How does one differentiate between the logically possible and the impossible?

Question

The term “married bachelor” seems to be an obvious contradiction. The very definition of bachelor and unmarried are with respect to each other and so it seems nonsensical to talk about a married bachelor.

But what about terms that seem to contradict each other given their implied meanings? For example, what about a “five armed human”? It is not hard to imagine a human with five arms so some might call this logically possible. But one could also argue a five armed human isn’t a human since humans are known to have two arms and thus through implication argue that this is logically impossible. So which is it??

Lastly, how does one differentiate between the logically impossible and meaningless? Isn’t part of what makes something meaningless also what we know to not be possible, such as a five armed human? And yet, clearly “Jr heheh rurjrbr rhrur” seems much “more” meaningless than a five armed human.

This question stemmed from discussions about God where many argue that some of His attributes (such as omni-knowledge and human free will) contradict each other. People have often tried to get away from this contradiction by saying that God attains knowledge from “outside time”. Is this a rightful evasion or meaningless?

Similarly, some physicalists have argued that a purely immaterial cause (I.e. a mind without a body for example) causing physical effects on the world is impossible. If it is, in what sense? Is it metaphysically possible, logically possible but metaphysically impossible, impossible in both senses, meaningless, or what?

Can one “redefine” terms or propose ways of rescuing these potential contradictions with the phrase “married bachelor” as well?

Answer 1

Logical impossibility can only be rigorously defined in a formal language, one built on consistent, well-defined rules, and a limited universe of discourse. In such a language, we can also distinguish easily between something like "A and NOT A" which is impossible-by-definition, and something like "A and NOT" which is meaningless.

For the kind of move you're looking to make, you would have to be able to map first back to a natural language (which would lose the rigorous definitions) and then to the general material world (which doesn't have well-defined members) and from there to the metaphysical world (which has an unlimited universe of discourse).

So your line of argument really falls prey to false analogy. You're taking a set of concepts from a highly structured realm and applying them somewhere that's at a significantly far remove, without any guarantee that the relevant characteristics are preserved across the transitions.

Answer 2

This question collects together a number of different, but related issues, so detailed responses to the specific cases are appropriate:-

Take your second paragraph -

But what about terms that seem to contradict each other given their implied meanings? For example, what about a “five armed human”?

This is a version of the "black swan" case. In such cases, there is always a choice to be made, whether to include the new case within the definition or not. So when black swans or five-armed humans are discovered, the concept can be expanded (as happened in the black swan case), or the new case can be defined as a new concept. The choice does not need to be arbitrary - the decision to include black swans as swans, for me, is perfectly rational. This just shows that definitions do not necessarily cater for all possible variations in advance and decisions sometimes need to be made.

Your third paragraph asks

... how does one differentiate between the logically impossible and meaningless? Isn’t part of what makes something meaningless also what we know to not be possible, such as a five armed human? And yet, clearly “Jr heheh rurjrbr rhrur” seems much “more” meaningless than a five armed human.

The difference between the logically impossible and the meaningless is often not well-defined in much philosophical discussion. For me, the difference is that something logically impossible can serve as the premiss in a reductio ad absurdum argument, which is undoubtedly meaningful. But letter (or, indeed, word) salads like "Jr heheh rurjrbr rhrur" cannot. There are varieties of nonsense. "'Twas brillig and the slithy toves/ did gyre and gimble in the wabe" is another one. See Jabberwocky

Your fourth paragraph asks

... many argue that some of His attributes (such as omni-knowledge and human free will) contradict each other. People have often tried to get away from this contradiction by saying that God attains knowledge from “outside time”. Is this a rightful evasion or meaningless?

On the face of it, "outside time" must be a metaphor, since time is not a spatial concept and consequently has no "outside". But the phrase seems to have some sort of meaning in mathematics. In addition, the "timeless present" is a firmly established concept - see, for example, Timeless present examples. Formal logic often relies on this use. One would have to find out what is meant by "outside time" before proceeding to argue about it. It may well be that the discussion will get bogged down in the preliminaries, but that itself would be a victory of sorts.

Your fifth paragraph is

Similarly, some physicalists have argued that a purely immaterial cause (I.e. a mind without a body for example) causing physical effects on the world is impossible. If it is, in what sense? Is it metaphysically possible, logically possible but metaphysically impossible, impossible in both senses, meaningless, or what?

This is the classic objection to dualism and seems conclusive to me. But clearly not everyone agrees, and the concept of a "field" in physics is an awkward case. However, if something can cause a physical effect, it is a physical something. So a field counts as physical, but not as material.

Finally, you ask:-

Can one “redefine” terms or propose ways of rescuing these potential contradictions with the phrase “unmarried bachelor” as well?

It isn't clear what you have in mind here, but if the use of a word can change over time, one couldn't rule out changes in the use of "bachelor" that would change the status of "unmarried bachelor".

Answer 3

Part of the problem with the typical examples of the "logically impossible" is that, they employ a natural language and are more about synonyms and antonyms. When a term lacks a univocal definition (that is, one can cleanly list the necessary conditions that are jointly sufficient, say in defining what a "square" is), it becomes a mess trying to say anything about logical necessity or impossibility. Let me apply this to the cases you are concerned with (God and the soul).

Outside of religious revelation, God is an imaginary or conceptual hypothesis. Various attributes about God are usually poorly defined, and so it's easy for someone to point out "contradictions" from thought-experiments. Since God's attributes are poorly defined, its easy for the theist to evade such critiques. Regarding meaning, one would have to clarify what we mean by "meaningless evasions," because the theist surely finds it meaningful. I typically focus on the nature of the argument for God at issue to set the scope of meaning: if it's a first cause argument, then it needs to be physically meaningful as defined by physics.

