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What is the basis for the definition of logical connectives? Are they just arbitrary convention? Or does it depend on the meaning of the constituent propositions? Does the individual meaning of two propositions determine or constrain what kind of logical connectives can be formed between them? In other words, are the definition given to logical connectives based on empirical observation and logical necessity or are they just arbitrarily defined concepts and rules(Axioms)?

To give a context for this question, I provide the following statements from Mathematics texts and my commentary of them:

In a discrete Mathematics book, the following definition was given to conjunction and implication.

(I)Conjunction- ''Let P and Q be propositions. The proposition ''P and Q'', is the proposition that is true when both P and Q are true and is false otherwise''.

Comment on the above definition: As per the above definition, I can consider any arbitrary pair of propositions and connect it by the ''Conjunction'', since it does not impose any constraint on the meaning of P and Q. So according to the definition, both of the following two examples seem reasonable. Example

(1). P=This car weighs 50 tons. Q=This car is green in color. P and Q = This car weighs 50 tons and is green in color.

(2). P= This car is green. Q=The milky way galaxy is 100,000 light years wide. P and Q = This car is green and the milky way galaxy is 100,000 light years wide.

As per the definition, both examples are valid cases of Conjunction. But it appears that both the examples do not have the same empirical and\or logical footing. In the former case, it is easy to see that the region of the Venn diagram corresponding to P and Q, is made of points corresponding to individual physical objects(car), that possess two properties: Weighing 2 tons and having a green color. In the latter case, however, the region of the Venn diagram pertaining to ''P and Q''is supposedly made of points, each corresponding to two distinct objects possessing two distinct properties. From an empirical standpoint, the second example looks artificial while the former looks more ''natural'', in the sense that there are arbitrarily many conjunctions that can be formed from any random pair of propositions but only a countable number of conjunctions formable from propositions corresponding to the properties of a Physical object.

Definition of implication from the same book

(II)Implication-''Let P and Q be two propositions. The implication ''If P then Q'' is the proposition that is false when P is true and Q is false, and true otherwise''.

Again this definition also does not impose any condition, based on meaning, on the constituent propositions. So the following two examples seem reasonable.

(1) P= x < 2, Q=x < 6

for all x in the set of real numbers.

If P then Q = If P<2, Q<6 for all x in the set of real numbers.

(2) P= This box is big, Q= Tomorrow it will rain,

If P then Q= If this box is big, it rained yesterday.

Commentary: According to the definition of ''implication'' logical connective, both the examples could be said to constitute valid examples of ''Implication''

Consider, in particular, the second example where the ''If P then Q'' assumes a truth value of F, only if The box is big and it did not rain yesterday(According to the definition). Clearly, this second example is not both logically and empirically on the same footing as the first example. The second example seems to have been constructed '' artificially'', based on the rules specified in the definition. On the other hand, the former example seems to have a logical necessity independent of the definition given to ''Implication''.

I am not completely sure if the second example in both, the case of Conjunction and the case of Implication are ''allowed''. If they are not allowed, what are the rules constraining the formation of logical connectives? It seems that the constraining rules depend on the meaning of the constituent propositions. On the other hand, if they are allowed, is not the arbitrariness (which was illustrated in the examples) troubling?

  • "Does the individual meaning of two propositions determine or constrain what kind of logical connectives can be formed between them?" No in classical logic. Yes in natural languages (but the constraints there are rather vague). "Are the definitions given to logical connectives based on empirical observation and logical necessity or are they just arbitrarily defined concepts and rules?" Both (except for "arbitrarily"). They incorporate some aspects of natural meaning, but clean up its idiosyncrasies and context restrictions to fit into a simple mechanizable calculus. – Conifold Mar 21 at 9:03
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Connectives are already present in natural language : they are part of syntax and their use is regulated by grammar.

Thus, logical connectives are not arbitrary (or conventional), but are a simplified model based on the way that grammatical connectives operate.

The truth-functional definition of some of them : conjunction, negation, is more close to the "natural" way of using their grammatical counterparts, while others : conditional, disjunction, present some "unnatural" aspects.


The debate about the adequacy of the truth-functional version of "if..., then..." started with formal logic itself : see Stoic Logic and is still open today; see Conditionals.

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Generally speaking, the meaning of sentences and words is a matter of syntax, semantics and pragmatics. Syntax is concerned with formal rules for forming and manipulating sentences and includes the rules of grammar and the rules of deduction that feature in logic. Semantics is concerned with conventional meaning and interpretation. Pragmatics is concerned with the more messy principles and guidelines that govern what a speaker is trying to achieve when making an utterance in a given context, on a given occasion and before a given audience. Mathematicians like formal proofs that are independent of context, so mathematics is big on syntax and avoids pragmatics. Ordinary language on the other hand is big on pragmatics: it is a major feature of understanding what people mean. Because of this, mathematical usage is not very typical of meanings as they feature in ordinary language, so it is best not to put too much weight on definitions found in math books.

