# Logical Necessity

Logical necessity is that which is necessary by definition. Otherwise it is sometimes considered to be that which cannot possibly be false, or cannot be otherwise metaphysically. The usual example is that all bachelors are unmarried men.

In a valid deductive argument where the conclusion necessarily follows from the premises - i.e. logical consequence - is that also called logical necessity?

So it's a logical necessity that all bachelors are unmarried men, but is the fact that the conclusion of the following argument necessarily follows from the truth of its premises also called logical necessity?

p1 All bachelors are unmarried

p2 All bachelors are men

c All bachelors are unmarried men

It's logical conseqence, but we also call the conclusion a logical necessity.

p1 All frogs are green (not sound - obviously)

p2 All biology students get a frog

c All biology students get a green frog

It's a valid unsound argument, and involves logical consequence, but is the conclusion what we call logically necessary or true by definition? It's not a logical necessity that all frogs are green, but is the conclusion a logical necessity since we have defined all frogs as green in the premise?

• "That which is necessary by definition" is usually called analytic necessity, and what "cannot be otherwise metaphysically" is called metaphysical necessity. Logical necessity covers only what logically follows from adopted premises. If the premises are not adopted what follows isn't necessary. "Bachelors are unmarried men" is usually classified as analytic but not logical, although some authors do not draw the distinction, and others (Quine) consider the concept of analytic to be altogether ill-defined. Jun 7, 2019 at 4:07
• Thanks @Conifold . Can you be clearer about what you mean by adopted premises? Use the frog example. In that example, is the conclusion a logical necessity relative to the argument? Jun 7, 2019 at 5:05
• There is no "relative". Since you call the argument unsound you do not consider the premises true, let alone logically true, so no. Jun 7, 2019 at 5:59

IIMx Philosophy Glossary Natural and Logical Necessity https://www.youtube.com/watch?v=kvicOsOddk4

• It might be best to summarize what is in that video that is relevant to the question in case the link breaks. However, referencing the link as a source is important and supports the answer. This also might be added as a comment or an edit to the original question. Regardless, best wishes. I find your question interesting and up-voted it, but I don't have an answer yet. Jun 10, 2019 at 12:13
• The notion of logical necessity is a modal one. The reason for the question was to forage for other uses of the nomenclature. As far as I knew, the term 'logical necessity' was almost comprehensively used for modal (possible worlds) logic, but I wanted to check for other use. Philosophy, philosophical logic, and logic are full of overloaded and polysemous terms. Conifold answered using the usual modal context, and answered correctly. Adoption in that context is about which world or state description is selected 'as actual'. Jun 11, 2019 at 10:10

The question of what constitutes "Logical Necessity" might be usefully approached starting by asking which of the two words in that composition takes priority. That is, are you interested in what is necessary, given logic, or are you interested in what is purely logical, given necessity?

In some ways, the idiom "Necessary by Definition" suggests the second, in that you have a modality over meanings. As Conifold notes in their comment, this is often thought of as Analyticity, and I quite like Rudolph Carnap's interpretation of this as operating within the language of scientific practice. Scientific practice, it is held, works in a logically rigorous way, and its language is similarly logically structured, but the modality involved here is about statements/propositions and their connections to observations of the world.

But it seems strange to call these statements themselves the logically necessary ones; you need to have meaning postulates, to have connecting bridge principles which tie propositions to things in the world. As such while analyticity has the flavour of logic, it is still not itself entitled to claim the full ontological status of logicality.

If we have a prior understood modality at work, it may make sense to interpret the logically necessary statements in terms of a logic of those modal operations. It is not logically true that all men are bachelors; however, given an interpretation of how analytic necessity, the universal quantifier and implication connect together, it might be logically true that if all men are bachelors, and if an object is a man, then that same object is a bachelor.

In order for this to work, we need to understand the language in question as having a logical structure characterised by connectives like "for all", "if", "and" etc. And, as Quine has argued, it is not simply a foregone conclusion that the language we have is indeed so structured, and you can't fall into the trap of begging the question against yourself as to the mathematical soundness of your language to be scrutinized. Not even mathematical language can characterize itself without significant difficulty - see e.g. Hilbert, Tarski, Godel.

So perhaps the other avenue might make more sense. Starting from logic, what kinds of modality might we introduce, and what necessary truths might fall out from the logics of necessity we establish?

This is the direction Saul Kripke took Necessity in the field of Modal Logic. In this, necessity is not so much as a prior given over which logic might be run but rather as a mathematically characterisable functor (or, more accurately, a plurality of such functors) over the domain of logical statements. The idea is that Necessary is just an operator, like "if" or "and", that can be built into a logic and using which we can describe some particularly rich structures of true statements. Or, you know, value 1 statements, or statements admitting of multiple valuations etc. etc.

Some such structures will admit of statements that are true purely in virtue of that logic - in fact, some such structures will admit of statements featuring the necessity operator that are true purely in virtue of that logic. These seem to be to be a more accurate place to be leaning on our understanding of Logical Necessity.

However, well, what functor of necessity are we using? Different interpretations of the modal operators may yield different accounts of the purely logical necessities.