Some physicalists do try to say it's a priori false that an immaterial entity can influence a physical system. Again, one can play with "immaterial," while also asking the physicalist to clearly state what counts as meaningless here. A good physicalist will eventually slip into a posteriori concerns about conservation of momentum, etc. From there you exit the debate over logical impossibility to what physics could allow.

In short: a lot of this arises over poorly defined terms, and confusing synonyms with tautologies.

Answer 4

You ask about logic (in particular in your title) but then stray into physics or physiology. They are not related: What is possible in nature is essentially unknown (even if nature science gives us a pretty good pragmatic idea what to actually expect).

Logic impossibility, by comparison, is really easy: You have two statements with clearly defined meanings (of course, the devil here lurks in the detail because we use natural language) and see whether they contradict each other.

• Define human body as having 2 arms

• Have two statements "Is human body" and "has 5 arms": Logically impossible.

• But, instead, have human body with number of arms left unspecified: No contradiction.

More interesting recently are statements like "is a male" and "has a vagina". Here, too, the logician simply asks for the definition of "male" (let's assume they are familiar with vaginas). If the definition of "male" includes "no vagina", then there is a contradiction; if it doesn't, then there isn't. All the squabbles are the result of people assigning different definitions to natural language terms, sometimes even on purpose.

The logician has no part in this; they simply get out the popcorn, lean back and watch until the dust has settled.

Answer 5

When we first learn logic, we just have to use our imagination as to what is logically possible. Later as we learn more, we can replace this naive idea with more concrete concepts from proof theory and formal semantics. This typically involves introducing formal systems of logic, so somewhere along the way, we transition from relying exclusively on natural language to using formal systems.

Once we have a formal logic, we can make the concept of logical possibility more concrete. Proof theory allows us to define the closely related concept of consistency: we cannot prove a contradiction from something. Formal semantics allows us to define the closely related concept of satisfiability: there is at least one interpretation under which it is true. We lose something along the way, because formal systems will never capture all the features of natural language, but we gain the ability to demonstrate that this follows from that in a way that is so simple even a computer can do it.

Expressions like 'married bachelor' or 'five armed human' are not contradictions in a strict sense. They have to be interpreted in order to know whether they are true or false. Meanings and linguistic conventions are slippery things that change over time and are not really a satisfactory basis for logic. That said, there is a tradition going back to Abelard that allows for a concept of 'material validity'. Some logicians continue to use it. Some logicians speak of analytic consequence or analytic implication.

In practice, with technical terms we tend to delegate their meanings to experts. If some fish were discovered with two tails, we would let biologists decide whether they were a distinct species or a subspecies or a mutant variation. 'Married bachelor' might become a useful term if 'bachelor' came to be used primarily to refer to people who live a particular lifestyle, rather than people who are single.

Whether a sentence is 'meaningless' is harder to judge. Within formal systems this concept takes concrete form by way of well-formed formulas. But given that natural language is much broader, it would be far too restrictive to say a sentence is meaningless just because it cannot be represented as a well-formed formula. If anything, the word 'meaningless' is thrown around far too much. I suspect this is a hangover from the bad old days of logical positivism when the meanings of sentences were identified with their verification conditions.

Concepts like immaterial cause are not logically impossible. Maybe they are physically impossible: it would be a reasonable methodological assumption to suppose so. To think about the universe from a perspective outside time is not unusual. Logicians often state propositions in a kind of 'timeless present'. The concept of eternalism or the block universe model is a respectable way to think about the universe and is common among some theoretical physicists.

Answer 6

Your definitions are not correct.

Metaphysics

Metaphysics includes logic, which is necessary for mathematics.

• Logically, possibility is just a predicate with no meaning:

P(a) means "a is (P)ossible". Meaning is provided by the reader.

• Mathematically, impossibility means p=0, and possibility is p>0.

• Pure language is not enough to define possibilities. At maximum you can express tautologies: bachelors are single and that is true, which is true, which is also true.... But in final terms, this is just logic (predicate logic, the first case above).

Experience

Empirically, all probabilities follow:

0<p<1

that is, nothing is impossible, and nothing is absolutely sure.

Answer 7

Here "logically impossible" is presumably equivalent to "contradictory", so the opposition is between "impossible" and "contradictory".  In fact, Leibniz makes precisely such a distinction!  Sometimes he qualifies "impossible" by "accidentally impossible" or "accidentally impossible".  He uses this distinction to explain things like negative numbers, imaginary roots, irrationals, and infinitesimals.  All these seem impossible in our world, but do not actually pose any contradiction: numerous fruitful results in mathematics require these tools and are impossible to ignore.  On the other hand, contradictory things like 2+2=5 (Leibniz's example) are absolutely impossible and useless in the practice of natural philosophy (including mathematics). 

Sometimes Leibniz speculates concerning possible worlds: while in our world, square root of -1 may be an impossibility, perhaps in other worlds it isn't.  The "possible worlds" theme is more obscure.  At any rate, when Leibniz speaks of infinitesimals as "fictions" he has precisely this in mind: they are only conditionally impossible, but not contradictory.  Elsewhere he makes it clear that his infinitesimals violate Euclid Book V definition 5 (definition 4 in modern editions), which is closely related to the Archimedean property (the latter seems necessarily satisfied in our world, but perhaps not in others...)  This is at any rate the interpretation we have pursued in a number of scholarly ariticles, including this publication in the British Journal for the History of Mathematics.  Other Leibniz scholars fiercely disagree; see here for a summary of the debate.