclusion is based on an addition that is easier to solve in arithmetic than in a logical language. Toulmin’s layout would satisfy this condition by adding a warrant-backing system that refers to the laws of arithmetic.
(v) Logical connectives capture only a small part of the linguistic role played by natural language connectives. The connector “&”requires only that the conjoined clauses be true. This property is common to many ordinary words, and, but, yet … and to all concessive words:
The circumstances which render the compound true are always the same, viz. joint truth of the two components, regardless of whether ‘and’, ‘but’ or ‘although’ is used. Use of one of these words rather than another may make a difference in naturalness of idiom and may also provide some incidental evidence to what is going on in the speaker’s mind, but it is incapable of making the difference between truth and falsehood of the compound. The difference in meaning between ‘and’, ‘but’, and ‘although’ is rhetorical, not logical. Logical notation, unconcerned with rhetorical distinctions, expresses conjunction uniformly. (Quine 1959, p. 40-41)
In other words, elementary logical theory has no adequate concepts to deal with the phenomena of argumentative orientation, and imposes no obligation in this respect. Quine’s argumentative strategy is to minimize the problem and delegate it to rhetoric, which is seen as a dumping ground for problems left unsolved by logical analysis.
The word « and » carries with it subtle semantic conditions, such as a sensitivity to temporal sequence. If “P & Q” is true, then “Q & P” is true. But these two statements do not contain the same information, and this is no longer a matter of rhetoric, whatever the meaning of that word means:
They married and had many children.
They had many children and married.
One might think that, under certain conditions, this logical analysis introduces a third proposition “The events succeeded in this order”. For other conditions affecting the use of and, see Composition and division.
3. No subordination, but bilateral relations
There is no ideal way to imagine the relationship between logical language and natural language; everything depends on the theoretical and practical goals of the researcher, whether he is building a conversational robot, or developing a formal syntax for ordinary language, or teaching second-level argumentation courses.
Logic is an autonomous mathematical language, that can be constructed from some chosen segments of ordinary language. From the beginning, the teaching of logic can draw more or less heavily on the resources of ordinary language. The same is true for the teaching of everyday argumentation in relation to the resources provided by the logical language. The teacher is free to make pedagogical choices, and possible alternative approaches should be judged by their results, according to the standard methods used to evaluate educational methods.