Fortunately, in mathematics, we don't necessarily care a whole lot about whether a model is practically applicable in order to say it exists - each of these concepts of Logical Necessity can be modelled, and so each may be worth investigating as a legitimate object in its own right.

But perhaps we still haven't quite got there. It's not so much that we're interested in how to model necessity in logic, but more what the modality of logic itself is supposed to be. What is it in virtue of which we say that from `A` and `A->B` follows `B` - what is the modal force of this kind of "following", this consequence?

A particularly interesting route into this discussion is to look at disputes around the applications of competing logical principles. We create characterisations of different logics that (for example) allow different patterns or principles of inference, and warrant different strengths of conclusions. Are there some statements in the world that are neither true nor false? What can we conclude when we introduce premises that might be both true and false? Do all of our conclusions have to be derivable in computably many steps from base deductive principles?

I am an advocate of Logical Pluralism - I think that "logic" is not an unequivocal term, but rather appeals to different structures of inference and interpretation depending on the domain of discourse. The domain of Mathematics does not admit of fuzzy truth, but the world of Politics and Commerce most definitely does, and Data scientists (my particular field of interest at the moment) need to be aware that one cannot straightforwardly infer conclusions using a one-size-fits-all model of logical consequence.

But the logical tools that we've created in the mathematics can still be useful and informative across domains, even if we deny the possibility of a singular catch-all analytic concept of logical necessity. Try some models, see what they tell you, see if that holds, and refine and keep trying. We're probably not going to get it exactly right, but that's the nature of the scientific method, I believe!

• Some of your sentences seem to have issues. What do you reveresed a premise. The original was all bachelors are unmarried. Your swap was that "it is not logically true that all men are bachelors. That was not stated but you address this. As it was written are all bachelors are unmarried remains to be true unless you can provide a case otherwise. You did not address the given proposition but substituted the original with a eaisly reputable proposition. Secondly, .modality was not introduced. Before you can introduce another possible answer one must show the current method insufficient. Jul 10, 2019 at 19:06

The term logically necessary can have more than one context. Usually the kind of logical necessity in valid arguments are defined to Express the impossibility of the premises being true and the conclusion false at the same time. For having a false conclusion along with true premises would make deductive reasoning a waste of time and basically unreliable and worthless. Thankfully that is not the case.

In the example, All bachelors are unmarried men this clearly is distinct from logical necessity as described in the case of validity and the conclusion being logically forced. In the bachelor example there is a linguistic element that is NOT in the validity context. In English certain syntax and grammar rules apply whether we like it or not. I just can't put words in any order I desire and still be said to make sense in ENGLISH. For instance, "ball brown the boy's is" is NOT a proper sentence in English. There are RULES of grammar and syntax. The term ANALYTIC can be defined differently by different individuals but I can tell you what I was taught. Analytic propositions express either something logically necessary or self contradictory. All bachelors are male is neccessary as the RULE IN English requires this to be the case. We don't look to science or the world to find out what a bachelor is. The world and our sensory experience don't define words for us. Different people can and often call the SAME THING a different name. In the south people call SODA "pop". I am like "what is pop? Oh you mean soda!" Why not just call it soda as multiple people not from the south recognize SODA. Even these people know what soda is but INSIST on saying something else. We define triangles as three sided shaped objects. There is no triangle that has more than three sides or less than three sides. This is because our LANGUAGE makes this so and not scientific laws, rules or our experiences. Analytic propositions requires us to have knowledge of how the language works. The knowledge of the world is NOT required. I would not be able to manually figure out some definitions for many words and what they imply just based on my experience.

There are also SYNTHETIC propositions to complement ANALYTICAL propositions. Synthetic propositions are propositions that ARE NOT based upon our knowledge of a language. My knowledge of any language and only language would not inform me that getting shot in the head could kill me. Maybe it will or maybe it won't. Language alone has no say about the world and what happens in it. I know when the stove is hot I should not touch the stove by experience. I know what boot camp is like in the U.S. military by experience, not by language. I know what it is like to be in a fist fight by experience. Even if I read all the books on every martial art that would not mean I would be any good at fighting. So SYNTHETIC propositions are those propositions that rely on sense verification: sight, hearing, taste, touch and smell. All of science relies on sense verification. ANALYTIC propositions do not rely on sense verification. By the way, your attempt at a syllogism failed. All bachelors are unmarried and All bachelors are males EXPRESS THE EXACT SAME THING. You have no syllogism there.

When I hear something is "logically necessary" I usually take that expression to mean that the result is unavoidable given a set of rules or laws in a context. In the game of chess this can be seen as a Mate in one. That is, no matter what person or what computer you put in front of me I will absolutely win IF I play the correct move. That means pick the best human chess player or the best computer at chess at any time frame, I will 100 percent win. There is no way to play the game, follow the rules and avoid the result. Well if you resign you STOP playing the game and you simply avoid the result by refusing to play. You cant have it both ways. You would need to break some of the rules if you can avoid the result. The same way you can get a speeding ticket traveling from one toll to another. There is a time stamp on the toll ticket you must hand in. Based on the distance your speed can be calculated. This is unavoidable. If you traveled from NY to Washington DC by car in 2 hours flat YOU MUST HAVE broken the speed laws at least once during the trip. The result is impossible to avoid within the information given.