Here are some examples for illustration. Your math books defines "P and Q" to be simply P is true and Q is true. But the meaning of the utterance, "eat that apple and you'll die" clearly expresses something much more: eating that apple will result in your death, perhaps either because it is poisonous, or because the apple is mine and I will kill you if you eat it. "Alice drove home and drank a beer" has a different meaning from "Alice drank a beer and drove home". In both cases the difference lies in the pragmatics of language. In the first example, "you'll die" would be irrelevant unless the speaker intends the audience to understand that it is the consequence of eating the apple. In the second example, it would be disorderly of the speaker to state what Alice did in the wrong sequence. Many of these pragmatic features can be explained by Paul Grice's theory of the co-operative principle and its use in implicatures. https://plato.stanford.edu/entries/implicature/

In the case of conditionals the matter is even more complex because ordinary language conditionals are quite extraordinarily messy and it is difficult to draw the line between semantics and pragmatics. The conditional in your math text, i.e. that "if P then Q" is false when P is true and Q is false, and true otherwise, goes back to the stoic philosopher Philo and is usually known as material implication, but it is not the only conditional, nor is it typical of ordinary English usage, though it definitely is useful in mathematics. Usually in English we utter a conditional in order to express a causal or evidential relation, or a constraint or a precondition, or any one of a number of other things.

So logical connectives are not arbitrary and often they have conventional meanings that abstract away from the meanings of the sentences they connect, but these meanings do not exhaust their use in ordinary English. Formal logic is an attempt to formulate the rules that describe the conventional component of linguistic meaning, but it only approximates meanings within natural languages.

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The answer is, according to me: logical connectives are abstract but not conventional.

Let me reconstruct the history of logical connectives.

(1) First there are connectives in natural languages: and, or, if...then, etc.

(2) Second, logicians noticed that the validity of some inferences depended on the meaning of these connectives. But they also noticed that this meaning was not precise enough. So they wanted to make them TRUTH-FUNCTIONNAL, in order to be able to COMPUTE the value of the resulting sentence solely on the basis of the truth-values of the atomic sentences linked by these connectives.

(3) At this moment, logic totally leaves the ground of natural language. "Connectives" become FUNCTIONS , like mathematical functions, such that f(x)= x², or f(x)=x; etc. A connective is now defined as a truth function, that is a function that takes truth values ( or couples of truth values) as INPUT and gives back truth values ( True or False) as OUTPUT.

Examples:

One of the possible "connective", is the following function: when the input is the value True, the output is the value False, when the input is False, the output is True.

T -------------> F

F--------------> T

Another possible connective is the following function : when the input is the couple (True, True), the output is is True, when the input is not (True, True), the output is the value False. It "works" as follow:

(T, T) -------------> T

(T, F)--------------> F

(F, T) -------------> F

(F, F) -------------> F

These connectives, or functions are not at all arbitrary. It can be proved that there are are only 2 possible unary connectives ( one input, one output), and only 16 possible binary connectives (Four couples as input, and two possible values as output). These functions were not created or invented by logicians, they were simply discovered.

(4) After that, the problem arises to give names to these functions ( i.e. connectives). And here the problem has to be correctly understood: we want ordinary language names to be as close as possible to the mathematical meanning of the connectives, not the other way round. So, for example, we attribute the name AND to the binary connective which gives True as output if and only if the input is the couple (True , True) although, in many cases, the ordinary meaning of the word "and" is not truth-functional. We give the name "OR" to the connective that gives True as output all the time, except when the input is the couple : (False, False) - although, in ordinary language, the word "or" often excludes the two sentences connected to be true at the same time.

(5) So what is conventional is not at all the connectives themselves, but the names we give to them: we could not have called " if ...then" the connective " --> " , but we have chosen to call if " if...then" ( implication) because " if ...then" is the ordinary language expression that is the closest to " ---> ".

(6) And the choices logicians have made as to names ( not as to connectives themselves) is not bad. For example, whith their abstract " --> " they manage to capture a part the ordinary meaning of implication, which usually involves necessity. So they say that " John is a pianist IMPLIES John is a musician" means not only that " John is a pianist --> John is a musician" , but that " Necessarily ( John is a pianist --> John is a musician)".

Because " --> " is so usefull to capture the meaning of " implies" in ordinary language (adding of course to ' --> ' the supplementary idea of necessity ) it was not at all a bad idea to call this connective " if ...then" . Even though, of course, it can sound strange to hear, when beginning logic, that

                  " If 2+2=4 then The Nazis have  lost WWII" 

is a true sentence, simply because both the first and the second proposition are true. One simply has to keep in ming that " -->" ( material implication) is just a building-block to define the concept of logical implication, and that " If 2+2=4 then Germany has lost WW2" absolutely doesn't mean that " The Nazis have lost WW2" is a logical consequence of "2+2=4". When logicians say : " If 2+2=4 then The Nazis have lost WW II" they simply mean : " it is not the case that (2+2=4 and that The Nazis have NOT lost WWII) ". And that's true, since, for the conjunction to be true, the second proposition ( " The Nazis have NOT lost WWII") should be false, and, as anyone knows, is isn't ( ... quite fortunately ).

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The less natural cases in your examples are allowed, as is any case involving any sentences whatsoever as long as they have a truth value.

This is a feature, not a bug: logic is supposed to be fully general. Having said that, typically, pragmatic (as opposed to semantic) considerations will guide our judgments about whatever a speaker means to say when they utter a sentence involving conjunctions or implications.

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The meaning of two propositions sometimes affects which logical connections can be made.

Premises can be arbitrary but sometimes the premises are dependent on each other's truth-value. That is, some premises come already associated with a logical connection. For example you can define Q as the negation of P, which implies an exclusive disjunction. Or you could define P as always being false and Q as being always true, which implies converse nonimplication.

Each of the 16 logical connections are mutually exclusionary with some other connections - you cannot claim that two premises are both always true and always false at the same time. Neither can you claim both a conjunction and contradiction. Neither can you claim simultaneously material implication and material nonimplication. If the premises have fixed truth-values there is only one valid logical connection to be made. Etcetera.

I will provide examples below based on the original post.

A Valid Conjunction

Regarding the second example given for conjunction:

Let P represent the premise "This car is green".

Let Q represent the premise "The Milky Way galaxy is 100,000 light years wide".

Therefore P∧Q means "This car is green and the Milky Way galaxy is 100,000 light years wide". Is this a valid logical statement? Yes: if both propositions (premises) are true, then the conjunction of these two is true by definition. There is no hidden premise here saying the color of the car is related to the width of the galaxy.

This can be plotted on a Venn diagram like so:

Venn diagram of a logical conjunction

You can still falsify this conjunction by contradicting the premises - usually with observations such as "this car is yellow". There is another case where the premises contradict themselves, which I will discuss below.

The original poster seemed hung up on the arbitrary choice of premises. As noted by Schiphol, this arbitrariness doesn't necessarily affect the validity of the conjunction. But the arbitrariness may undermine the utility of the conclusion. Indeed one can make an infinite number of logical conjunctions between valid, arbitrary premises... but these conclusions do not necessarily imply any meaningful correlation.

This car is green and the Milky Way galaxy is 100,000 light years wide. So what?

An Invalid Conjunction

As per the above definition, I can consider any arbitrary pair of propositions and connect it by the ''Conjunction'', since it does not impose any constraint on the meaning of P and Q.

This assumption does not hold if the premises are mutually exclusive. Consider the following counterexample:

Let P represent the premise "This car is green".

Let Q represent the premise "This car is not green". Q may also be represented as ¬P, or "not P".

This implies a logical connection: exclusive disjunction. The truth-value of P is always different from the truth-value of Q. In other words, there is a third hidden premise: "It is either exclusively true that this car is green, or it is exclusively true that this car is not green." That may sound drawn out and verbose but it is less ambiguous than colloquial English.

Now you can not make the conjunction P∧Q. Here P∧Q resolves to "This car is green and it is false that this car is green" which is a blatant contradiction of the exclusive disjunction above. A Venn diagram will have no overlapping space, and the conjunction is invalid on its face.

Venn diagram of an invalid conjunction

Material Implication

Next consider the box and rain example:

Let P represent the premise "This box is big".

Let Q represent the premise "It will rain tomorrow".

I assert, P→Q "If this box is big, then it will rain tomorrow". This confuses a lot of people but material implication does not imply causality. In logic the words "If... then..." do not mean there is a causal relationship between the two. It only means if one is true, the other is true.

For example, the above statement may be true if I work in a weather station and we bring out a big box every time we forecast rain. The box doesn't cause rain, but it still logically implies rain.

Ultimately you need to resort to observations to decide whether this material implication comes true or not. But unless assumed that it will never rain tomorrow, or that "this box" has nothing to do with the tomorrow's weather, the statement is valid on its face. Context matters.

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